Priprava Teorija Mat4

Predpostavka P3: \( f(A) \) ni povezana.

1. A1: \( U, V \in \mathbb{C} \) odprti

   A2: \( U \cap V = \emptyset \)

   Dokaži:

   \[ f(A) = \left( U \cap f (A) \right) \cup \left( V \cap f(A) \right) \]

   kjer sta preseka neprazna.

   Dokaz: Po predpostavki P3 ter definiciji povezanosti.

2. A3: \( U' = f^{-1} (U) \)

   A4: \( V ' = f^{-1} (V) \)

   Dokaži: \( U', V' \) odprti

   Dokaz: Funkcija \( f \) je zvezna, za \( f: U \to \mathbb{C} \) je zvezna \( \iff \) \( f^{-1} (V) \) je odprta za vsako odprto množico \( V \subseteq C \)

3. \( U' \cap V' = \emptyset \)

   Dokaz: Brez dokaza.

4. \( A \cap U' \ne \emptyset \) in \( A \cap V' = \emptyset \)

   Dokaz: Brez dokaza.

5. \( A = \left( A \cap U' \right) \cup \left( A \cap V ' \right) \)

   Dokaz: Po definiciji povezanosti ter točkah <1>3 in <1>4.

6. QED

   Dokaz: Točka <1>5 je v protislovju s predpostavko P2, da je množica \( A \) povezana.

1. Tangentni vektor na tir poti \( \gamma \) v \( z_0 \) v polarnem zapisu

   \[ \dot{\gamma} (t_0) = \left| \dot{\gamma}(t_0) \right|e^{\mathrm{i} \alpha} \]

2. Tangentni vektor na tir pot \( \delta \) v \( z_0 \) v polarnem zapisu

   \[ \dot{\delta} (s_0) = \left| \dot{\delta}(s_0) \right| e^{\mathrm{i} \beta} \]

3. Kot med \( \delta \) in \( \gamma \) v \( z_0 \) je \( \alpha - \beta \)

   D: Kot \( \gamma \) in \( \delta \) je definiran kot kot med tangentnima vektorja, iz točk <1>1 in <1>2 sledi \( \alpha - \beta \).

4. \( f'(z_0) = \left| f'(z_0) \right| e^{\mathrm{i} \phi}\)

5. \( \frac{\mathrm{d} }{\mathrm{d} t} f\circ \gamma (t_0) = f' (z_0) \dot{\gamma} (t_0) \)

   D: Definicija tangentnega vektorja na pot \( f \circ \gamma \).

6. \( f'(z_0) \dot{\gamma} (t_0) = \left|f (z_0) \right| \left| \dot{\gamma} (t_0) \right| e^{\mathrm{i} \left( \alpha + \phi \right)}\)

   D: Točke <1>1 in <1>5

7. \( \frac{\mathrm{d} }{\mathrm{d} t} \left( f \circ \delta \right) (s_0) = f' (z_0) \dot{\delta}(s_0) \)

   D: Definicija tangentnega vektorja na pot \( f \circ \delta \).

8. \( f' (z_0) \dot{\delta}(s_0) = \left| f'(z_0) \right| \left| \dot{\delta}(s_0) \right| e^{\mathrm{i} (\phi + \beta)} \)

9. Kot med \( f\circ \gamma \) in \( f \circ \delta \)

   \[ \alpha - \beta \]

   D: Kot med potema je kot med tangentnima vektorjema.

10. QED

    D: V točki <1> 9 smo dokazali, da je kot med \( f\circ \gamma \) in \( f \circ \delta \) v \( z_0 \) enak kotu <1>3

1. A1: \( R' \) konvergenčni radij \( g(z) \)

   \[ \frac{1}{R'} = \limsup_{n \to \infty} \sqrt[n]{\left| n a_n \right|} \]

   D: Definicija konvergenčnega polmera

2. \[ \limsup_{n \to \infty} \sqrt[n]{\left| n a_n \right|} = \limsup_{n \to \infty} \sqrt[n]{n} \sqrt{\left| a_n \right|} \]

   D: Produkt limit je limita produkta

3. \[ \limsup_{n \to \infty} \sqrt[n]{n} \sqrt{\left| a_n \right|} = \limsup_{n \to \infty} \sqrt{\left| a_n \right|} = \frac{1}{R} \]

   D: Limita \( \sqrt[n]{n} \) gre proti 1, ko gre \( n \to \infty \) ter definicija konvergenčnega radija vrste \( f(z) \)

4. QED

   D: Dokazali smo, da sta konvergenčna radija \( g(z) \) in \( f(z) \) enaka.

1. A1: \( \rho < R \)

   A2: \( w, z \in \bar{D} (0, \rho) \)

   \[ \left| \frac{f(w) - f(z)}{w - z} - g(z) \right| \to 0, \ w \to z \]

   1. \[ \left| \frac{f(w) - f(z)}{w - z} - g(z) \right| = \left| \sum\limits_{n = 0}^{\infty} a_n \frac{w^n - z^n}{w - z} - \sum\limits_{n = 0}^{\infty} n a_n z^{n - 1} \right| \]

      D: Definicija potenčne vrste

   2. \[ \left| \sum\limits_{n = 0}^{\infty} a_n \left[ \frac{\left( w - z \right) \left( w ^{ n - 1} + w^{n - 2} z + \ldots + z^{n - 2} w + z^{ n- 1} - n z^{n- 1} \right)}{w - z} \right] \right| \]

      D: Razlika \( n \) potenčnikov

   3. \[ \left| \sum\limits_{n = 0}^{\infty} a_n \left( w^{n - 1} - z^{n - 1} + w^{n - 2} z - z^{n - 1} + \ldots + wz^{n- 2} - z^{n - 1} + z^{n- 1} - z ^{n -1} \right) \right| \]

      D: \( z^{n - 1} n = z^{n - 1} + \ldots + z^{n - 1} \)

   4. \[ \left| \sum\limits_{n = 0}^{\infty} a_n \left[ (w - z) (w^{n - 2} + w^{n - 3} z + \ldots + w z^{n - 3} + z^{n - 2}) + z (w - z) \left( w^{n - 3} + w^{n - 4} z + \ldots + w z ^{n - 4} + z^{n -3} \right) + \ldots z^{n -2} (w - z) \right] \right| \]

      D: Razlika \( n - i \) potenčnikov

   5. \[ \left| (w - z) \sum\limits_{n = 0}^{\infty} a_n \left( w^{n - 2} + 2 w^{n - 3}z + 3 w^{n -4} z^2 + \ldots + (n -1 ) z^{n - 2} \right) \right| \]

      D: izpostavili smo \( (w - z) \) in sešteli člene z istimi potencami

   6. \[ \le \left| w - z \right| \sum\limits_{n = 0}^{\infty} \left| a_n \right| \left[ \left| w \right| ^{n - 2} + 2 \left| w \right| ^{n - 3} \left| z \right| + \ldots + (n - 1) \left| z \right| ^{n - 2} \right] \]

      D: \( \left| a + b\right| \le \left| a \right| + \left| b \right|\)

   7. \[ \le \left| w - z \right| \sum\limits_{n = 0}^{\infty} \left| a_n \right| \left( \rho^{ n - 2} + 2 \rho^{n - 2} + 3 \rho^{n - 2} + \ldots + (n - 1) \rho^{n - 2} \right) \] D: A2 ter ocena navzgor, \( \left| w \right| , \left| z \right| \le \rho \)

   8. \[ \left| w - z \right| \sum\limits_{n = 0}^{\infty} \left| a_n \right| \rho^{n - 2} \left( 1 + 2 + \ldots + n - 1 \right) \]

      D: izpostavljen \( \rho^{n - 2} \)

   9. \[ \frac{\left| w - z \right|}{2} \sum\limits_{n = 0}^{\infty} n (n - 1) \left| a_n \right| \rho^{n - 2} \]

      D: Vsota \( n \) naravnih členov.

   10. \[ \sum\limits_{n = 0}^{\infty} n (n - 1) \rho^{n -2} \]

       konvergira \( \mathcal{D} (0, R) \)

       D: P3 in A1 sledi, da vrsta konvergira.

   11. A3 : \( M = \sum\limits_{n = 1}^{\infty} \left| a_n \right| n (n - 1) \rho^{ n - 2}\)

       \[ \lim_{z \to w} \left| \frac{f(w) - f(z)}{w - z} - g(z) \right| \le \frac{1}{2} M \lim_{z \to w} \left| z - w \right| = 0 \]

   12. QED D: Razlika \( f'(z) \) in \( g(z) \) je enaka 0, torej sta enaka
2. QED

Dokaz z indukcijo.

1. A1: Riemannova vstoa \( R = \sum\limits_{j = 1}^n f(\xi_j) \left( x_j - x_{j- 1} \right) \)

   \[ \left| R \right| \le \sum\limits_{j = 1}^n \left| f \left( \xi_j \right) \right| \left( x_j - x_{j -1} \right) \]

   D: Trikotniška neenakost

2. QED

   D: Ko gre \( n \to \infty \), velja.

1. \[ \left| \int\limits_{\gamma}^{} f(z) \, \mathrm{d} z \right| = \left| \int\limits_a^b f(\gamma(t)) \dot{\gamma}(t) \, \mathrm{d} t \right| \]

   D: Definicija 1.18

2. \[ \left| \int\limits_a^b f(\gamma(t)) \dot{\gamma}(t) \, \mathrm{d} t \right| \le \int\limits_a^b \left| f(\gamma(t)) \dot{\gamma}(t) \right| \, \mathrm{d} t \]

   D: Trditev 1.15

3. \[ \int\limits_a^b \left| f(\gamma(t)) \dot{\gamma}(t) \right| \, \mathrm{d} t = \int\limits_a^b \left| f(\gamma(t)) \right| \left| \dot{\gamma}(t) \right| \, \mathrm{d} t \]

   D: \( \left| a \cdot b \right| = \left| a \right| \left| b \right|\)

4. A1: \( \left| \dot{\gamma}(t) \right| \, \mathrm{d} t = \mathrm{d} z\)

   \[ \int\limits_a^b \left| f(\gamma(t)) \right| \left| \dot{\gamma}(t) \right| \, \mathrm{d} t = \int\limits_a^b \left| f(z) \right| \, \mathrm{d} z \]

   D: A1 ter <1>3

5. QED

1. \[ \int\limits_{\gamma\circ \phi}^{} f(z) \, \mathrm{d} z = \int\limits_c^d f(\gamma(\phi(t))) \dot{\gamma}(\phi(t)) \dot{\phi}(t) \, \mathrm{d} t \]

   D: Definicija integrala funkcije \( f(z) \) po poti \( \gamma \circ \phi \)

2. C1: nova spremenljivka \( u = \phi(t) \)

   \[ \int\limits_c^d f(\gamma(\phi(t))) \dot{\gamma}(\phi(t)) \dot{\phi}(t) \, \mathrm{d} t = \int\limits_{a}^b f (\gamma(u)) \dot{\gamma}(u) \, \mathrm{d} u = \int\limits_{\gamma}^{} f(z) \, \mathrm{d} z \]

   D: C1 ter definicija ločnega elementa

3. QED

   D: V <1>2 dokažemo, da je enak <1>1

1. \[ \int\limits_{\gamma^{-}}^{} f(z) \, \mathrm{d} z = \int\limits_a^b f(\gamma(a + b - t)) \dot{\gamma}(a + b - t) \, \left(-\mathrm{d}t \right) \]

   D:Definicija funkcije \( f(z) \) po krivulji \( \gamma^- \)

2. C1: \( u = a + b- t \) nova spremenljivka

   \[ \int\limits_b^a f(\gamma(a + b - t)) \dot{\gamma}(a + b - t) \, \mathrm{d} t = - \int\limits_a^b f(\gamma(u)) \dot{\gamma}(u) \, \mathrm{d} u \]

   D: C1 ter <1>1

3. \[ \int\limits_b^a f(\gamma(u)) \dot{\gamma}(u) \, \mathrm{d} u = - \int\limits_a^b f(\gamma(u)) \dot{\gamma}(u) \, \mathrm{d} u = - \int\limits_{\gamma}^{} f(z) \, \mathrm{d} z \]

   D: Definicija funkcije \( f(z) \) po krivulji \( \gamma \).

4. QED

1. \[ \int\limits_{\gamma}^{} f'(z) \, \mathrm{d} z = \int\limits_a^b f'(\gamma(t)) \dot{\gamma}(t) \, \mathrm{d} t \]

   D: Definicija integrala funkcije \( f'(z) \) po poti \( \gamma \).

2. \[ \frac{\mathrm{d} }{\mathrm{d} t} \left( f \circ \gamma \right) = f' \left( \gamma(t) \right) \dot{\gamma}(t) \]

   D: Verižno pravilo velja tudi v kompleksnem prostoru.

3. \[ \int\limits_a^b \frac{\mathrm{d} }{\mathrm{d} t} \left( f \circ \gamma \right) \, \mathrm{d} t = \left. \left( f \circ \gamma \right) \right|_a^b \]

   D: <1>1 in <1>2

4. QED

   D: <1>3

1. A1: fiksna točka \( a \in D \)

   A2: poljubna točka \( z \in D \)

   Obstaja neka pot \( \gamma \) med \( a \) in \( z \).

   D: Območje \( D \) je povezano.

2. A3: definiramo \[ F(z) = \int\limits_{\gamma}^{} f(w) \, \mathrm{d} w \]

   C3 je dobro definiran.

   1. C3Definicija A4: \( \gamma_1, \gamma_2 \) različni poti od \( a \) do \( z \)

      A5: \( \gamma_2^- \) je obratna pot od \( \gamma_2 \)

      \[ \int\limits_{\gamma_1 \cup \gamma_2^-}^{} f(w) \, \mathrm{d} w = \int\limits_{\gamma_1}^{} f(w) \, \mathrm{d} w - \int\limits_{\gamma_2}^{} f(w) \, \mathrm{d} w = 0 \]

      D: Integral funkcije \( f(w) \) po spoju poti, integral po obratni poti \( \gamma_2 \) ter predpostavka P3

   2. \[ \int\limits_{\gamma_1}^{} f(w) \, \mathrm{d} w = \int\limits_{\gamma_2}^{} f(w) \, \mathrm{d} w \]

      D: <1>1

   3. QED

      D: \( F(z) \) je dobro definiran, saj je neodvisen od izbire poti.

3. \[ \left| \frac{F( + h ) - F(z)}{h} - f(z) \right| \to 0, \ h \to 0 \]

   1. \( h \) je tak, da je \( z + h\in D \)

      D: \( D \) je območje = odprto + povezano

   2. A7: \( \gamma_1 \) je pot od \( a \) do \( z \)

      A8: \( \gamma_2 \) je daljica \( z \) do \( z + h \)

      A9: \( \gamma_3 \) je pot od \( a \) do \( z + h \)

      \[ \left| \frac{F(z + h) - F(z) }{h} - f(z) \right| = \left| \frac{1}{h} \left( \int\limits_a^{z + h} f(w) \, \mathrm{d}w - \int\limits_a^z f(w) \, \mathrm{d} w \right) - f(z) \right| \]

      D: <1>2, sklenjena krivulja po poti \( \gamma_1 \cup \gamma_2 \cup \gamma_3 \) ter A7 - A9

   3. \[ = \left| \frac{1}{h} \int\limits_z^{z + h} f(w) \, \mathrm{d} w - f(z) \right| \]

      D: <1>2 ter sklenjena pot \( \gamma_1 \cup \gamma_2 \cup \gamma_3 \)

   4. \[ f(z) = \frac{1}{h} \int\limits_z^{ z + h} f(z) \, \mathrm{d} w \]

      D: \( f(z) \) je konstantna

   5. \[ \left| \frac{1}{h} \int\limits_z^{ z + h} \left( f (w) - f(z) \right) \, \mathrm{d }w \right| \le \frac{1}{\left| h \right|} \int\limits_z^{ z+ h} \left| f(w) - f(z) \right| \, \mathrm{d}w \]

   6. A10: \( \epsilon, \delta > 0 \)

      \[ \frac{1}{ \left| h \right|} \int\limits_z^{ z + h} \left| f(w) - f(z) \right| \, \mathrm{d} w = \frac{1}{\left| h \right|}\int\limits_z^{z + h} \epsilon \, \mathrm{d} w = \epsilon \]

      D: \( f \) je zvezna, kar pomeni, da za \( w \in D \), da je \( \left| w - z \right| < \delta \), kar pomeni, da je \( \left| f(w) - f(z) \right| < \epsilon \) ter integral konstante.

   7. QED

      Razlika v <1>3 je manjša od poljubnega \( \epsilon > 0 \).

4. QED

   D: \( F' = f \) iz <1>3

1. A1: \( \alpha \in \mathbb{C} \setminus \gamma^{\ast} \)

   Obstaja \( \mathcal{D}(\alpha, r) \subset \mathbb{C} \setminus \gamma^{\ast} \)

   D: \( \mathbb{C} \setminus \gamma^{\ast} \) je odprta množica, saj je komplement kompaktne krivulje.

2. A2: \( z \in \mathcal{D}(\alpha, r) \)

   A3: \( \xi \in \gamma^{\ast} \)

   \[ \frac{1}{\xi - z} = \frac{1}{\xi - \alpha + \alpha - z} = \frac{1}{\xi - \alpha} \left( 1 + \frac{z - \alpha}{\xi - \alpha} \right)^{-1} = \sum\limits_{n = 0 }^{\infty} \frac{\left( z -\alpha \right) ^{n}}{\left( \xi - \alpha \right)^{n + 1}} \]

   Dokaz:

   1. \( \left| z - \alpha \right| < r \)

      D: A2

   2. \( \left| \xi - \alpha \right| > r \)

      D: A3, ki leži izven \( \mathcal{D}(\alpha, r) \)

   3. \( q = \left| \frac{z - \alpha}{\xi- \alpha} \right| < 1 \)

      D: <2>1, <2>2

   4. QED

      D: Geometrijska vrsta je definirana kot

      \[ \frac{1}{1 + q} = \sum\limits_{n = 0}^{\infty} q^n \]

3. \[ F(z) = \frac{1}{2\pi \mathrm{i}} \int\limits_{\gamma}^{} \sum\limits_{n = 0}^{\infty} \frac{f(\xi)}{\left( \xi - \alpha \right)^{n + 1}} \left( z - \alpha \right)^n \, \mathrm{d} \xi \]

   D: Definicija \( F \) ter <1>2

4. \[ \frac{1}{2\pi \mathrm{i}} \int\limits_{\gamma}^{} \sum\limits_{n= 0}^{\infty} \frac{f(\xi)}{\left( \xi - \alpha \right)^{n + 1}} \left( z - \alpha \right)^n \, \mathrm{d} \xi = \frac{1}{2 \pi \mathrm{i}} \sum\limits_{n = 0}^{\infty} \int\limits_{\gamma}^{} \frac{f(\xi)}{\left( \xi - \alpha \right)^{n + 1}} \left( z - \alpha \right)^n \, \mathrm{d} \xi \]

   D: Vrsta enakomerno konvergira

   1. \[ \left| \frac{f(\xi)}{\left( \xi - \alpha \right)^{n + 1}} \left( z - \alpha \right)^n \right| = \frac{\left| f(\xi) \right|}{\left| \xi - \alpha \right|} \left| \frac{z - \alpha}{\xi - \alpha} \right| \]

      D: \( \left| a b \right| = \left| a \right| \left| b \right| \)

   2. \[ M = \max_{\gamma^{\ast}} \left| f \right| \]

      D: Po P2 je zvezna na kompaktni množici in doseže globalni maksimum

   3. \[ \left| \xi - \alpha \right| \ge d (\alpha, \gamma^{\ast}) \]

      D: A3 in definicija razdalje

   4. \[ \left| \frac{f(\xi)}{\left( \xi - \alpha \right)^{n + 1}} \left( z - \alpha \right)^n \right| \le \frac{M}{d(\alpha, \gamma^{\ast})} q^n \]

      D: <1>2<2>3, <2>3, <2>2

   5. QED

      D: Vrsta konvergira, ker je \( q < 1 \)

5. Funkcijo \( F \) je holomorfna.

   D: Funkcijo smo razvili v vrsto \( \implies \) holomorfna.

6. QED

   Funkcija je holomorfna in razvili smo jo v vrsto.

1. A1:

   \[ \mathrm{ind}_{\gamma} (z) = \frac{1}{2\pi \mathrm{i} } \int\limits_{\gamma}^{} \frac{\mathrm{d} \xi}{\xi - z} \]

   Funkcija \( \mathrm{ind}_{\gamma} (z) \) je zvezna.

   D: Po lemi 1.23 je \( \mathrm{ind}_{\gamma}(z) \) holomorfna funkcija za \( z \in \mathbb{C} \setminus \gamma^{\ast} \), torej je tudi zvezna. Uporabiš \( f(z) = 1 \) v lemi 1.23

2. Ime je Indeks=konstanta Indeks je konstanta na vsaki povezani komponenti \( D \)

   1. A2: definiramo

      \[ F(t) = \left( \gamma(t) - z \right) \exp \left(- \int\limits_a^t \frac{\dot{\gamma}(u) }{\gamma(u) - z} \, \mathrm{d} u \right) \]

      \( F \) je konstantna.

      Dokaz:

      1. \[ \dot{F}(t) = \dot{\gamma} (t) \exp \left( - \int\limits_a^t \frac{\dot{\gamma}(u)}{\gamma(u) - z} \, \mathrm{d} u \right) + \left( \gamma(t) - z \right) \exp \left( - \int\limits_a^t \frac{\dot{\gamma}(u)}{\gamma(u) - z} \, \mathrm{d} u \right) \cdot \left( - \frac{\dot{\gamma}(t)}{\gamma(t) - z} \right) \]

         D: Odvod produkta

      2. QED

         D: Odvod funkcije \( F(t) \) je 0, kar pomeni, da je funkcija \( F \) konstantna.

   2. \( F(a) = F(b) \)

      D: <1>2 velja za \( \forall t \in [a, b] \)

   3. \[ F(a) = \gamma(a) - z = \gamma(b) - z \]

      D: A2 za \( t = a \) ter <1>3

   4. \[ F(b) = \left( \gamma(b) - z \right) \exp \left( - \int\limits_a^b \frac{\dot{\gamma}(u)}{\gamma(u) - z} \, \mathrm{d} u \right) \]

      D: A2 za \( t = b \)

   5. \[ \exp \left( - \int\limits_a^b \frac{\dot{\gamma}(u)}{\gamma(u) - z} \, \mathrm{d} u \right) = 1 \]

      D: <1>3, <1>4 in <1>5

   6. \[ - \int\limits_a^b \frac{\dot{\gamma}(u)}{\gamma(u) - z} \, \mathrm{d} u = 2 \pi \mathrm{i} k, \ k \in \mathbb{Z} \]

      D: Rešitev enačbe \( e^z = 1 \) v kompleksnem prostoru.

   7. \[ \frac{1}{2\pi \mathrm{i} } \oint \limits_{\gamma} \frac{1}{\xi - z} \, \mathrm{d} \xi \in \mathbb{Z} \]

      D: \( \xi = \gamma(u) \) ter <2>6

   8. QED

      D: \( \mathrm{ind}_{\gamma} (z) \) je zvezna in slika v \( \mathbb{Z} \), je funkcija konstantna na vsaki povezani komponenti od \( D \).

3. \( \mathrm{ind}_{\gamma}(z) = 0 \) na neomejeni ploskvi

   Dokaz:

   1. A3: \( z \) komponenta neomejene komponente množice D A4: \( M = \max_{t \in [a, b]} \dot{\gamma}(t) \)

      \[ \left| \mathrm{ind}_{\gamma} (z) \right| = \left| \frac{1}{2\pi \mathrm{i} } \left( \int\limits_a^b \frac{\dot{\gamma}(u)}{\gamma(u) - z} \, \mathrm{d} u \right) \right| \]

   2. \[ \le \frac{1}{2\pi} \int\limits_a^b \frac{\left| \dot{\gamma}(u) \right|}{\left| \gamma(u) - z \right|} \, \mathrm{d} u \]

      D: \( \left| a b \right| = \left| a \right| \left| b \right|\)

   3. \[ \le \frac{M}{2 \pi} \int\limits_a^b \frac{1}{\left| \gamma(u) - z \right|} \, \mathrm{d} u \]

      D: \( A4 \)

   4. \[ \le \frac{M}{2\pi d(z, \gamma^{\ast})} (b - a) \]

      D: \( \left| \gamma(u) - z \right| \ge d(z, \gamma^{\ast}) \)

   5. QED

      D: \( M \), \( 2\pi \) ter \( b - a \) so konstante. \( d(z, \gamma^{\ast}) \) ni omejena. Za veliko razdaljo, je vrednost <2>4 in posledično <2>1 poljubno majhna. Zaradi zveznosti to pomeni, da če velja za zelo oddaljene točke, velja tudi za zelo majhne.

4. QED

1. A1: \( u = \Re f \)

   A2: \( v = \Im f \)

   A3: \( f(z) = u + \mathrm{i} v \)

   A3.5: \( \gamma = \gamma_1 + \mathrm{i} \gamma_2 \)

   \[ \int\limits_{\gamma}^{} f(z) \, \mathrm{d} z = \int\limits_{\gamma}^{} \left( u \, \mathrm{d} x - v \, \mathrm{d} y \right) + \mathrm{i} \int\limits_{\gamma}^{} \left( v \, \mathrm{d} x + u \, \mathrm{d} y \right) \]

   D: definicija integrala funkcije \( f \) po kompleksni poti \( \gamma = \gamma_1 + \mathrm{i} \gamma_2 \).

   \[ \int\limits_{\gamma}^{} f(z) \, \mathrm{d} z = \int\limits_{\gamma_1}^{} (u + \mathrm{i} v) \, \mathrm{d} x + \mathrm{i} \int\limits_{\gamma_2}^{} (u + \mathrm{i} v) \, \mathrm{d} y \]

2. A4: \( \gamma \) omejuje množico \( \Delta \)

   \[ \int\limits_{\gamma}^{} \left( u \, \mathrm{d} x - v \, \mathrm{d} v \right) + \mathrm{i} \int\limits_{\gamma}^{} \left( v \, \mathrm{d} x + u \, \mathrm{d} y \right) = \iint\limits_{\Delta}^{} \left( -v_x - u_y \right) \, \mathrm{d} x \mathrm{d} y + \mathrm{i} \int\limits_{\Delta}^{} \left( u_x - v_y \right) \, \mathrm{d} x \mathrm{d} y \]

   D: Greenova formula

   \[ \oint_{\gamma} P \, \mathrm{d} x + Q \, \mathrm{d} y = \iint\limits_{\Delta}^{} \left( Q_x - P_y \right) \,\mathrm{d x} \mathrm{d}y \]

   ki jo lahko uporabimo zaradi P3

3. \[ \int\limits_{\gamma}^{} f(z) \, \mathrm{d} z = 0 \]

   D: Uporabimo CR enačbe, vstavimo v <1>2.

1. A1: \( \gamma_1 \) krožnica s polmerom \( r \) in središčem v \( z \), orientirana v nasprotni smeri od \( \gamma \)

   A2: \( z \in D \setminus \gamma^{\ast} \)

   A3:

   \[ g(\xi) = \frac{f(\xi) - f(z)}{\xi - z} \]

   Funkcija \( g(z) \) je holomorfna na \( D \setminus \left\{ z \right\} \)

   D: A3

2. A4: \( \gamma' = \gamma \cup \gamma_1 \)

   \[ \int\limits_{\gamma'}^{} g(\xi) \, \mathrm{d} \xi = 0 \]

   Dokaz: Cauchyjev izrek

3. \[ = \int\limits_{\gamma}^{} g(\xi) \, \mathrm{d} \xi - \int\limits_{\gamma_1}^{} g(\xi) \, \mathrm{d} \xi \]

   D: A4, ter integral funkcije \( g \) po spoju krivulj ter nasprotna orientacija \( \gamma_1 \) A1

4. \[ = \int\limits_{\gamma}^{} \frac{f(\xi) - f(z)}{\xi - z} \, \mathrm{d} \xi - \int\limits_{\gamma_1}^{} \frac{f(\xi) - f(z)}{\xi - z} \, \mathrm{d} \xi \]

   D: A3

5. \[ = \int\limits_{\gamma}^{} \frac{f(\xi)}{\xi - z} \, \mathrm{d} \xi - f(z) \cdot 2 \pi \mathrm{i} \, \mathrm{ind}_{\gamma} (z) - \int\limits_{\gamma_1}^{} \frac{f(\xi) - f(z)}{\xi - z} \, \mathrm{d} \xi \]

   D: Ovojno število, definicija

6. \[ \left| \int\limits_{\gamma_1}^{ } \frac{f(\xi) - f(z)}{\xi - z} \, \mathrm{d} \xi \right| \overset{r \to 0}{\longrightarrow} 0 \]

   Dokaz:

   1. A5: Parametrizacija \( \gamma_1 (t) = \xi = z + r e^{\mathrm{i} t}, \ t \in [2\pi, 0] \)

      \[ = \left| \int\limits_{2\pi}^0 \frac{f(z + r e^{\mathrm{i} t}) - f(z)}{re^{\mathrm{i} t} } r \mathrm{i} e^{\mathrm{i} t} \, \mathrm{d} t \right| \]

      D: A5

   2. \[ \le \int\limits_{2\pi}^0 \left| f(z + re^{\mathrm{i} t}) - f(z) \right| \, \mathrm{d}t \overset{r \to 0}{\longrightarrow} 0 \]

      D: Trditev 1.15

   3. QED

7. QED

   D: Dobim željeno enačbo.

Cauchyjevo formulo odvajamo kot integral s parametrom \( n \)-krat.

1. A1: \( \mathcal{D}(\alpha, r) \subseteq \mathcal{D}(\alpha, R) \subseteq D \)

   \[ f(z) = \frac{1}{2\pi \mathrm{i} } \int\limits_{\partial \mathcal{D}(\alpha, r)}^{} \frac{f(\xi)}{\xi - z} \, \mathrm{d} \xi \]

   D: Cauchyjeva formula.

2. Funkcijo \( f(z) \) lahko razvijemo v potenčno vrsto.

   D: Lema 1.23

3. Formula za koeficiente sledi iz posledice 1.27

1. A1: \( \rho < r \)

   A2: \( \gamma \) krožicna s polmerom \( \rho \) in središčem v \( a \)

   Case 1:

   Parametrizacija \( \gamma \) je

   \[ \gamma = \xi = a + \rho e^{\mathrm{i} t} \ t \in [0, 2\pi] \]

2. \[ \left| f^{(n)} (a) \right| \le \frac{n!}{2\pi \rho^n} \int\limits_0^{2\pi} \frac{\left| f(a + \rho e^{\mathrm{i} t}) \right|}{\left| \rho^n e^{\mathrm{i} n t} \right|} \, \mathrm{d} t \]

   D: Cauchyjeva formula za območja.

3. \[ \le \frac{M n!}{2\pi \rho^n } \int\limits_0^{2\pi} \, \mathrm{d} t \]

   D: P2

4. \[ = \frac{M n!}{\rho^n} \]

   D: <1>2

5. QED

   D: <1>2-<1>4 velja za poljuben \( \rho < r \). Ko gre \( \rho \to r \), dobimo neenakost.

1. A1: \( a \in \mathbb{C} \)

   A2: \( f'(a) = 0 \)

   A3: \( \left| f(z) \right| \le M , \ \forall z \in \mathbb{C}\) zaradi omejenosti

   A4: \( \mathcal{D}(\alpha, r) \)

   Velja

   \[ \left| f'(a) \right| \le \frac{M}{r} \]

   D: Cauchyjeva ocena

2. Velja za vsak \( r> 0 \).

   D: \( f \) holomorfna povsod.

3. \[ \left| f'(a) \right| \to 0 \]

   D: \( r \to \infty \) za <1>1

4. QED

   D: Odvod je ničeln, kar pomeni, da je funkcija konstantna.

1. A1: \( p(z) \) nekonstanten polinom s koeficienti iz \( \mathbb{C} \) in brez kompleksne ničle.

   Dokaži: \( p(z) \) je konstanten.

2. Funkcija

   \[ f(z) = \frac{1}{p(z)} \]

   je definirana povsod in cela.

   D: A1 nima ničel, ter je holomorfna.

3. \( f(z) \) je omejena funkcija

   1. A2:

      \[ p(z) = a_n z^n + a_{n - 1} z^{n - 1} + \ldots + a_1 z + a_0, \ a_n = 0 \]

      A3: \( a_n = 1 \) brez izgube splošnosti.

   2. \[ \left| f(z) \right| = \left| \frac{1}{p(z)} \right| \]

   3. \[ = \frac{1}{\left| z^n \left(1 + a_{n - 1} z^{n - 1} + \ldots + a_1 z + a_0 \right) \right|} \]

      D: <2>2, A2, A3

   4. \[ \le \frac{1}{\left| z^n \right|} \frac{1}{\left| 1 - \frac{\left| a_{n - 1} \right|}{\left| z \right|} - \ldots - \frac{\left| a_1 \right|}{\left| z^{n - 1} \right|} - \frac{\left| a_0 \right|}{\left| z^n \right|} \right|} \]

      D: \( \left| a + b \right| \ge \left| a \right| - \left| b \right|\)

   5. \[ \frac{1}{\left| z \right|^n } \overset{\left| z \right| \to \infty}{\longrightarrow} 0 \]

      <2>4

   6. QED

4. \( f(z) \) je konstantna.

   D: Liouvilleov izrek in <1>3

5. QED

   D: <1>4 pove, da je polinom \( p(z) \) tudi konstanten, kar je v protislovju z <1>1.

1. A1: \( A \) množica stekališč \( Z(f) \) v \( D \)

   A2: \( A = \emptyset \)

   Vsaka ničla je izolirana.

   D: \( A \) ni stekališče, kar pomeni, da po definiciji stekališča ne obstajajo točke, ki so v \( A \) in v vsaki okolici ničle, ampak niso ničla.

2. \( f \ne 0 \)

   1. A3: \( \forall a \in Z (f) \)

      A4: \( m_a \in \mathbb{N} \)

      \[ f(z) = (z - a)^{m_a} h(z) \]

      D: Funkcija je holomorfna.

   2. QED

3. A5: \( A \ne \emptyset \)

   \( A \) je zaprta.

   Dokaz:

   1. Stekališče ničel funkcije \( f \) je ničla.

      D: Zveznost.

   2. QED

      D: stekališče stekališč je stekališče, \( A \) je zaprta.

4. \( A \) je odprta.

   Dokaz:

   1. A6: \( a \in A \)

      A7: Obstaja \( r > 0 \)

      \( \mathcal{D} (a, r) \subseteq D \)

      D: Množica \( D \) je odprta.

   2. \( a \) ne more biti izolirana ničla.

      D: \( a \) je po A6 stekališče.

   3. Na \( \mathcal{D}(a, r) \) razvijemo \( f \) v potenčno vrsto.

      D: \( f \) je holomorfna.

   4. A8: \( \mathcal{D}(a, \rho), \rho < r \).

      \( f = 0 \) na \( \mathcal{D} (a, \rho) \)

      D: \( \mathcal{D}(a, \rho) \subseteq \mathcal{D}(a, r) \) in \( a \) je ničla.

   5. Razvijemo \( f \) v potenčno vrsto na \( \mathcal{D}(a, \rho) \).

      D: \( f \) je holomorfna.

   6. \( f = 0 \) na \( \mathcal{D}(a, r) \)

      D: Koeficienti vrst <2>3 ter <2>5 so enaki.

   7. \( A \) neprazna odprta množica v \( D \)

      D: \( \mathcal{D}(a, \rho) \subseteq A \)

   8. QED

5. \( A = D \)

   D: \( A \) je odprta in zaprta.

6. QED

   \( f = 0 \) na celotni množici \( D \), ker je \( D \) povezana in \( A = D \).

1. A1: \( h = f - g \)

   \( A \subseteq Z (h) \)

   D: \( \left. h \right|_A = 0 \) zaradi P4

2. \( h = 0 \) na celotni množici \( D \)

   D: \( A \) je stekališče \( h \).

3. QED

1. \[ f(z) = \frac{1}{2\pi \mathrm{i} } \int\limits_{\partial A}^{} \frac{f(\xi)}{\xi - z} \, \mathrm{d} \xi \]

   D: Cauchyjeva formula za območja.

2. \[ f(z) = \frac{1}{2 \pi \mathrm{i} } \int\limits_{\gamma_2}^{} \frac{f(\xi)}{\left( \xi - z \right) \left( 1 - \frac{z - a}{\xi - z} \right)} \, \mathrm{d} \xi + \frac{1}{2 \pi \mathrm{i} } \int\limits_{\gamma_1}^{} \frac{f(\xi)}{(z - a) \left( 1 - \frac{\xi - a}{ z - a} \right)} \, \mathrm{d} \xi \]

   D: \( \xi - a + a - z \), ter izpostaviš pravilen člen, ter \( \partial A = \gamma_1 \cup \gamma_2 \)

3. A1: \( \xi \in \gamma_2 \), integracija po \( \gamma_2 \)

   \[ \left| \frac{z - a}{\xi - a} \right| < 1 \]

   D: P3 in velja \( \left| \xi - a \right| < \left| z - a \right|\)

4. A2: \( \xi \in \gamma_1 \), integracija po \( \gamma_1 \)

   \[ \left| \frac{\xi - a}{z - a} \right| < 1 \]

   D: P4 in velja \( \left| \xi - a \right| > \left| z - a \right|\)

5. \[ f(z) = \frac{1}{2 \pi \mathrm{i} } \int\limits_{\gamma_2}^{} \frac{f(\xi)}{\xi - a} \sum\limits_{n = 0}^{\infty} \left( \frac{z - a}{\xi - a} \right) ^n \, \mathrm{d} \xi + \frac{1}{2 \pi \mathrm{i} } \int\limits_{\gamma_1}^{} \frac{f(\xi)}{z - a} \sum\limits_{n = 0}^{\infty} \left( \frac{\xi - a}{z - a} \right)^n \, \mathrm{d} \xi \]

   D: geometrijska vrsta

6. \[ \sum\limits_{n = 0}^{\infty} \left( \frac{1}{2 \pi \mathrm{i} } \int\limits_{\gamma_2}^{} \frac{f(\xi)}{\left( \xi - a \right)^{n + 1} } \, \mathrm{d} \xi \right) \left( z - a \right)^n + \sum\limits_{n = 0}^{\infty} \frac{1}{2 \pi \mathrm{i} } \int\limits_{\gamma_1}^{} f(\xi) \left( \xi - a \right)^n \, \mathrm{d} \xi \left( z - a \right)^{- n - 1} \]

   D: Kot pri lemi 1.23, vrsta konvergira enakomerno

7. \[ \sum\limits_{n = 0}^{\infty} \left( \frac{1}{2 \pi \mathrm{i} } \int\limits_{\gamma_2}^{} \frac{f(\xi)}{\left( \xi - a \right)^{n + 1} } \, \mathrm{d} \xi \right) \left( z - a \right)^n + \sum\limits_{n = -\infty}^{-1} \left( \frac{1}{2 \pi \mathrm{i } }\int\limits_{\gamma_1}^{} \frac{f(\xi)}{\left( \xi - a \right)^{n + 1} } \right) \left( z - a \right)^{n} \]

   D: Menjava \( m = -1 - n \).

8. QED

1. Razvoj \( f(z) \) v Laurentovo vrsto

   \[ f(z) = \sum\limits_{n =- \infty}^{\infty} c_n \left( z - a \right)^n \]

   D: P2

2. A1: \( \delta > 0 \)

   \( c_{-n} = 0 \ \forall n \in \mathbb{N} \)

   1. \( f(z) \le M \) za vsak \( z \in \mathcal{D}(a. r) \)

      D: P3

   2. A2: parametrizacija \( \gamma = a + \delta e^{\mathrm{i } t }, \ t \in [0, 2\pi) \)

      \[ \left| c_{-n} \right| \le \left| \frac{1}{2 \pi \mathrm{i} } \int\limits_0^{2\pi} \frac{f \left( a + \delta e^{\mathrm{i} t} \right)}{\left( \delta e^{\mathrm{i} t} \right)^{-n + 1} } \delta \mathrm{i} e^{\mathrm{i} t} \, \mathrm{d} t \right| \]

      D: Formula za koeficiente z negativnim indeksom pri Laurentovem razvoju.

   3. \[ \le \frac{1}{2 \pi} \int\limits_0^{2\pi} \frac{\left| f \left( a + \delta e^{\mathrm{i} t} \right) \right|}{\delta^{-n } \left| e^{\mathrm{i} n t} \right|} \, \mathrm{d} t \]

   4. \[ \le \frac{M}{2\pi} \int\limits_0^{2\pi} \delta^n \, \mathrm{d} t = M \delta^n \]

      D: <1>2

   5. QED

      D: \( \delta \to 0 \), je \( c_{-n} = 0 \)

3. QED

1. A1: \( w \in \mathbb{C} \)

   A2: \( \epsilon, \delta > 0 \)

   Za vsak \( z \), za katerega velja \( \left| z - a \right| < \delta\), \( z \ne a \), sledi \( \left| f(z) - w \right| \ge \epsilon \).

2. A3: \[ g(z) = \frac{1}{f(z) - w} \]

   Funkcija \( g(z) \) je holomorfna na \( \mathcal{D}'(a,r ) \)

   D: \( f(z) \) je holomorfna

3. \[ \left| g(z) \right| = \frac{1}{\left| f(z) - w \right|} \le \frac{1}{\epsilon} \]

   D: <1>1

4. \( g(z) \) je omejena v okolici \( a \)

   D: <1>3

5. \( g(z) \) razvijemo v potenčno vrsto okoli \( a \) ter jo zapišemo kot produkt

   \[ g(z) = \left( z - a \right)^m h(z) \]

   kjer je \( h(z) \) holomorfna funkcija.

   D: \( a \) je odpravljiva singularnost za \( g(z) \) po P3

6. \[ f(z) = \frac{1}{g(z)} + w = \frac{1}{\left( z - a \right)^m h(z)} + w \]

   D: A3

7. \( z = a \) je pol stopnje \( m \) za \( f \)

   D: <1>6

8. QED

   D: <1>7 je v protisljovu s predpostavko, da je \( a \) bistvena singularnost.

Opomba, da je na realnih množicah logaritem definiran kot

\[ \int\limits_{}^{} \frac{f'(x)}{f(x)} \, \mathrm{d} x = \log f(z) + C \]

1. A1: fiksen \( z_0 \in D \)

   A2: poljubna točka \( z \in D \)

   A3: \( \gamma \) povezuje \( z_0 \) in \( z \)

   Definiramo

   \[ g(z) = z_0 + \int\limits_{\gamma}^{} \frac{f'(\xi)}{f(\xi)} \, \mathrm{d} \xi \]

   kjer je integrand holomorfen.

   D: \( f(z) \ne 0 \) po P2

2. Definicija <1>1 je neodvisna od izbire \( \gamma \).

   D: \[ \int\limits_{\gamma}^{} \frac{f(\xi)}{\xi} \, \mathrm{d} \xi \]

   je 0 po vsaki sklenjeni poti zaradi Cauchyjevega izreka ter P3.

3. \[ e^{- g(z)} f(z) = 1 = G(z) \]

   je konstantna.

   1. \[ G' (z) = -e^{- g(z)} g'(z) f(z) + e^{- g(z)} f'(z) \]

      D: Odvajanje produkta ter verižno pravilo.

   2. \[ 0 = - e^{- g(z)} \frac{f'(z)}{z} f(z) + e^{- g(z)} f'(z) \]

      D: odvod <1>1 ter <2>1

   3. QED

      D: Funkcija je konstantna, ker je njen odvod enak \( 0 \).

4. A4: \( G(z_0) = 1 \)

   \[ e^{- g(z_0)} f(z_0) = e^{- z_0} f(z_0) \]

   D: <1>1 za \( z_0 \)

5. A5: \( f(z_0) = w_0 \)

   \[ e^{z_0} = w_0 \]

6. \( z_0 = \log w_0 \)

   D: <1>5

7. A6: \( e^{g(z)} = f(z) \)

   A7: \( e^{h(z)} = f(z) \)

   \[ g(z) - h(z) = 2 \pi k \mathrm{i} \]

   D: \( e^{g(z) - h(z)} = 1 \)

8. QED

   D: v <1>6 smo dokazali obstoj in v <1>7, da velja razlika.

1. A1: \( q_k (z) \) glavni del Laurentove vrste funkcije \( f \) okoli točke \( a_k \)

   Funkcija

   \[ f(z) - q_1 (z) - \ldots - q_n (z) \]

   je holomorfna na celotnem območju \( D \).

   D: Funkcija ima odpravljive singularnosti.

2. \[ \oint\limits_{\gamma}^{} \left( f(z) - \sum\limits_{k = 1}^n q_k (z) \right) \, \mathrm{d} z = 0 \]

   D: Cauchyjev izrek.

3. \[ \int\limits_{\gamma}^{} f(z) \, \mathrm{d} z = \sum\limits_{k = 1}^n \int\limits_{\gamma}^{} q_k (z) \, \mathrm{d} z \]

   D: Menjava možna zaradi končne vsote.

4. \[ \int\limits_{\gamma}^{} q_k (z) \, \mathrm{d} z = \sum\limits_{n = - \infty}^{-2} \int\limits_{\gamma}^{} c_n \left( z - a_k \right)^n \, \mathrm{d} z + \int\limits_{\gamma}^{} \frac{c_{-1}}{z - a_k} \, \mathrm{d} z \]

   D: A1

5. \[ \sum\limits_{n = - \infty}^{-2} \int\limits_{\gamma}^{} c_n \left( z - a_k \right)^n \, \mathrm{d} z = 0 \]

   D: Cauchyjev izrek, saj območje \( D \) vsebuje tudi singularnosti funkcije \( f \) ter je \( f \) holomorfna.

6. \[ \int\limits_{\gamma}^{} q_k (z) \, \mathrm{d} z = 2 \pi \mathrm{i} \, c_{-1} \mathrm{ind}_{\gamma} (a_k) \]

   D: Cauchyjeva formula

7. QED

   D: Upoštevamo definicijo residuuma

   \[ \int\limits_{\gamma}^{} f(z) \, \mathrm{d} z = \sum\limits_{k = 1}^n 2 \pi \mathrm{i} \, \mathrm{ind}_{\gamma}\left( a_k \right) \mathrm{Res} \left( f, a_k \right) \]

1. \[ f(z) = \frac{c_{-n}}{\left( z - a \right)^n} + \ldots + \frac{c_{-1}}{z - a} + c_0 + c_1 (z - a) + \ldots \]

   D: Laurentov razvoj funkcije \( f \) okoli \( a \) za \( n \)-ti pol

2. \[ f(z) \left( z - a \right)^n = c_{-n} + c_{-n + 1} (z - a) + \ldots + c_{-1} (z - a)^{n -1} + c_0 (z - a)^n + c_1 (z- a)^{n + 1} \]

   je \( a \) odpravljivo singularnost.

   D: ker so vsi koeficienti z negativnimi indeksi enaki 0 pri \( a \)

3. \[ \left( (z - a)^n f(z) \right) ^{(n - 1)} = \left( n - 1 \right)! c_{-1} + n! c_0 (z - a) + (n + 1)! c_1 (z - a)^2 + \ldots \]

   D: \( (n - 1) \) odvajanje.

4. \[ c_{-1} = \frac{\lim_{z \to a} \left(\left( z - a \right)^n f(z)\right) ^{(n - 1)} }{(n -1)!} \]

   D: \( z \to a \)

5. QED

   D: upoštevamo definicijo residuuma.

1. A1: \( a_1, a_2, \ldots, a_r \) različne ničle funkcije \( f \) znotraj \( \mathcal{D}(a, r) \) z večkratnostmi \( m_1, m_2, \ldots, m_r \)

   A2: \(b_1, b_2, \ldots, b_s\) različni poli \( f \) znotraj \( \mathcal{D}(a, r) \) z večkratnostmi \( n_1, n_2, \ldots, n_s \)

   C1:

   \[ f(z) = \frac{\left( z - a_1 \right)^{m_1} \left( z - a_2 \right)^{m_2} \ldots \left( z - a_r \right)^{m_r} }{\left( z - b_1 \right)^{n _1} \left( z - b_2 \right) ^{n _2} \ldots \left( z - b_s \right)^{n_s} } g(z) \]

   Funkcija \( g(z) \) nima niti ničel niti polov v \( \mathcal{D}(a, r) \)

2. A3: \[ f(z) = (z - a)^k g(z) \]

   \[ \frac{f'(z)}{f(z)} = \frac{k}{z - a} + \frac{g'(z)}{g(z)} \]

   Dokaz: A3 odvajaš po \( z \), ter deliš z \( (z - a)^k g(z) \)

3. \[ \frac{f'(z)}{f(z)} = \frac{m_1}{z - a_1} + \frac{m_2}{z - a_2 } + \ldots + \frac{m_r}{z - a_r } - \frac{n_1}{ z - b_1} - \frac{n_2}{ z - b_2} - \ldots - \frac{n_s}{ z - b_s} + \frac{g'(z)}{g(z)} \]

   D: <1>2

4. \[ \frac{1}{2 \pi \mathrm{i} } \int\limits_{\partial \mathcal{D}(a, r)}^{} \frac{f'(z)}{f(z)} \, \mathrm{d} z = m_1 \frac{1}{2 \pi \mathrm{i} } \int\limits_{\partial \mathcal{D}(a, r)}^{} \frac{\mathrm{d} z}{z - a_1} + \ldots - n_1 \frac{1}{2 \pi \mathrm{i} } \int\limits_{\partial \mathcal{D}(a, r)}^{} \frac{\mathrm{d} z}{z - b_1} - \ldots + \frac{1}{2 \pi \mathrm{i} } \int\limits_{\partial \mathcal{D}(a, r)}^{} \frac{g'(z)}{g(z)} \, \mathrm{d} z \]

   D: Integracija.

5. \[ \int\limits_{\partial \mathcal{D}(a, r)}^{} \frac{f'(z)}{f(z)} \, \mathrm{d} z = m_1 + \ldots + m_r - n_1 - \ldots - n_s = \mathcal{N} - \mathcal{P} \]

   D: Cauchyjeva formula ter Cauchyjev izrek pri \( g(z) \), saj nima polov na \( \mathcal{D}(a, r) \).

1. A1: \( U \) odprta množica v \( D \)

   A2: slika funkcije \( f \) je \( f(u) = \left\{ f(u); \ u \in U \right\} \)

   A3: \( b \in f(U) \)

   Obstaja tak \( a \in U \), da je \( f(a) = b \)

   D: A3

2. A4: \( g = f(z) - b \)

   A5: \( \alpha \) je \( n \)-kratna ničla

   A6: \( \delta, \epsilon > 0 \)

   A7: \( \forall w \in \mathcal{D}( b, \epsilon ) \), \( w \ne b \)

   Enačba \( f(z) = w \) ima natanko \( n \) rešitev v \( \mathcal{D}(a, \delta) \)

   D: No proof.

3. \[ \mathcal{D}(b, \epsilon) \subseteq f \left( \mathcal{D}(a, \delta) \right) \subseteq f(U) \]

   D: <1>2 ter \( \mathcal{D}(a, \delta) \subseteq \), ker je \( U \) odprta.

4. QED

   \( f(U) \) je odprta po definiciji, ker je za nek \( \epsilon > 0 \) in vsak \( b \in f(U) \) \( \mathcal{D}(b, \epsilon) \subseteq f(U) \)

1. A1: \( z_0 \in D \)

   A2: \( \left| f(z_0) \right| = D \) ima maksimum na \( D \).

   Dokaži: protislovje.

   1. Množica \( f(D) \) je odprta.

      D: \( D \) odprta in izrek o odprti preslikavi.

   2. A3: \( r > 0 \)

      \( \mathcal{D}(f (z_0), r) \subseteq f(D) \)

      D: <1>1

   3. A4: \( w = f(z_1) \)

      A5: \( w \in \mathcal{D}(f(z_0), r) \)

      A6: \( w \) kot na sliki

      

      Velja \( \left| w \right| > \left| f (z_0) \right| \)

   4. QED

      Po <2>3 imamo protislovje.

2. A7: \( f \) nima ničel

   A8: \( \left| f(z) \right| \) zavzame minimum v točki \( z_0 \in D \)

   A9: \[ g(z) = \frac{1}{f(z)} \]

   A10: \( \left| g(z_0) \right| \) ima maksimalno vrednost.

   Dokaži: narobe

   D: po <1>1 to ne more držati.

3. QED

1. A1: \( \forall z \in \mathcal{D}(0, 1) \)

   \[ \left| f(z) \right| \le \left| z \right| \]

   1. A2: \( f(z) = a_0 + a_1 z + a_2 z ^2 + \ldots \)

      \( a_0 = 0 \)

      D: P2

   2. \[ g(z) = \frac{f(z)}{z} = a_1 + a_2 z + \ldots \]

      ima odpravljive singularnosti pri \( z = 0 \).

      D: Koeficienti z negativnimi indeksi so enaki \( 0 \).

   3. A3: \( g(0) = f'(0) = a_1 \)

      \( g \) je holomorfna.

      D: Z A3 imamo razvoj v potenčno vrsto.

   4. A4: \( \left| r \right| < 1\)

      \[ g(z) \le \max_{\left| \xi \right| = r} \left| g(\xi) \right| \]

      D: princip maksimuma

   5. A5: \( \xi_0 \in \mathcal{D}(0, r) \)

      A6: \( \left| g \left( \xi_0 \right) \right| \) je maksimum funkcije

      \[ \left| g(z) \right| \le \left| g(\xi_0) \right| \]

      D: <2>4

   6. \[ \frac{\left| f(z) \right|}{\left| z \right|} \le \frac{\left| f(\xi_0) \right|}{\left| \xi_0 \right|} = \frac{\left| f(\xi_0) \right|}{r} \]

      D: <2>2 ter <2>5

   7. \( \left| f(\xi_0) \right| = 1 \) (ali pa konstanta)

      D: A3

   8. QED

      Za \( r \to 1 \) sledi \( \frac{\left| f(z) \right|}{\left| z \right|} \le 1 \)

2. \[ \left| f'(0) \right| = \left| a_1 \right| = \left| g(0) \right| \le 1 \]

   D: A3 ter <1>1<2>8

3. A7: \( z_0 \in \mathcal{D}(0, 1) \)

   A8: \( \left| f(z_0) \right| = \left| z_0 \right| \)

   A9: \( \left| f'(0) \right| = 1 \)

   \( \forall z \in \mathcal{D}(0, 1) \)

   \[ f(z) = \alpha z \]

   \( \alpha \in \mathbb{C}, \alpha = \mathrm{konst}, \ \left| \alpha \right| = 1 \)

   1. \( \left| g(0) \right| = 1 \)

      D: A9

   2. \( g \) zavzame maksimum v \( z = 0 \) ali

   3. V \( z_0 \) funkcija \( \left| g(z) \right| \) zavzame maksimum.

      D: \( \left| g(z_0) \right| = 1 \) in definicija \( g(z) \) <1>1 <2>2

   4. \( g(z) \) konstantna na \( \mathcal{D}(0, 1) \).

      D: Princip maksima

   5. \( g(z) = \alpha \)

      D: <2>2

   6. \[ f(z) = \alpha z, \ \alpha = g(z_0) \implies \left| \alpha \right|= \left| g(z_0) \right| = 1 \]

      D: <2>3 ter <2>1

1. A1: \( d(\alpha, 0) < 1 \)

   Pol funkcije je \( z = \frac{1}{{\bar{\alpha}} \)

   D: P2

2. \[ \left| z \right| = \frac{1}{\left| \bar{\alpha} \right|} = \frac{1}{\left| \alpha \right|} \]

   D: Absolutna vrednost kompleksnega števila.

3. \( \left| z \right| > 1 \)

   D: <1>2

4. QED

Dokazujemo:

A1: \( \left| z \right| = 1\)

\[ \left| f_{\alpha} (z) \right| = 1 \]

1. \[ \left| f_{\alpha} \left( z \right) \right| = \left| \frac{z - \alpha}{1 - \bar{\alpha} z} \right| \]

   D: P2

2. \[ = \left| \frac{z - \alpha}{1 - \alpha \bar{z}} \right| \]

   D: \( \frac{1}{1 - \bar{\alpha} z } = \frac{1}{\overline{1 - \bar{\alpha} z} } \)

3. \[ = \left| \frac{z - \alpha}{1 - \frac{\alpha}{z} } \right| \]

   D: \( \bar{z} = \frac{1}{z} \), saj \( z \bar{z} = 1 \)

4. \[ \left| \frac{z (z - \alpha)}{z - \alpha} \right| =1 \]
5. QED

1. A1: \( z_0 \in \mathcal{D}(0, 1) \) A2: \( \left| f_{\alpha} \left( z_0 \right) \right| \ge 1 \)

   \( \left| f_{\alpha} (z) \right| \) zavzame maksimum v \( \mathcal{D}(0, 1) \).

   D: A2

2. \( f_{\alpha} \) je konstantna za vsak \( z \in \mathcal{D} (0, 1) \)

   D: Princip maksimuma

3. <1>2 ne drži.

   D: Hkrati velja \( f_{\alpha}(\alpha) = 0 \) P2, ter A2

4. QED

   D: Sledi \( \left| f_{\alpha} \right| < 1 \)

1. \[ f_{-\alpha} \left( \partial \mathcal{D}(0, 1) \right) \subseteq \partial \mathcal{D}(0, 1) \]

   D: \( \left| - \alpha \right| < 1 \) ter trditev 1.52a

2. \[ f_{-\alpha} \left( \mathcal{D}(0, 1) \right) \subset \mathcal{D}(0, 1) \]

   D: \( \left| - \alpha \right| < 1\) ter trditev 1.52b

3. \[ \left( f_{\alpha} \circ f_{-\alpha} \right) (z) = z \]

   Dokaz:

   1. \[ = f_{\alpha} \left( \frac{z + \alpha}{1 + \bar{\alpha} z} \right) \]

      D: P2 za \( -\alpha \)

   2. \[ = \frac{\frac{z + \alpha}{1 + \bar{\alpha} z }- \alpha}{1 - \bar{\alpha} \frac{z + \alpha}{1 + \bar{\alpha} z } } \]

      D: <2>1

   3. \[ = \frac{z + \alpha - \alpha - \left| \alpha \right| ^2 z}{1 + \bar{\alpha} z - \bar{\alpha} z - \left| \alpha \right| ^2} \] D: <2>2
   4. \[ = \frac{1 - \left| \alpha \right| ^2}{1 - \left| \alpha \right| ^2} z = z \]
   5. QED

4. \[ \left( f_{-\alpha} \circ f_{\alpha}\right) (z) = z \]

   Dokaz:

   1. \[ = f_{-\alpha} \left( \frac{z - \alpha}{1 - \bar{\alpha} z } \right) \]

      D: P2

   2. \[ = \frac{\frac{z + \alpha}{1 + \bar{\alpha} z} - \alpha}{1 - \bar{\alpha} \frac{z + \alpha}{1 + \bar{\alpha} z}} \]

      D: <2>1

   3. \[ = \frac{z + \alpha - \alpha - \left| \alpha \right| ^2 z}{1 + \bar{\alpha} z - \bar{\alpha} z - \left| \alpha \right| ^2} \]

      D: <2>2

   4. \[ = z \frac{1 + \left| \alpha \right| ^2}{1 + \left| \alpha \right| ^2} = z \]

      D: <2>3

5. QED

   \[ f_{\alpha} \circ f_{-\alpha} = f_{-\alpha} \circ f_{\alpha} = \mathrm{id}_{\bar{\mathcal{D} }(0, 1)} \]

1. \[ \partial \mathcal{D}(0, 1) = f_{\alpha} \left( f_{-\alpha} \left( \partial \mathcal{D}(0, 1) \right) \right) \]

   D: \( f_{\alpha} \) je bijektivna.

2. \[ f_{\alpha} \left( f_{-\alpha} \left( \partial \mathcal{D}(0, 1) \right) \right) \subseteq f_{\alpha} \left( \partial \mathcal{D}(0, 1) \right) \]

   D: Trditev 1.52a za \( \left| - \alpha \right| < 1 \)

3. \[ f_{\alpha} \left( \partial \mathcal{D}(0, 1) \right) \subseteq \partial \mathcal{D}(0, 1) \]

   D: Trditev 1.52a

4. QED

   D: <1>1 je enaka <1>3

1. \[ \mathcal{D}(0, 1) = f_{\alpha} \left( f_{-\alpha} \left( \mathcal{D}(0, 1) \right) \right) \]

   D: \( f \) je bijektivna.

2. \[ f_{\alpha} \left( f_{-\alpha} \left( \mathcal{D}(0, 1) \right) \right) \subseteq f_{\alpha} \left( \mathcal{D}(0, 1) \right) \]

   D: Trditev 1.52b za \( \left| -\alpha \right| < 1 \)

3. \[ f_{\alpha} \left( \mathcal{D}(0, 1) \right) \subseteq \mathcal{D}(0, 1) \]

   D: Trditev 1.52b

4. QED

   D: <1>1 in <1>3 sta enaki.

1. A1: \( f(0) = 0 \)

   A2: \( \forall z \in \mathcal{D}(0, 1) \)

   \[ \left| f(z) \right| \le \left| z \right| \]

   D: Schwarzeva lema

2. A3: \( \forall z \in \mathcal{D}(0, 1) \)

   \[ \left| f(z) \right| = \left| z \right| \]

   1. \( f^{-1} : \mathcal{D}(0, 1) \to \mathcal{D}(0, 1) \) je bijektivna in holomorfna.

      D: \( f: \mathcal{D}(0, 1) \to \mathcal{D}(0, 1) \) je bijektivna in holomorfna.

   2. A4: \( \forall z \in \mathcal{D}(0, 1) \)

      \[ \left| f^{- 1} (z) \right| \le \left| z \right| \]

      D: Schwarzeva lema

   3. \[ z = f \left( f^{-1} (z) \right) \]

      D: Identiteta funkcija

   4. \[ \left| f \left( f^{-1} (z) \right) \right| \le \left| f(z) \right| \le \left| z \right| \]

      D: <2>2 ter <1>1

   5. QED

3. A5: \( \forall z \in \mathcal{D}(0, 1) \)

   A6: \( \left| \alpha \right| = 1 \)

   \[ f(z) = \alpha z = \alpha f_0 \left( z \right) \]

   D: Definicija \( f_{\alpha} \)

4. A7: \( \beta \in \mathcal{D}(0, 1) \)

   A8: \( f(0) = \beta \)

   A9: \[ g:= f_{\bamma} \circ f \]

   Funkcija \( g: \mathcal{D}(0,1) \to \mathcal{D}(0, 1) \) je bijektivna.

   D: Funkciji kompozituma sta biholomorfni

5. \( g(0) = 0 \)

   D: \[ g(0) = f_{\beta} \left( f (0) \right) = f_{\beta} (\beta) = 0 \]

   in prvi primer

6. \[ g(z) = \alpha z \]

   D: Prvi primer

7. \[ f(z) = \alpha f_{-\gamma } \]

   Dokaz:

   1. \[ f_{\beta} \left( f(z) \right) \alpha z \]

      D: Definicija \( g \) ter <1>6

   2. \[ f(z) = f_{-\beta} \left( \alpha z \right) \]

      D: Inverz funkcije \( f_{\beta} \)

   3. \[ f(z) = \frac{\alpha z + \beta}{1 - \bar{\beta} \alpha z} \]

      D: <2>2

   4. \[ f(z) = \alpha \frac{z + \bar{\alpha} \beta}{ 1 + \bar{\beta} \alpha z } = \alpha f_{-\bar{\alpha} \beta } (z) \]

      D: definicija \( f_{\alpha} \) ter <2>3

   5. QED
8. QED

1. \[ \left( f_A \circ f_B\right) (z) = f_A \left( f_B (z) \right) \]

   D: Definicija kompozituma.

2. \[ \left( f_A \circ f_B \right) (z) = \frac{a \cdot \frac{uz + v}{wz + t} + b}{c \frac{uz + v}{wz + t} + d} \]

   D: Definicija ulomljene linearne transformacije

3. \[ \left( f_A \circ f_B \right) (z) = \frac{\left( au + bw \right)z + av + bt}{\left( cu + dw \right) z + cv + dt} \]

   D: <2>2

4. \[ C = AB = \begin{bmatrix} au + bw & av + bt \\ cu + dw& cv + dt \end{bmatrix} \]
5. QED

1. \[ f_A \circ f_{A^{-1} } = f_{AA^{-1} } \]

   D: Lema 1.54

2. \[ f_{AA^{-1} } = f_I = z = \mathrm{id } \]
3. \[ f_{A^{-1}} \circ f_A = f_{A^{-1}A} \]
4. \[ f_{A^{-1}A} = f_I = z = \mathrm{id } \]

1. \( \impliedby \) A1: \( \det A \ne 0 \)

   \( f_A \) ni konstantna.

   D: \( A \) obrnljiva.

2. \( \implies \) A2: \( ad = bc \)

   \[ f(z) = \mathrm{konst} \]

   1. A3: \( c \ne d \ne 0 \)

      \[ \frac{a}{b} = \frac{c}{d} \]

      D: A3

   2. \[ f(z) = \frac{\frac{a}{c} z + \frac{b}{c} z}{z + \frac{d}{c} } = \frac{b}{d} \left( \frac{z + \frac{d}{c} }{ z + \frac{d}{c} } \right) \]

   3. QED

      D: <2>2

   4. A4: \( c \ne 0, \ d = 0 \)

      \[ f(z) = \frac{az + 0 }{cz + 0} = \frac{a}{c} \]

      D: \( d = 0 \) implicira \( b = 0 \)

   5. A5: \( c = 0, \ d \ne 0 \)

      \[ f(z) = \frac{0z + b}{0z + d} = \frac{b}{d} \]

      D: \( c = 0 \) implicira \( a = 0 \)

   6. A6: \( c = 0 = d \)

      Funkcija \( f(z) \) ne obstaja.

   7. QED
3. QED

1. A1: \[ f(z) = \frac{az + b}{cz + d} \]

   A2: \( c = 0 \)

   A3: \( d \ne 0 \)

   \( f(z) \) je kompozitum translacije in raztega.

   D: \[ z \mapsto \frac{a}{d} z \mapsto \frac{a}{d} z + \frac{b}{d} \]

2. A4: \( c \ne 0 \)

   A5: \( d = 0 \)

   \( f(z) = \frac{az + b}{cz} \) je kompozitum inverza, raztega in translacije.

   D: \[ z \mapsto \frac{1}{z} \mapsto \frac{b}{c} \frac{1}{z} \mapsto \frac{a}{c} + \frac{b}{c} \frac{1}{z} \]

3. A6: \( c \ne d \ne 0 \)

   \( f(z) = \frac{az + b}{cz + d} \) je kompozitum raztega, translacije in inverza.

   D: \[ z \mapsto cz \mapsto cz + d \mapsto \frac{1}{cz + d} \mapsto \left( b - \frac{ad }{c} \right) \frac{1}{cz + d} \mapsto \frac{a \left( b - \frac{ad }{c} \right)}{cz + d} = \frac{az + b}{cz + d} \]

1. A1: \( z = x + \mathrm{i} y \)

   A2: \( \beta = a + \mathrm{i} b \)

   \[ \alpha \left( x ^2 + y ^2 \right) + \left( x + \mathrm{i} y \right) \left( a + \mathrm{i} b \right) + \left( x - \mathrm{i} y \right) \left( a - \mathit{i}b \right) + \gamma = 0 \]

   D: A1, A2 ter enačba.

2. \[ \alpha \left( x ^2 + y ^2 \right) + 2(ax - by) + \gamma = 0 \]

   D: <1>1

3. A3: \( \alpha = 0 \)

   Enačba predstavlja premico.

   D: \[ -2by = -2ax - \gamma \]

4. A4: \( \alpha \ne 0 \) A5: \( \left| \beta \right| ^2 > \alpha \gamma \)

   Enačba je krožnica in A5 zagotovi, da množic točk, ki jo opisuje enačba ni prazna.

5. QED

1. Premik slika krožnice v krožnice ter premice v premice.
2. Raztegi slikajo krožnice v krožnice ter premice v premice.
3. Rotacije slikajo krožnice v krožnice ter premice v premice.
4. Inverzija ohranja družino premic in krožnic.

   1. A1: \( f(z) = \frac{1}{z} \)

      A2: \( \alpha \left| z \right| ^2 + \beta z + \bar{\beta} \bar{z} + \gamma = 0 \)

      \[ \frac{\alpha}{\left| z \right| ^2} + \frac{\beta}{z} + \frac{\bar{\beta} }{\bar{z} } + \gamma = 0 \]

   2. \[ \alpha + \beta \bar{z} + \bar{\beta} z + \gamma \left| z \right| ^2 = 0 \]

   3. QED

      D: za \( \alpha ' = \gamma \), \( \beta ' = \bar{\beta} \) ter \( \gamma' = \alpha \) dobimo prvotno enačbo.

1. A1: \( w_1 = 0 \)

   A2: \( w_2 = 1 \)

   A3: \( w_3 = \infty \)

   \[ f(z) = \frac{z - z_1}{z - z_3} \frac{z_2 - z_3}{z_2 - z_1} \]

   ustreza sliki \( f(z_1) = 0 \), \( f(z_2) = 1 \) in \( f(z_3) = \infty \).

2. Obstajata enolični transformaciji \( f, g : \hat{\mathbb{C} } \to \hat{\mathbb{C} } \), da je \( f(z_1) = g(w_1) = 0 \), \( f(z_2) = g(w_2) = 1 \) ter \( f(z_3) = g(w_3) = \infty \)

3. Preslikava \( h = g^{-1} \circ f : \hat{\mathbb{C} } \to \hat{\mathbb{C} }\) je ulomljna linearna preslikava.

   D: inverz in kompozitum vsake ulomljene linearne preslikave je ulomljena linearna preslikava.

4. Velja

   \[ h(z_1) = g^{-1} \left( f(z_1) \right) = g^{-1} (0) = w_1 \]

   D: <1>2, <1>3 ter predpostavke.

5. Velja

   \[ h(z_2) = g^{-1} \left( f(z_2) \right) = g ^{-1} (1) = w_2 \]

6. Velja

   \[ h (z_3) = g^{-1} \left( f(z_3) \right) = g^{-1} (\infty) = w_3 \]

7. \( h \) je enolično določen

   1. A4: \( h_1 : \hat{\mathbb{C} } \to \hat{\mathbb{C} } \)

      A5: \( h_1 \) ima enake lastnosti kot \( h \)

      \[ g \circ h = g \circ h_1 \]

   2. Velja

      \[ \left( g \circ h \right) \left( z_1 \right) = \left( g \circ h_1 \right) (z_1) = 0 \]

   3. Velja

      \[ \left( g \circ h \right) (z_2) = \left( g \circ h_1 \right) (z_2) = 1 \]

   4. Velja

      \[ \left( g \circ h \right) (z_3) = \left( g \circ h_1 \right) (z_3) = \infty \]

   5. QED
8. QED

1. A1: \( z \in D \)

   A2: \( \partial \mathcal{D}(z, r) \subset D \)

   \[ f_n(z) = \frac{1}{2 \pi \mathrm{i} } \int\limits_{\partial \mathcal{D} \left( z, r \right)}^{} \frac{f_n(\xi)}{\xi - z} \, \mathrm{d} \xi \]

   D: Po P2 so \( f_n \) holomorfne.

2. A3: \( n \to \infty \)

   \[ f(z) = \frac{1}{2 \pi \mathrm{i} } \int\limits_{\partial \mathcal{D}(z, r)}^{} \frac{f(\xi)}{\xi - z} \, \mathrm{d} \xi \]

   D: P3

3. QED

   D: Po Cauchyjevi formuli je funkcija \( f \) holomorfna na \( \mathcal{D}(z, r) \).

1. A1: \[ F_n (z) = \int\limits_{\frac{1}{n} }^n t^{ z - 1} e ^{-t} \, \mathrm{d} t \]

   Funkcija \( F_n (z) \) so holomorfna.

   D: Integrand \( t^{z - 1} e^{-t} \) je zvezen.

2. Funkcije \( F_n (z) \) so zvezno odvedljive.

   D: Odvod \( \frac{\partial }{\partial z} \left( t^{z - 1} e^{- t} \right) \) je zvezen na \( t \in \left[ \frac{1}{n}, n \right] \)

3. A2: \( F_n (z) \overset{n \to \infty}{\longrightarrow} \Gamma(z) \)

   Konvergena je enakomerna na kompaktnih množicah \( K \) na \( \left\{ z \in \mathbb{C}, \ \Re z > 0 \right\} \)

   1. A3: \( \delta = \min d \left( z \in K, \ \Re z = 0 \right) \)

      A4: \( M = \max d \left( z \in K, \ \Re z = 0 \right) \)

      \[ \int\limits_0^{\frac{1}{n } } t^{z - 1} e^{-t} \, \mathrm{d} t \overset{n \to \infty}{\longrightarrow} 0 \]

      1. \[ t^{z - 1} = e^{(z - 1) \ln t} \]

         D: Definicija logaritma.

      2. A5: \( z = x + \mathrm{i} y \)

         \[ t^{z - 1} = e^{(x - 1)\ln t} \cdot e^{\mathrm{i } y \ln t} \]

         D: A5 ter <3>1

      3. \[ \left| t^{z - 1} \right| = t^{\Re z - 1} \]
      4. \[ \left| \int\limits_0^{\frac{1}{n} } t^{z - 1} e^{- t} \right| \le \int\limits_0^{\frac{1}{n} } \left| t^{z - 1} e ^{-t} \right| \]
      5. \[ = \int\limits_0^{\frac{1}{n} } t^{\Re z - 1} e^{- t} \, \mathrm{d} t \]

      6. \[ \le \int\limits_0^{\frac{1}{n} } t^{M - 1} e ^{-t} \, \mathrm{d} t \]

         D: A4

      7. \[ \le \int\limits_0^{\frac{1}{n} } t^{M - 1} \, \mathrm{d} t \]

         D: za \( t \in \left[ 0, \frac{1}{n} \right] \) je \( e^{-t} < 1 \)

      8. \[ = \left. \frac{t^M}{M} \right|_0 ^{\frac{1}{n} } = \frac{\left( \frac{1}{n} \right)^M}{M} \overset{n \to \infty}{\longrightarrow} 0 \]
      9. QED
   2. \[ \int\limits_n^{\infty} t^{z - 1} e^{-t} \, \mathrm{d} t \overset{n \to \infty}{\longrightarrow} 0 \]
      1. \[ \left| \int\limits_n^{\infty} t^{z - 1} e^{- t} \, \mathrm{d} t \right| \le \int\limits_n^{ \infty} t^{M - 1} e^{-t} \, \mathrm{d} t \]

      2. A6: \( g(t) = t^{M - 1} e^{- \frac{t}{2} } < m \)

         \[ \le m \int\limits_n^{\infty} e^{-\frac{t}{2} } \, \mathrm{d} t \]

      3. \[ = \left. - 2m e^{-\frac{t}{2} } \right|_n ^{\infty} = 2m e ^{- \frac{n}{2} } \overset{n \to \infty}{\longrightarrow} 0 \]
      4. QED
   3. QED
4. QED

1. Funkcija \( f(x + \mathrm{i} y) = u(x, y) + \mathrm{i} v(x, y) \) je holomorfna.

2. A1: \( (x, y) \in D \)

   A2: \( (a, b) \in D \)

   A3: \( \gamma_1 \) je ena pot, ki povezuje \( (x, y) \) in \( (a, b) \)

   A4: \( \gamma_2 \) je druga pot, ki povezuje \( (x, y) \) in \( (a, b) \)

   Za \( \gamma = \gamma_1 \cup \gamma_2^- \) je

   \[ v(x, y) = \int\limits_{\gamma}^{} \left( -u_y \, \mathrm{d} x + u_x \, \mathrm{d} y \right) \]

   dobro definiran.

   Dokaz:

   1. \[ \int\limits_{\gamma}^{} \left( - u_y \, \mathrm{d} x + u_x \, \mathrm{d} y \right) = \int\limits_{\gamma_1 }^{} \left( - u_y \, \mathrm{d} x + u_x \, \mathrm{d} y \right) - \int\limits_{\gamma_2}^{} \left( - u_y \, \mathrm{d} x + u_x \, \mathrm{d} y \right) \]

      D: Integral po spoju poti.

   2. \[ \int\limits_{\gamma}^{} \left( -u_y \, \mathrm{d} x + u_x \, \mathrm{d} y \right) = \iint\limits_D^{} \left( u_{xx } - \left( - u_{yy } \right) \right) \,\mathrm{d x dy} = \iint\limits_D^{} \nabla ^2 u \,\mathrm{d x dy} \]

      D: Greenova formula.

   3. \[ \int\limits_{\gamma_1}^{} \left( - u_y \, \mathrm{d} x + u_x \, \mathrm{d} y \right) = \int\limits_{\gamma_2}^{} \left( - u_y \, \mathrm{d} x + u_x \, \mathrm{d} y \right) \]

      D: Predpostavka P2, <2>1 ter <2>2

   4. QED

3. Vektorsko polje

   \[ \begin{bmatrix} -u_y \\ u_x \end{bmatrix} \]

   je potencialno.

   D: \[ \mathrm{rot} \begin{bmatrix} -u_{y} \\ u_{x} \end{bmatrix} = 0 \]

4. Obstaja \( v \in \mathcal{C}^1 \), da velja

   \[ \begin{bmatrix} -u_y \\ u_x \end{bmatrix} = \mathrm{grad} v = \nabla v = \begin{bmatrix} v_x \\ v_y \end{bmatrix} \]

   D: <1>3

5. QED

   D: \( u \) je diferenciabilen, držijo CR enakosti.

1. A1: \( a \in U \)

   A2: \( \mathcal{D}(a, r) \subseteq U \)

   Funkcija: \( u \) je realni del holomorfne funkcije \( f \) na okolici \( a \in U \).

   D: Trditev 2.1 ter enostavna povezanost kroga.

2. \( u \) je neskončnokrat odvedljiva.

   D: Holomorfna funkcija \( f \) je neskončnokrat odvedljiva.

1. Krog \( \bar{\mathcal{D}}(a, r) \) je enostavno povezan

   D: krog je enostavno povezan

2. A1: \( f \) holomorfen na \( \bar{\mathcal{D}}(a, r) \)

   \[ u = \Re f \]

   D: Trditev 2.1

3. \[ f(a) = \frac{1}{2 \pi \mathrm{i} } \oint\limits_{\partial \mathcal{D}(a, r)}^{} \frac{f(\xi)}{\xi - a} \, \mathrm{d} \xi \]

   D: Cauchyjeva formula.

4. A2: parametrizacija \( \xi = a + r e^{\mathrm{i} \phi} \)

   \[ f(a) = \frac{1}{2 \pi} \int\limits_0^{2\pi} f\left(a + re ^{\mathrm{i} \phi}\right) \, \mathrm{d} \phi \]

   D: <1>3 ter A2

5. \[ u(a) = \Re f(a) = \frac{1}{2 \pi} \int\limits_0^{2 \pi} u \left( a + re ^{\mathrm{i} \phi} \right) \, \mathrm{d} \phi \]

   D: <1>2 ter <1>4

1. A1: \( a \in D \)

   A2: \( u \) v točki \( a \) zavzame maksimum

   \[ u(a) = M \]

   A3: Množica vseh točk v \( D \),kjer \( u \) zavzame maksimum:

   \[ U = \left\{ z \in D, u(z) = M \right\} \]

   Množica \( U \) je neprazna.

   D: A2

2. A4: označimo prasliko \( U = u^{-1} \left( \left\{ M \right\} \right) \)

   \( U \) je zaprta.

   D: \( u \) je zvezna in \( \left\{ M \right\} \) je zaprta.

3. \( U \) je odprta

   1. A6: \( R > 0 \)

      A7: \( b \in U \)

      A8: \( \mathcal{D} (b, R) \setminus \left\{ b \right\} = \bigcup_{\rho < R} \partial \mathcal{D}(b, \rho) \)

      \[ \mathcal{D}(b, R) \subseteq U \]

      oz.

      \[ \partial \mathcal{D}(b, \rho) \subseteq U \]

      1. \[ M = u(b) = \frac{1}{2 \pi} \int\limits_0^{2\pi} u \left( b + \rho e^{\mathrm{i} \phi} \right) \, \mathrm{d} \phi \]

         D: Izrek o povprečni vrednosti.

      2. Velja \( u \left( b + \rho e^{\mathrm{i} \phi} \right) \le M \)

         D: \( u(b) \) je maksimum

      3. \[ 0 = \int\limits_0^{2\pi} \left( M - u(b + \rho e^{\mathrm{i} \phi}) \right) , \mathrm{d} \phi \]

         D: <3>2

      4. \[ M = u \left( b + \rho e^{\mathrm{i} \phi} \right), \forall \phi \]

         D: <3>3 ter zveznost

      5. \[ b + \rho e^{\mathrm{i} \phi} \in U, \forall \phi \]

         D: <3>4 in A3

      6. QED
   2. QED

4. \[ U = D \]

   D: <1>1, <1>2 ter <1>3 ter povezanost \( D \)

5. \( u \) je konstantna na \( D \)

   D: A3, <1>4

6. QED

   D: <1>5 je v protislovju z P2

Dokažeš z \( z = r e^{\mathrm{i} \theta} \)

1. \( P_r \) je zvezna.

   1. Pol je pri

      \[ 1 - 2r \cos \theta + r ^2 = 0 \]

      D: Definicija jedra.

   2. \[ D = 4 r ^2 \cos ^2 \theta - 4 r ^2 \le 0 \]

      D: Definicija diskriminante.

   3. \( D = 0 \) za \( \theta = k \pi, \ k \in \mathbb{Z} \)

      D: \( \cos ^2 \theta = 1 \) ter <2>1

   4. A1: \( \theta = 0 \)

      Pol je pri \( r= 1 \)

      D: <2>1

   5. QED
2. \( P_r > 0 \)

   1. \( 1 - r ^2 > 0 \)

      D: Definicija Poissonovega jedra \( r < 1 \)

   2. \( 1 - 2r \cos \theta + r ^2 > 0 \)

      D: \( r < 1 \) ter <1>1

3. \( P_r (-\theta) = P_r (\theta) \)

   D: Sodost funkcije kosinus.

4. \( P_r \) je periodična s periodo \( 2\pi \)

   D: Funkcija kosinus je periodična z \( 2 \pi \)

5. A2: \( 0 \le \delta \le \theta \le \pi \)

   \[ P_r (\theta) \le P_r (\delta) \]

   1. \( \cos \theta < \cos \delta \)

      D: A2

   2. A3: za fiksen \( r\in [0, 1) \)

      \[ 1 - 2r \cos \theta + r ^2 > 1 - 2r \cos \delta + r ^2 \]

      D: Množenje z minusom.

   3. <2>2 na potenco \( -1 \) obrne predznak.
   4. QED
6. \( \lim_{r \nearrow 1} P_r (0) = \infty \)

   1. \[ \lim_{r \nearrow 1} P_r (0) = \lim_{r \nearrow 1} \frac{1 -r ^2}{1 - 2r + r ^2} \]

      D: Definicija Poissonovega jedra

   2. \[ \lim_{r \nearrow 1} \frac{1 + r}{1 - r} = \infty \]
   3. QED
7. \( \forall \delta > 0 \) je \( \lim_{r \nearrow 1} P_r (\theta) = 0 \) enakomerno za \( 0 < \delta \le \left| \theta \right| \le \pi \).

   1. A3: fiksen \( \delta > 0 \)

      A4: \( 0 \le \delta \le \theta \le \pi \)

      A5: \( \epsilon > 0 \)

      A6: \( r_0 \le r < 1 \)

      \[ P_r < \epsilon, \forall r \]

      D: Točka 5 ter A5, A6

   2. Limita \( \lim_{r \nearrow 1} P_r (\theta) = 0 \) enakomerno za \( \delta \le \left| \theta \right| \le \pi \)

      D: \( 0 < P_r (\theta) \le P_r (\delta) < \epsilon \)

   3. QED
8. QED

1. A1: \( u \) je harmonična na neki odprti okolici \( \bar{\mathcal{D}}(0, 1) \) Funkcija \( u \) je na okolici \( \mathcal{D}(0, 1) \) realni del neke holomorfne funkcije \( f \).

   D: A3

2. A2: \( z = re^{\mathrm{i} \phi} \in \mathcal{D}(0, 1) \) A3: \( \left| \xi \right| = 1, \ \xi \in \partial \mathcal{D}(0, 1) \)

   \[ f(z) = \frac{1}{2 \pi \mathrm{i}} \int\limits_{\partial \mathcal{D}(0, 1)}^{} \frac{f(\xi)}{\xi - z} \, \mathrm{d} \xi \]

   D: Cauchyjeva formula

3. \[ f(z) = \frac{1}{2 \pi \mathrm{i} } \int\limits_{\partial \mathcal{D}(0, 1)}^{} \frac{f(\xi)}{1 - z \bar{\xi} } \, \frac{\mathrm{d} \xi}{\xi} \]

   D: Izpostavimo \( \xi \) ter \( \xi \bar{\xi} = 1 \)

4. \[ \int\limits_{\partial \mathcal{D}(0, 1)}^{} \frac{f(\xi)}{1 - \bar{z} \xi} \, \mathrm{d} \xi \]

   nima pola na \( \mathcal{D}(0, 1) \) in je holomorfen.

   Dokaz:

   1. Pol je pri \( \xi = \frac{1}{z} \)

   2. \[ \left| \xi \right| = \frac{1}{\left| z \right|} > 1 \]

      D: A1

   3. \( \left| \xi \right| = 1 \)

      D: A3

   4. QED

      D: <2>2 ter <2>3 sta v protislovju.

5. \[ f(z) = \frac{1}{2 \pi \mathrm{i} } \int\limits_{\partial \mathcal{D}(0, 1)}^{} f(\xi) \left( \frac{1}{1 - z \bar{\xi}} + \frac{1}{1 - \bar{z} \xi} - 1 \right) \, \frac{\mathrm{d} \xi}{\xi} \]

   Dokaz:

   1. \[ \int\limits_{\partial \mathcal{D}(0, 1)}^{} f(\xi) \left( \frac{1}{1 -\bar{z} \xi} - 1 \right) \, \frac{\mathrm{d} \xi}{\xi} = \int\limits_{\partial \mathcal{D}(0, 1)}^{} \frac{\bar{z} \xi f(\xi)}{1 - \bar{z} \xi} \, \frac{\mathrm{d} \xi}{\xi} \]

   2. \[ = \bar{z} \int\limits_{\partial \mathcal{D}(0, 1)}^{} \frac{f(\xi)}{1 - \bar{z} \xi} \, \mathrm{d} \xi = 0 \]

      D: Cauchyjev izrek ter <1>4

   3. QED

6. \[ = \frac{1}{2 \pi \mathrm{i} } \int\limits_{\partial \mathcal{D}(0, 1)}^{} f(\xi) \frac{1 - \left| z \right| ^2}{\left| 1 - \bar{z} \xi \right| ^2} \, \frac{\mathrm{d} \xi}{\xi} \]

   D: A2

7. A4: \( z = re^{\mathrm{i} \phi} \)

   A5: parametrizacija \( \xi = e^{\mathrm{i} \theta} \)

   \[ = \frac{1}{2 \pi} \int\limits_0^{2\pi} f \left( e^{\mathrm{i} \theta} \right) \frac{1 - r ^2}{\left| 1 - re^{\mathrm{i} (\theta - \phi)} \right| ^2} \, \mathrm{d} \theta \]

   D: A4, A5 ter <1>6

8. \[ = \frac{1}{2 \pi}\int\limits_0^{2 \pi} u \left( e^{\mathrm{i} \theta}\right) P_r \left( \theta - \phi \right) \, \mathrm{d} \theta \]

   D: Definicija Poissonovega jedra ter <1>7

   Dokazali smo na neki odprti okolici, želimo pa na odprti okolici zaprtega enotskega diska.

9. A6: \( 0 < \rho < 1 \)

   A7: \( u_{\rho} (z) \equiv u \left( \rho z \right) \)

   Funkcija \( u_{\rho} \) je harmonična za \( \rho < 1 \)

   1. \( \left| z \right| < 1\)

      D: \( u \) je harmonična na \( \mathcal{D}(0, 1) \)

   2. \[ \left| \rho z \right| < 1 \]

      D: A7

   3. \[ \left| z \right| < \frac{1}{\rho} \]

      D: <2>2

   4. \[ \rho < 1 \]

      D: <2>3

   5. QED

10. \[ u_{\rho} \left( re ^{\mathrm{i} \phi} \right) = \frac{1}{2 \pi} \int\limits_0^{2 \pi} P_r \left( \theta - \phi \right) u_{\rho} \left( e^{\mathrm{i} \theta} \right) \, \mathrm{d} \theta \]

    D: <1>9 ter <1>8

11. A8: \( \rho \nearrow 1 \)

    \( u_{\rho} (z) \) postane \( u(z) \)

    D: zveznost

12. \[ u \left( re ^{\mathrm{i} \phi} \right) = \frac{1}{2 \pi} \int\limits_0^{2\pi} P_r \left( \theta - \phi \right) u \left( e^{\mathrm{i} \theta} \right) \, \mathrm{d} \theta \]

    D: Enakomerna zveznost ter <1>10

13. QED

1. A1: \( u = 1 \)

   A2: \( \phi = 0 \)

   \[ 1 = \frac{1}{2 \pi} \int\limits_0^{2 \pi} P_r (\theta) \, \mathrm{d } \theta \]

   Dokaz: Poissonovo jedro

2. QED

1. \( u \) je enolična

   1. A1: \( u_1 \) rešitev Dirichletovega problema

      A2: \( u_2 \) druga rešitev Dirichletovega problema

      A3: \( u = u_1 - u_2 \)

      \[ \nabla ^2 u = \nabla ^2 u_1 - \nabla ^2 u_2 = 0 \]

      D: linearnost odvodov

   2. \[ \left. u \right|_{\partial \mathcal{D}(0, 1)} = \left. u_1 \right|_{\partial \mathcal{D}(0, 1)} - \left. u_2 \right|_{\partial \mathcal{D}(0, 1)} = g - g = 0 \]
   3. \( u = 0 \)

      1. \( u \) harmonična na \( \mathcal{D}(0, 1) \)

         D: Predpostavka.

      2. \( u \) zavzame maksimum ali minimum na robu kroga \( \partial \mathcal{D}(0, 1) \)

         D: Princip maksimuma in minimuma (\( u \) zavzame maksimum/minimum na zaprtem krogu) in po <2>2 je \( \left. u \right|_{\partial \mathcal{D}(0, 1)} = 0 \)

      3. QED

   4. QED

      D: \( u_1 = u_2 \) iz <1>3 ter A3

2. \( u \) obstaja.

   1. \( \left. u \right|_{\partial \mathcal{D}(0, 1)} = g \)

      D: Po predpostavki.

   2. \( u \) je harmonična na \( \mathcal{D}(0, 1) \)

      Dokaži: \( u \) je realni del neke holomorfne funkcije

      1. A4: \( z = re ^{\mathrm{i} \theta} \)

         A5: Poissonovo jedro

         \[ P_r (\theta) = \Re \left( \frac{1 + z}{1 - z} \right) \]

         A6: \( \left| r \right| < 1\)

         \[ u \left( re^{\mathrm{i} \phi} \right) = \frac{1}{2 \pi} \int\limits_0^{2\pi} P_r \left( \theta - \phi \right) g \left( e^{\mathrm{i}\theta} \right) \, \mathrm{d} \theta \]

         \[ u \left( re ^{\mathrm{i} \phi} \right) = \frac{1}{2 \pi} \int\limits_0^{2 \pi} \Re \left( \frac{1 + re^{\mathrm{i} \left( \vartheta - \phi \right)}}{1 - re ^{\mathrm{i} \left( \theta - \phi \right)}} \right) g \left( e^{\mathrm{i} \theta} \right) \, \mathrm{d} \theta \]

         D: A5 ter definicija Poissonovega jedra

      2. \[ = \frac{1}{2\pi} \int\limits_0^{2\pi} \Re \left( \frac{e^{\mathrm{i} \theta} + re ^{\mathrm{i} \phi}}{e^{\mathrm{i} \theta} - re^{\mathrm{i}\phi}} \right) g \left( e^{\mathrm{i} \theta} \right) \, \mathrm{d} \theta \]

      3. \[ \Re \left( \frac{1}{2 \pi} \int\limits_0^{2\pi} \frac{e^{\mathrm{i} \theta} + z }{e^{\mathrm{i} \theta} - z} g \left( e^{\mathrm{i} \theta} \right) \, \mathrm{d} \theta \right) = \Re f(z) \]

         D: Integral realnega dela kompleksne funkcije je enak realnemu delu integrala kompleksne funkcije

      4. \( f(z) \) je odvedljiva

         D: integrand je zvezen, parcialno zvezno odvedljiv po \( z \)

      5. \( f(z) \) je holomorfna

         D: <3>4

      6. QED

   3. \( u \) je zvezna na \( \bar{\mathcal{D}} (0, 1) \)

      Dokaži:

      \[ u \left( re^{\mathrm{i} \phi} \right) \to g \left( e^{\mathrm{i} \Psi} \right) \]

      1. A7: \( u \left( re^{\mathrm{i} \phi} \right)\overset{r \nearrow 1}{\longrightarrow} g \left( e^{\mathrm{i} \phi} \right) \)

         \[ \left| u \left( re^{\mathrm{i} \phi} \right) - g \left( e^{\mathrm{i} \phi} \right) \right| = \left| \frac{1}{2 \pi} \int\limits_0^{2\pi} P_r \left( \theta - \phi \right) g \left( re ^{\mathrm{i} \theta} \right) \, \mathrm{d} \theta - g \left( e^{\mathrm{i} \phi} \right) \frac{1}{2 \pi} \int\limits_0^{2\pi} P_r \left( \theta - \phi \right) \, \mathrm{d} \theta \right| \]

         D: A6 ter Poissonovo jedro za konstanto

      2. \[ = \frac{1}{2 \pi} \left| \int\limits_0^{2\pi} P_r \left( \theta - \phi \right) \left[ g \left( e^{\mathrm{i} \theta} \right) - g \left( e^{\mathrm{i} \phi} \right) \right] \, \mathrm{d} \theta \right| \]

      3. \[ P_r \left( \theta - \phi \right) \le M \]

         D: \( P_r \left( \theta - \phi \right) \) je omejen, ker je \( \theta- \phi \) omejen z \( \delta \).

      4. A9: \( \epsilon > 0 \)

         \[ \left| g \left( e^{\mathrm{i} \theta} \right) - g \left( e^{\mathrm{i} \phi} \right) \right| < \frac{\epsilon}{2} \]

         D: \( \left| \theta - \phi \right| < \delta\)

      5. \[ \frac{1}{2 \pi} \left| \int\limits_0^{2 \pi} P_r \left( \theta - \phi \right) \left[ g \left( e^{\mathrm{i}\theta} \right) - g \left( e ^{\mathrm{i} \phi} \right) \right] \, \mathrm{d} \theta\right| \le \frac{1}{2\pi} \int\limits_0^{2\pi} P_r \left( \theta - \phi \right) \left| g \left( e^{\mathrm{i} \theta} \right)- g \left( e^{\mathrm{i} \phi} \right) \right| \]

      6. \[ = \frac{1}{2 \pi} \left( M \int\limits_{\left| \theta -\phi \right|< \delta}^{} \left| g \left( e^{\mathrm{i} \theta}\right) - g \left( e^{\mathrm{i} \phi} \right) \, \mathrm{d} \theta \right| + \int\limits_{\left| \theta - \phi \right| \ge \delta}^{} \left| g \left( e^{\mathrm{i} \theta} \right)- g \left( e^{\mathrm{i}\phi} \right) \right| P_r \left( \theta - \phi \right) \, \mathrm{d} \theta\right) \]

         D: <3>3 ter razdelitev intervala \( [0, 2\pi] \) na manjše in večje od \( \delta \)

      7. \[ < \frac{1}{2\pi} \left( M \frac{\epsilon}{2} + M_1 P_r (\delta) \right) \]

         D: <3>4 ter \( P_r \left( \theta - \phi \right) \le P_r (\delta) \)

      8. A10: \( r \nearrow 0 \)

         \[ = 0 \]

         D: Lastnost 8 Poissonovih jeder ter za vsak \( \epsilon \) večji od 0.

      9. \[ \left| u \left( re^{\mathrm{i} \phi} \right) - g \left( e^{\mathrm{i} \Psi} \right) \right| = \left| u \left( re^{\mathrm{i} \phi} \right) - g \left( e^{\mathrm{i} \phi} \right) + g \left( e^{\mathrm{i} \phi} \right) - g \left( e^{\mathrm{i} \Psi} \right) \right| \]

      10. \[ \le \left| u \left( re^{\mathrm{i} \phi} \right) - g \left( e^{\mathrm{i} \phi} \right) \right| + \left| g \left( e^{\mathrm{i} \phi} \right) - g \left( e^{\mathrm{i} \Psi} \right) \right| = 0 + 0 \]

          D: <3>8

3. QED

1. \[ \iint\limits_{\partial D}^{} v \partial_{\vec{n}} u \,\mathrm{d S} = \iiint\limits_D^{} \left( v \nabla ^2 u + \nabla u \nabla v \right) \,\mathrm{d V} \]

   Dokaz:

   1. \[ \iint\limits_{\partial D}^{} v \, \partial_{\vec{n}} u \,\mathrm{d S} = \iint\limits_{\partial D}^{}v \nabla u \cdot \vec{n} \, \mathrm{d} S \]

      D: Normalni odvod funkcije \( u \)

   2. \[ = \iint\limits_{\partial D}^{} v \nabla u \,\mathrm{d } \vec{S} \]

      D: \( \vec{n} \, \mathrm{d} S = \mathrm{d} \vec{S} \)

   3. \[ = \iiint\limits_{D}^{} \mathrm{div} \left( v \nabla u \right) \,\mathrm{d V} \]

      D: Gaussov izrek

      \[ \iint\limits_{\partial D}^{} \vec{F} \,\mathrm{d } \vec{S} = \iiint\limits_{}^{} \mathrm{div} \vec{F} \,\mathrm{d V} \]

   4. \[ \mathrm{div} \left( v \nabla u \right) = \nabla v \nabla u + v \nabla ^2 u \]

      Dokaz:

      1. \[ v \nabla u = v \begin{bmatrix} \frac{\partial u}{\partial x} \\ \frac{\partial u}{\partial y} \\ \frac{\partial u}{\partial z} \end{bmatrix} \]

         D: Definicija gradienta

      2. \[ \mathrm{div}\, v \left( \nabla u \right) = \frac{\partial }{\partial x} v u_x + \frac{\partial }{\partial y} v u_y + \frac{\partial }{\partial z} v u_z \]

         D: Definicija divergence ter <3>1

      3. \[ = \frac{\partial v}{\partial x} u_x + \frac{\partial v}{\partial y} u_y + \frac{\partial v}{\partial z} u_z + v \left( u_{x x} + u_{yy }+ u_{zz } \right) \]

         D: <3>2

      4. QED
   5. QED

2. \[ \iint\limits_{\partial D} \left( v \, \partial_{\vec{n}} u - u \, \partial_{\vec{n}} v \right) \,\mathrm{d S} = \iiint\limits_D^{} \left( v \nabla ^2 u - u \nabla ^2 v \right) \,\mathrm{d V} \]

   D:

   1. Zamenjaš \( u \) in \( v \) iz prve Greenove identitete.
   2. QED

1 Greenova identiteta za poljuben \( u \) ter \( v \equiv 1 \), kjer je potem \( \nabla ^2 u = 0 \) zaradi harmoničnosti ter \( \nabla v = 0 \).

3 Greenova identita, kjer je \( D \) naša krogla \( \bar{\mathcal{K}} \)

1. \[ u \left( \vec{r}_0 \right) = \frac{1}{4 \pi} \iint\limits_{\partial \mathcal{K}( \vec{r}_0, R )}^{} \left( \frac{1}{\left\lVert \vec{r} - \vec{r}_0 \right\rVert} \partial_{\vec{n}} \left( \vec{r} \right) - u \left( \vec{r} \right) \partial_{\vec{n}} \frac{1}{\left\lVert \vec{r} - \vec{r}_0 \right\rVert} \right) \,\mathrm{d S} - \frac{1}{4 \pi} \iiint\limits_{\mathcal{K} \left( \vec{r}_0, R \right)}^{} \frac{\nabla ^2 u}{\left\lVert \vec{r} - \vec{r}_0 \right\rVert} \,\mathrm{d V} \]

   \3. Greenova identiteta

2. \[ \iiint\limits_{\mathcal{K} \left( \vec{r}_0, R \right)}^{} \frac{\nabla ^2 u}{ \left\lVert \vec{r} - \vec{r}_0 \right\rVert} \,\mathrm{d V} = 0 \]

   D: Harmoničnost funkcije \( u \)

3. \[ \iint\limits_{\partial \mathcal{K} \left( \vec{r}_0, R \right)}^{} \frac{1}{R} \partial_{\vec{n}} u \left( \vec{r} \right) \,\mathrm{d S} = 0 \]

   D: Posledica 2.10

4. \[ \iint\limits_{\partial \mathcal{K} \left( \vec{r}_0, R \right)}^{} u \left( \vec{r} \right) \left( - \frac{1}{R ^2} \right) \,\mathrm{d S} = - \iint\limits_{\partial \mathcal{K} \left( \vec{r}_0, R \right)}^{} \frac{u}{R ^2} \,\mathrm{d S} \]

   D: \( \left\lVert \vec{r} - \vec{r}_0 \right\rVert = R\) ter odvod.

5. QED

1. A1: \( h > 0, \ h \in \mathbb{R} \) \[ \left| \hat{f}(\xi + h) - \hat{f}(\xi) \right| = 0 \]

   1. \[ = \frac{1}{\sqrt{2\pi}} \left| \int\limits_{-\infty}^{\infty} f(x) e^{- \mathrm{i} x (\xi + h)} \, \mathrm{d} x - \int\limits_{- \infty}^{\infty} f(x) e^{- \mathrm{i} x \xi} \, \mathrm{d} x \right| \]

      D: Definicija Fourierove transformacije

   2. \[ = \frac{1}{\sqrt{2 \pi}} \left| \int\limits_{- \infty}^{\infty} f(x) e^{- \mathrm{i} \xi x} \left( e^{- \mathrm{i} h x} - 1 \right) \, \mathrm{d} x \right| \]
   3. \[ \le \frac{1}{\sqrt{2 \pi}} \int\limits_{-\infty}^{\infty} \left| f(x) \right| \left| e^{- \mathrm{i} hx } - 1 \right|\, \mathrm{d} x \]

   4. A2: \( A > 0 \)

      \[ \int\limits_{\left| x \right| > A}^{} \left| f(x) \right| \, \mathrm{d} x \]

      ki je majhen

      D: \[ \int\limits_{- \infty}^{\infty} \left| f(x) \right| \, \mathrm{d} x \le \infty \]

   5. A3: \( \epsilon > 0 \)

      \[ = \frac{1}{\sqrt{2 \pi}} \left[ \int\limits_{\left| x \right| \ge A}^{} \left| f(x) \right| \left| e^{- \mathrm{i} h x} - 1 \right| \, \mathrm{d} x + \int\limits_{- A}^A \left| f(x) \right| \left| e^{-\mathrm{i}hx} - 1 \right| \, \mathrm{d} x \right] < \epsilon \]

      Dokaz:

      1. \[ \int\limits_{\left| x \right| \ge A}^{} \left| f(x) \right| \left| e^{- \mathrm{i} h x} - 1 \right| \, \mathrm{d} x < \frac{\epsilon}{2} \]

         Dokaz:

         1. A4: za vsak \( \epsilon > 0 \) obstaja \( A > 0 \)

            \[ \int\limits_{\left| x \right| \ge A}^{} \left| f(x) \right| \, \mathrm{d} x \le \frac{\epsilon \sqrt{2\pi}}{4} \]

            D: \( f \in L^1 (\mathbb{R}) \)

         2. \[ \left| e^{- \mathrm{i} h x} - 1 \right| \le 2 \]

            D: \( \le \left| e^{- \mathrm{i} h x} \right| + \left| 1 \right| \le 2 \)

         3. QED
      2. \[ \int\limits_{- A}^A \left| f(x) \right| \left| e^{- \mathrm{i} h x} - 1 \right| \, \mathrm{d} x < \frac{\epsilon}{2} \]

         1. A5: \( \delta > 0 \)

            \[ \left| e^{- \mathrm{i} \delta x} - 1 \right| \le \frac{\epsilon \sqrt{2\pi}}{2 \left\lVert f \right\rVert_1} \]

            D: Funkcija \( e^{- \mathrm{i} x} \) je zvezna

         2. A6: \( \left| xh \right| < A \left| h \right| < \delta \)

            \[ \left| e^{- \mathrm{i} hx} - 1 \right| < \frac{\epsilon \sqrt{2 \pi}}{2 \left\lVert f \right\rVert_1} \]

            D: <4>2

         3. \[ \int\limits_{-A}^A \left| f(x) \right| \, \mathrm{d} x \le \left\lVert f \right\rVert_1 \]
         4. QED
      3. QED
   6. QED
2. \[ \left| \hat{f} (\xi) \right| = \frac{1}{\sqrt{2\pi}} \left\lVert f \right\rVert_1 \]

   1. \[ \le \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty }^{\infty} \left| f(x) \right| \left| e^{- \mathrm{i} \xi x} \right| \, \mathrm{d}x \]

      D: Definicija Fourierove transformacije

   2. \[ = \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^{\infty} \, \mathrm{d} x \]

   3. \[ = \frac{1}{\sqrt{2 \pi}} \left\lVert f \right\rVert _1 \]

      D: Definicija \( L^1 (\mathbb{R}) \) norme

   4. QED
3. \[ \hat{f'} (\xi) = \mathrm{i} \xi \hat{f}(\xi) \]

   1. \[ \hat{f '} (\xi) = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} f'(x) e^{- \mathrm{i} \xi x} \, \mathrm{d} x \]

      D: Definicija Fourierove transformacije

   2. \[ = \frac{1}{\sqrt{2 \pi}} \left( \left. e^{- \mathrm{i} \xi x} f(x) \right) \right|_{- \infty} ^{\infty} + \int\limits_{- \infty}^{\infty} f(x) \mathrm{i} \xi e^{- \mathrm{i} \xi x} \, \mathrm{d} x \]

      D: per partes, ter \( e^{- \mathrm{i} \xi x} f(x) \in L^1 (\mathbb{R}) \), kar pomeni, da je manjši od \( \infty \)

   3. \[ = \mathrm{i} \xi \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} f(x) e^{- \mathrm{i} \xi x} \, \mathrm{d} x \]

      D: <2>2

   4. QED

      D: \( = \mathrm{i} \xi \hat{f}(\xi) \)

1. A1: \( f, g \in \mathcal{C}_C (\mathbb{R}) \)

   \[ \widehat{f \ast g} (\xi) = \sqrt{2 \pi} \hat{f} (\xi) \hat{g}(\xi) \]

   1. \[ = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} \left( f \ast g \right) (x) e^{- \mathrm{i} \xi x} \, \mathrm{d} x \]

      D: Definicije Fourierove transformacije

   2. \[ = \frac{1}{\sqrt{2\pi}} \int\limits_{- \infty}^{\infty} \, \mathrm{d} x \int\limits_{-\infty}^{\infty} f(x - t) g(t) e^{- \mathrm{i} \xi x} \, \mathrm{d} t \]

      D: Definicija konvolucije

   3. \[ = \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^{ \infty} \, \mathrm{d} t \int\limits_{- \infty}^{\infty} f(x - t) e^{- \mathrm{i }\xi x} g(t) \, \mathrm{d}t \]

      D: Fubinijev izrek, ker je integrand \( \in \mathcal{C}_C (\mathbb{R}) \)

   4. A2: nova spremenljivka \( x - t = y \)

      \[ \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^{ \infty} \, \mathrm{d} t \int\limits_{- \infty}^{\infty} f(y) e^{- \mathrm{i}y \xi} g(t) e^{- \mathrm{i} \xi t} \, \mathrm{d} y \]

   5. \[ = \int\limits_{-\infty}^{ \infty} e^{- \mathrm{i} \xi t} g(t) \cdot \hat{f}(\xi) \, \mathrm{d} t \]

      D: Definicija Fourierove transformacije

   6. QED

      \[ = \sqrt{2 \pi} \hat{g}(\xi) \hat{f}(\xi) \]

2. A3: \( f, g \in L^1 (\mathbb{R}) \)

   \[ \widehat{f \ast g} (\xi) = \sqrt{2 \pi} \hat{f} (\xi) \hat{g}(\xi) \]

1. Obstaja zaporedje \( \left( f_n \right)_{n \in \mathbb{N}} \), da \( \lim_{n \to \infty}f_n = f \)

   D: A3

2. Obstaja zaporedje \( \left( g_n \right)_{n \in \mathbb{N}} \), da \( \lim_{n \to \infty}g_n = g \)

   D: A3

3. A4: \( \forall n \in \mathbb{N} \)

   \[ \widehat{f_n \ast g_n} (\xi) = \sqrt{ 2\pi} \hat{f}_n (\xi) \hat{g}_n(\xi) \]

   D: <2>1 in <2>2

4. QED

   D: \( n \to \infty \)

1. A1: \( f \in \mathcal{S}(\mathbb{R}) \)

   \[ \int\limits_{-\infty}^{\infty} \left| f(x) \right| \, \mathrm{d} x < \infty \]

   D: omejenost funkcije

2. Funkcija \( x \mapsto f(x) \cdot \left( 1 + x ^2 \right) \) je omejena

   D: A1

3. A3: \( \left| f(x) \left( 1 + x ^2 \right) \right| \le M \) za vsak \( x \in \mathbb{R} \)

   \[ \int\limits_{-\infty}^{\infty} \left| f(x) \right| \, \mathrm{d} x \le \int\limits_{-\infty}^{ \infty} \frac{M}{1 + x ^2} < \infty \]

   D: A3

4. QED

   D: V <1>3 smo dokazali, da je tudi poljubna funkcija iz Schwarzevega razreda omejena in iz razreda \( L^1 (\mathbb{R}) \).

1. \[ \hat{g}_0 (\xi) = g_0 (x) \]

   1. \[ \hat{g}_0 (\xi) = \frac{1}{\sqrt{2 \pi}} \int\limits_{-\infty}^{\infty} e^{- \frac{x ^2}{2}} e^{- \mathrm{i} \xi x} \, \mathrm{d} x \]

      D: Definicija Fourierove transformacije

   2. \[ = \frac{1}{\sqrt{2 \pi}} e^{- \frac{\xi ^2}{2}} \int\limits_{- \infty}^{\infty} e^{- \frac{1}{2} \left( x + \mathrm{i} \xi \right) ^2} \]

      D: Dopolnitev do popolnega kvadrata

   3. \[ e^{- \frac{\xi ^2}{2}} \frac{1}{\sqrt{2 \pi}} \lim_{A \to \infty} \int\limits_{- A}^A e^{- \frac{1}{2} \left( x + \mathrm{i} \xi \right)} \, \mathrm{d} x \]

   4. A1: Nova spremenljivka \( z = x + \mathrm{i} \xi \)

      \[ e^{- \frac{\xi ^2}{2}} \frac{1}{\sqrt{2 \pi}} \lim_{A \to \infty} \int\limits_{- A + \mathrm{i} \xi}^{A + \mathrm{i} \xi } e^{- \frac{1}{2} z ^2} \, \mathrm{d} x \]

      D: A1

   5. \[ \oint\limits_{\gamma}^{} e^{- \frac{z ^2}{2}} \, \mathrm{d} z= 0 \]

      D: Cauchyjev izrek za celo funkcijo brez singularnosti

   6. \[ = \int\limits_{- A}^A e^{- \frac{z ^2}{2}} \, \mathrm{d} z + \int\limits_{\gamma_1}^{} e^{- \frac{z ^2}{2}} \, \mathrm{dz} - \int\limits_{- A + \xi \mathrm{i}}^{A + \xi \mathrm{i}} e^{- \frac{z ^2}{2}} \, \mathrm{d} z + \int\limits_{\gamma_2}^{} e^{- \frac{ z ^2}{2}} \, \mathrm{d} z \]

      D: Zaključena zanka

   7. \[ \int\limits_{\gamma_1}^{} e^{- \frac{z ^2}{2}} \, \mathrm{d} z = \int\limits_{\gamma_2}^{} e^{- \frac{z ^2}{2}} \, \mathrm{d}z = 0 \]

   8. \[ \lim_{A \to \infty} \int\limits_{- A + \mathrm{i} \xi}^{A + \mathrm{i} \xi} e^{- \frac{ z ^2}{2}} \, \mathrm{d} z = \int\limits_{- \infty}^{ \infty} e^{- \frac{ z ^2}{2}} = \sqrt{2\pi} \]

      D: <2>6 ter \( \Gamma \) funkcija preko substitucije \( u = \frac{x ^2}{2} \)

   9. QED

      D: <2>4 ter <2>8

2. \[ \widehat{ g_{0[a]} } (\xi) = \frac{1}{a} \hat{g}_0 \left( \frac{\xi}{a} \right) \]

   1. \[ = \frac{1}{a} e^{- \left( \frac{\xi}{a} \right)^2 \frac{1}{2}} \]

      D: <1>1

   2. QED
3. QED

1. A1: \( f: \mathbb{R} \to \mathbb{C} \) zvezna

   A2: \( f \) omejena

   \[ f \ast g_{(\delta)} (x) \overset{\delta \to 0}{\longrightarrow} f \]

   oz.

   \[ \left| \left( f \ast g_{(\delta)} \right)(x) - f(x) \right| = 0 \]

   1. A3: \( x \in [a, b ] \)

      \[ = \left| \int\limits_{-\infty}^{\infty} f(x - t) g_{(\delta)} (t) \, \mathrm{d} t - \int\limits_{-\infty}^{\infty} f(x) g_{(\delta)}(t) \, \mathrm{d} t \right| \]

      D: Definicija konvolucije \( g_{(\delta)} \) ter to, da je \( \int\limits_{-\infty}^{\infty} f(x) g_{(\delta)} (t) \, \mathrm{d} t = 1 \cdot f(x) \)

   2. \[ = \left| \int\limits_{-\infty}^{\infty} \left[ f(x - t) - f(x) \right] g_{(\delta)} (t) \, \mathrm{d} t \right| \]

      D: <2>1

   3. \[ = \int\limits_{- \infty}^{\infty} \left| f(x - t) - f(x) \right| \frac{1}{\delta} \left| g \left( \frac{t}{\sigma} \right) \right| \, \mathrm{d} t \]

      D: <2>2

   4. \( M = \sup f(x) \)

      D: Po A2 je \( f \) omejena

   5. \[ = \int\limits_{\left| t \right|< \delta}^{} \left| f( x - t ) - f(x) \right| \frac{1}{\delta} \left| g \left( \frac{t}{\delta} \right) \right| \, \mathrm{d} t + \int\limits_{\left| t \right| \ge \delta}^{} \left| f(x - t) - f(x) \right| \frac{1}{\delta} \left| g \left( \frac{t}{\delta} \right) \right| \, \mathrm{d} t = 0 \]
      1. \[ \int\limits_{\left| t \right| < \delta}^{} \left| f(x - t) - f(x) \right| \frac{1}{\delta} \left| g \left( \frac{t}{\delta} \right) \right| \, \mathrm{d} t = 0 \]

         1. A4: \( \left| t \right| < \delta \)

            A5: \( \forall \epsilon > 0 \)

            \[ \left| f(x - t) - f(t) \right| < \frac{\epsilon}{2 \left\lVert g_{(\delta)} \right\rVert_1} \]

            D: A1, funkcija \( f \) je (enakomerno) zvezna

         2. \[ \le \frac{\epsilon}{2 \left\lVert g \right\rVert_1} \int\limits_{\left| t \right| < \delta}^{} \frac{1}{\delta} \left| g \left( \frac{t}{\delta} \right) \right| \, \mathrm{d} t \]
         3. \[ \int\limits_{\left| t \right|< \delta}^{} \frac{1}{\delta} \left| g \left( \frac{t}{\delta} \right) \right| \, \mathrm{d} t \le \left\lVert g \right\rVert_1 \]
         4. \[ = \frac{\epsilon}{2} \]
         5. QED
      2. \[ \int\limits_{\left| t \right| \ge \delta}^{} \left| f(x - t) - f(x) \right| \frac{1}{\delta} \left| g \left( \frac{t}{\delta} \right) \right| \, \mathrm{d} t = 0 \]

         1. \[ \left| f(x - t) - f(x) \right| \le \left| f(x - t) \right| + \left| f(x) \right| \le 2 M \]

            D: <2>4

         2. A6: \( \delta \to 0 \)

            \[ \int\limits_{\left| t \right| \ge \delta}^{} \frac{1}{\delta} \left| g \left( \frac{t}{\delta} \right) \right| \, \mathrm{d} t \le \frac{\epsilon}{4 M} \]

            D: \( g \in L^1 (\mathbb{R}) \)

         3. QED
      3. QED
   6. QED

1. A1: \( g_{(\delta)} (x) = \frac{1}{\delta} \frac{1}{\sqrt{2\pi}} e ^{- \frac{x ^2}{\delta ^2 2}} \)

   A2: \( g_{[\delta]} (x) = \frac{1}{\sqrt{2\pi}} e^{- \frac{\delta ^2 x ^2}{2}} \)

   \[ \int\limits_{-\infty}^{\infty} \hat{f}(\xi) e^{- \xi x} g_{[\delta]} (\xi) \, \mathrm{d} \xi = \frac{1}{\sqrt{2\pi}} \int\limits_{- \infty}^{\infty} \, \mathrm{d} \xi \int\limits_{-\infty}^{\infty} f (t) e^{- \mathrm{i} \xi t} \, \mathrm{d} t \, e^{\mathrm{i} \xi x} g_{[\delta]} (\xi) \]

   D: ko \( \delta \to 0 \) je \( g_{[\delta]} (\xi) = 1 \)

2. \[ =\frac{1}{\sqrt{2 \pi}} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} f(t) e^{- \mathrm{i} \xi ( t - x )} g_{[\delta]} (\xi) \, \mathrm{d} \xi \, \mathrm{d} t \]

   D: Integrand je absolutno integrabilen in lahko uporabimo Fubinijev izrek.

3. \[ = \int\limits_{-\infty}^{\infty} f(t) \widehat{g}_{[\delta]} (t - x) \, \mathrm{d} t \]

   D: Definicija Fourierove transformacije

4. \[ = \int\limits_{-\infty}^{\infty} g_{(\delta)} (t - x) f(t) \, \mathrm{d} t \]

   D: velja \( \widehat{g_{[\delta]}} (\xi) = g_{(\delta)} (x) \)

5. \[ \int\limits_{-\infty}^{\infty} g_{(\delta)}(t - x) f(t) \, \mathrm{d} t \]

   D: Funkcija \( g \) je soda.

6. \[ = g_{(\delta)} \ast f (x) = f \ast g_{(\delta)} \]

   D: Definicija in lastnost konvolucije

7. QED

   D: \( \delta \to 0 \), trditev 3.11 lastnost a

1. \[ \left\langle f, g \right\rangle = \int\limits_{-\infty}^{\infty} f(x) \overline{g(x)} \, \mathrm{d} x \]

   D: P2

2. \[ \int\limits_{-\infty}^{\infty} \overline{g(x)} \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} \hat{f}(\xi) e^{\mathrm{i} \xi x} \, \mathrm{d} \xi \, \mathrm{d} x \]

   D: Definicija inverza Fourierove transformacije

3. \[ = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \hat{f}(\xi) \overline{g(x)} e^{\mathrm{i} x \xi} \, \mathrm{d} x \, \mathrm{d} \xi \]

   D: Fubinijev izrek

4. \[ = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} \hat{f}(\xi) \int\limits_{-\infty}^{\infty} \overline{g(x) e^{-\mathrm{i} \xi x}} \, \mathrm{d} x \, \mathrm{d} \xi \]

   D: Negacije kompleksne vrednosti

5. \[ = \int\limits_{-\infty}^{\infty} \hat{f}(\xi) \overline{g(\xi)} \, \mathrm{d} \xi = \left\langle \hat{f}, \hat{g} \right\rangle \]
6. QED

1. A1: \( \tilde{F}: L^2 (\mathbb{R}) \to L ^2 (\mathbb{R}) \)

   \( \tilde{F} \) je razširitev oz.

   \[ \tilde{F} (f) = \hat{f}(\xi) \]

   in razširimo Fourierovo transformacijo iz \( \mathcal{S}(\mathbb{R}) \) na \( L ^2 (\mathbb{R}) \)

   1. A2: \( f \in L ^2 (\mathbb{R}) \)

      A3: \( f= \lim_{n \to \infty} f_n, \ f_n \in \mathcal{S}(\mathbb{R}) \)

      Zaporedje \( \hat{f}_n \) je konvergentno.

      Dokazujemo, da je Cauchyjevo zaporedje konvergentno, saj je \( L ^2 (\mathbb{R}) \) poln prostor.

      1. \[ \left\lVert \hat{f}_n - \hat{f}_m \right\rVert = \left\langle \hat{f}_n - \hat{f}_m, \hat{f}_n - \hat{f}_m \right\rangle \]

         D: Definicija skalarnega produkta

      2. \[ = \left\langle \widehat{f_n - f_m}, \widehat{f_n - f_m} \right\rangle \]

         D: Lastnost Fourierove transformacije

      3. \[ = \left\langle f_n - f_m, f_n - f_m \right\rangle \]

         D: Trditev 3.18

      4. \[ = \left\lVert f_n - f_m \right\rVert _2 \overset{m, n \to \infty}{\longrightarrow} 0 \]

         D: zaporedje \( \left( f_n \right)_{n \in \mathbb{N}} \) je konvergentno v \( L ^2 (\mathbb{R}) \)

      5. QED

   2. Definicija \( \hat{f} \) je dobra

      oziroma

      A4: \( \hat{f} = \lim_{n \to \infty}f_n = f_0 \)

      A5: \( f = \lim_{n \to \infty} g_n \)

      \( \hat{g}_n \) tudi konvergira k \( f_0 \)

      1. \[ \left\lVert \hat{f}_n - \hat{g}_n \right\rVert \le \left\lVert \hat{f}_n - f_0 \right\rVert + \left\lVert f_0 - \hat{g}_n \right\rVert \to 0 \]

         D: Trikotniška neenakost in posamezen člen konvergira k \( 0 \)

      2. \[ \left\lVert f_n - g_n \right\rVert \le \left\lVert f_n - f \right\rVert + \left\lVert f - g_n \right\rVert \to 0 \]

         D: Trikotniška neenakost in posamezen člen konvergira k \( 0 \)

      3. QED

         D: \[ \left\lVert \hat{f}_n - \hat{g}_n \right\rVert = \left\lVert f_n - g_n \right\rVert \to 0 \]

   3. QED
2. \( \tilde{F} \) je unitarna
   1. \( \tilde{F} \) je izometrična

      1. A6: \( f \in L ^2 (\mathbb{R}) \)

         A7: \( g \in L ^2 (\mathbb{R}) \)

         \[ \left\langle \hat{f}, \hat{g} \right\rangle = \left\langle \lim_{n \to \infty} \hat{f}_n, \lim_{n \to \infty} \hat{g}_n \right\rangle \]

         D: Cauchyjevo zaporedje je konvergentno

      2. \[ = \lim_{n \to \infty} \left\langle \hat{f}_{n}, \hat{g}_n \right\rangle \]

      3. \[ = \lim_{n \to \infty} \left\langle f_n, g_n \right\rangle \]

         D: Trditev 3.18

      4. \[ = \left\langle \lim_{n \to \infty} f_n, \lim_{n \to \infty} g_n \right\rangle \]

      5. \[ = \left\langle f, g \right\rangle \]

         Po A6 in A7 sta Cauchyjevo konvergentni na \( L ^2 (\mathbb{R}) \)

      6. QED
   2. \( \tilde{F} \) je surjektivna
      1. Case: \( F^4 \) je surjektivna v \( \mathcal{S}(\mathbb{R}) \)
      2. \[ \lim_{n \to \infty} F (f_n) = \mathrm{id} \]
      3. QED
   3. QED

1. A1: \( f: \chi_{[a, b]} \)

   \[ \hat{f}(\xi) = \frac{e^{- \mathrm{i} b \xi} - e^{-\mathrm{i} a \xi}}{- \sqrt{2\pi} \mathrm{i}\xi} \]

   D: Definicija konvolucije

2. \[ \left| \hat{f}(\xi) \right| \le \frac{2}{\sqrt{2\pi} \left| \xi \right|} \]

   D: \( \left| e^{- \mathrm{i} \xi a} \right| = 1 \)

3. A2: \( f \in \mathbb{C}_C (\mathbb{R}) \)

   A3: aproksimiramo jo z \( g = \sum\limits_{}^{} c_i \chi_{[a_i, b_i]} \)

   \[ \hat{g}(\xi) \to 0, \, \xi \to \infty \]

   D: <1>2

1. A1:

   \[ J_{-m} (z) = \sum\limits_{n = 0}^{\infty} \frac{(-1)^n}{2^{-m + 2n} \, n! \, \Gamma (-m + n + 1)} z^{2n - m} \]

   A2:

   \[ J_m (z) = \sum\limits_{n = 0}^{\infty} \frac{(- 1)^n}{2^{m + 2n} \, n! \, \Gamma(m + n + 1)} z^{2n + m} \]

   Za \( n \le m- 1 \) je \( \Gamma (-m + n + 1) = \infty \).

   D: Definicija \( \Gamma \) funkcije v kompleksnem prostoru.

2. \[ \sum\limits_{n = 0}^{\infty} \frac{(-1)^n}{2^{-m 2n} \, n! \, \Gamma (-m + n + 1)} z^{2n - m} = \sum\limits_{n = m}^{\infty} \frac{(-1)^n}{2^{-m + 2n} \, n! \, \Gamma(-m + n + 1)} z^{2n - m} \]

   D: <1>1

3. A3: preindeksacija \( k = n - m \)

   \[ = \sum\limits_{k = 0}^{\infty} \frac{(- 1)^{m + k}}{2^{m + 2k} \, (m + k)! \, \Gamma( k + 1 )} z^{m + 2k} \]

   D: A3 ter <1>2

4. \[ = (-1)^m \sum\limits_{k = 0}^{\infty} \frac{(-1)^k}{2^{m + 2k} \, (m + k)! \, k! }z^{m + 2k} \]

   D: <1>3

5. \[ (-1) ^m \sum\limits_{n = 0}^{\infty} \frac{(-1)^n}{2^{m + 2n} \, n! \, \Gamma(m + n + 1)} z^{m + 2n} = J_m (-1)^m \] D: \( n = k \) ter \( (m + n)! = \Gamma (m + n + 1) \)

1. Vrsto

   \[ z^{\nu} J_{\nu} (z) = \sum\limits_{n = 0}^{\infty} \frac{(-1)^n}{2^{\nu + 2 n} \, n! \, \Gamma(\nu + n +1)} z^{2n + 2\nu} \]

   odvajamo po \( z \).

2. Analogen postopek
3. \[ J_{\nu} '(z) + \frac{\nu}{z} J_{\nu}(z) = J_{\nu - 1} (z) \]

   1. \[ \frac{\mathrm{d} }{\mathrm{d} z} \left[ z^{\nu} J_{\nu} (z) \right] = z^{\nu} J_{\nu - 1} (z) \]

      D: Točka \( a \)

   2. \[ z^{\nu} J_{\nu - 1} (z) = \nu z^{\nu - 1} J_{\nu} (z) + z^{\nu} J_{\nu} ' (z) \]

      D: Odvod produkta

   3. QED

      D: delimo z \( z^{\nu} \)

4. Analogno kot <1>3 samo za 3. točko
5. odštejemo 3 in 4
6. seštejemo 3 in 4

Skica

1. \[ e^{\frac{z}{2} \left( t - t^{-1} \right)} = e^{\frac{z}{2} t} \cdot e^{\frac{z}{2} t^{-1}} \]

   D: operacije z eksponenti

2. \[ = \sum\limits_{k = 0}^{\infty} \frac{1}{k!} \left( \frac{z}{2} t \right) ^k \cdot \sum\limits_{n = 0}^{\infty} \frac{1}{n!} (- 1)^n \left( \frac{z}{2} t^{-1} \right)^n \]

   D: Razvoj v Taylorjevo vrsto

3. \[ = \sum\limits_{m- \infty}^{\infty} J_m t^m \]

   1. A1: preindeksacija \( m = k - n \)

      \[ \sum\limits_{k = 0}^{\infty} \frac{1}{k!} \left( \frac{z}{2} \right)^k \cdot \sum\limits_{n = 0}^{\infty} \frac{1}{n!} (-1)^n \left( \frac{z}{2} \right)^n = \sum\limits_{k = 0}^{\infty} \frac{(-1)^n}{2^{m + n} \cdot 2^n \, n! \, (m + n)!} z^{m + n} \cdot z^n \]

   2. \[ = J_m(z) \]

      D: Definicija Besselove funkcije 1. reda

4. QED

1. A1: \( w = -z \)

   A2: \( m = 0 \)

   \[ J_0 (0) = 1 \]

2. \[ 1 = J_0 ^2 (z) + 2 \sum\limits_{k = 1}^{\infty} (-1)^k J_k (z) J_k (-z) \]

   D: Trditev 4.10

3. \[ 1 = J_0 ^2 (z) = 2 \sum\limits_{k = 1}^{\infty} J_k ^2(z) \] D: Velja \( J_k (-z) = (-1)^k J_k (z) \)
4. QED

1. \[ e^{\frac{z}{2} \left( t - t^{-1} \right)} = \sum\limits_{m = - \infty}^{\infty} J_m (z) t^m \]

2. A1: \( z = x \in \mathbb{R} \)

   A2: \( t = e^{\mathrm{i} \phi} \)

   \[ e^{\frac{x}{2} \left( e^{\mathrm{i} \phi} - e^{-\mathrm{i}\phi} \right)} = \sum\limits_{k = -\infty}^{\infty} J_k (x) e^{\mathrm{i} k \phi} \]

   D: A1 in A2 ter <1>1

3. \[ e^{\mathrm{i} x \sin \phi} = \sum\limits_{k = -\infty}^{\infty} J_k (x) e^{\mathrm{i}k \phi} \]

   D: Definicija sinusa v kompleksnem prostoru

4. \[ \cos (x \sin \phi) + \mathrm{i} \sin \left( x\sin \phi \right) = \sum\limits_{k = -\infty}^{\infty} J_k (x) \cos (k\phi) + \mathrm{i} \sum\limits_{k = -\infty}^{\infty} J_k (x) \sin (k \phi) \]

   D: Definicija Eulerjevega števila

5. Realni del je

   \[ \cos \left( x \sin \phi \right) = \sum\limits_{k = -\infty}^{\infty} J_k (x) \cos \left( k\phi \right) \]

   D: <1>4

6. Imaginarni del

   \[ \sin \left( x \sin \phi \right) = \sum\limits_{k = -\infty}^{\infty} J_k (x) \sin \left( k\phi \right) \]

7. \[ \cos \left( m \phi - x \sin \phi \right) = \cos \left( x \sin \phi \right) \cos \left( m \phi \right) + \sin \left( x \sin \phi \right) \sin \left( m \phi \right) \]

   D: Kosinus vsote dveh kotov

8. \[ \cos \left( m \phi - x \sin \phi \right) = \sum\limits_{k = -\infty}^{\infty} \left[ J_k (x) \cos \left( k\phi \right) \cos \left( m \phi \right) + J_k (x) \sin \left( k \phi \right) \sin \left( m \phi \right) \right] \]

   D: <1>5-7

9. \[ = \sum\limits_{k = -\infty}^{\infty} J_k (x) \cos \left[ \left( k - m \right)\phi \right] \]

   D: <1>8 in vsota kotov v kosinusu

10. \[ \int\limits_0^{\pi} \cos \left( m \phi - x \sin \phi \right) \, \mathrm{d} \phi = J_m (x) \pi \]

    D: <1>9, integracija ter

    \[ \int\limits_0^{\pi} \cos \left( \left[ k - m \right]\phi \right) \, \mathrm{d} \phi = 0, \ k \ne m \]

11. QED

1. A1: \( \nu \ge 0 \)

   A2: \( J_{\pm \nu} (z) \) rešita Besselovo DE

   \[ z ^2 \left( \frac{\partial y}{\partial \nu} \right)'' + z \left( \frac{\partial y}{\partial \nu} \right) ' + \left( z ^2 - \nu ^2 \right) \frac{\partial y}{\partial \nu} = 2 \nu y \]

   D: odvod po \( \nu \)

2. \[ z ^2 \left( \frac{\partial J_{\nu}}{\partial \nu} \right)'' + z \left( \frac{\partial J_{\nu}}{\partial \nu} \right)' + \left( z ^2 - \nu ^2 \right) \frac{\partial J_{\nu}}{\partial \nu} = 2 \nu J_{\nu} \]

   D: A2

3. \[ z ^2 \left( \frac{\partial J_{-\nu}}{\partial \nu} \right)'' + z \left( \frac{\partial J_{-\nu}}{\partial \nu} \right)' + \left( z ^2 - \nu ^2 \right) \frac{\partial J_{-\nu}}{\partial \nu} = 2 \nu J_{-\nu} \]

   D: A2

4. \[ z ^2 \left( \frac{\partial J_{\nu}}{\partial \nu} - \left( -1 \right)^m \frac{\partial J_{-\nu}}{\partial \nu} \right)'' + z \left( \frac{\partial J_{\nu}}{\partial \nu} - \left( -1 \right)^m \frac{\partial J_{-\nu}}{\partial \nu} \right) ' + \left( z ^2 - \nu ^2 \right) \left( \frac{\partial J_{\nu}}{\partial \nu} - \left( -1 \right) ^m \frac{\partial J_{-\nu}}{\partial \nu} \right) &= 2\nu \left( J_{\nu} - \left( -1 \right) ^m J_{-\nu} \right) \]

   D: <1>2 - <1>3

5. \[ z ^2 Y_m '' + z Y_m ' + \left( z ^2 - m ^2 \right) Y_m = \frac{2m}{\pi} \left( J_m - (-1)^m J_m \right) = 0 \]

   1. A1: \( \nu = m \)

      \[ Y_m (z) = \frac{1}{\pi} \lim_{\nu \to m} \left( \frac{\partial J_{\nu}}{\partial \nu} (z) - (-1)^m \frac{\partial J_{-\nu}}{\partial \nu}(z) \right) \]

   2. QED

      D: Formulo za \( Y_{\nu} (z) \) odvajamo po L’Hopitalovem pravilu

6. QED

1. \[ P_n (z) = \frac{1}{2^n} \sum\limits_{k = 0}^{\lfloor \frac{n}{2} \rfloor} (- 1)^k \frac{(2n - 2k!)}{2^n \, k! \, (n - k)! \, (n - 2k)!} z^{n - 2k} \]

   D: Definicija Legendrovih polinomov

2. \[ = \frac{1}{2^n} \sum\limits_{k = 0}^{\lfloor \frac{n}{2} \rfloor} (- 1)^k \frac{1}{k! \, (n - k)!} \frac{\mathrm{d} ^n}{\mathrm{d} z^n} (z^{2n - 2k}) \]

   Dokaz

   1. \[ \frac{\mathrm{d} ^n}{\mathrm{d} z^n} (z^{2n - 2k}) = (2n - 2k) (2n - 2k - 1) \ldots (2n - 2k - (n - 1)) z^{n - 2k} \]

      D: \( n \) odvajanj funkcije \( z^{2n - 2k} \)

   2. \[ = (2n - 2k) (2n - 2k - 1) \ldots (n - 2k + 1) z^{n - 2k} \]

      D: <2>1

   3. \[ \frac{(2n - 2k)!}{(n - 2k)!} z^{n - 2k} \]
   4. QED

3. \[ = \frac{1}{2^n n!} \sum\limits_{k = 0}^{\lfloor \frac{n}{2} \rfloor} (-1)^k \binom{n}{k} \frac{\mathrm{d} ^n}{\mathrm{d} z^n} \left( z^2 \right) ^{n - k} \]

   D: Definicija binomskega koeficienta ter množenje z \( \frac{n!}{n!} \)

4. \[ = \frac{1}{2^n n!} \frac{\mathrm{d} ^n}{\mathrm{d} z ^n} \sum\limits_{k = 0}^{n} \binom{n}{k} (-1)^k \left( z ^2 \right)^{n - k} \]

   D: za preskok iz \( \lfloor n \rfloor \) na \( n \)?

5. \[ = \frac{1}{2^n n!} \frac{\mathrm{d} ^n}{\mathrm{d} z ^n} \left[ \left( z ^2 - 1 \right)^n \right] \]

   D: \( (a + b)^n = \sum\limits_{k = 0}^n \binom{n}{k} a^k b^{n - k} \)

6. QED

Ideja dokaza:

1. \[ \left( 1 - 2zt + t ^2 \right)^{- \frac{1}{2}} = \sum\limits_{m = 0}^{\infty} (-1)^m \binom{-\frac{1}{2}}{m} \left( 2zt - t ^2 \right)^m \]

   D: Posplošena binomska vrsta za \( \left| t \right| < 1 \)

   \[ \left( 1 + t \right)^{\alpha} = \sum\limits_{n = 0}^{\infty} \binom{\alpha}{n} t^n \]

   kjer je

   \[ \binom{\alpha}{n} = \frac{\alpha (\alpha - 1) \ldots (\alpha - n + 1)}{n!}, \ \alpha \in \mathbb{R}, \ n \in \mathbb{N} \]

   in \( \left| 2zt - t ^2\right| < 1\)

2. \[ \binom{- \frac{1}{2}}{m} = \frac{(-1)^m (2m)!}{2^{2m} \left( m! \right)^2} \]

   D:

   1. \[ \binom{- \frac{1}{2}}{m} = \frac{\left( - \frac{1}{2} \right)\left( - \frac{1}{2} - 1 \right) \ldots \left( - \frac{1}{2} - m + 1 \right)}{m!} \]

      D: Definicija posplošenega binomskega simbola

   2. \[ = \frac{\left( -1 \right)^m \cdot 1 \cdot 3 \cdot \ldots \cdot \left( 2m - 1 \right)}{2^m m!} \]

      D: <2>1

   3. \[ = \frac{(-1)^m (2m - 1)!! }{2^m m!} \]

      D:<2>2

   4. \[ = \frac{(-1)^m}{2^m m!} \frac{(2m)!}{\left( m! \right)^2 2^m} \]

      D: \[ (2m - 1)!! = \frac{(2m)!}{(2m)!!} = \frac{(2m)!}{2^m m!} \]

      Eliminirati je potrebno vse sode faktorje, ki se pojavijo v \( (2m)! \)

Nedokončano.

1. \[ \frac{1}{\sqrt{1 - 2zt + t ^2}} = \sum\limits_{n = 0}^{\infty} P_n (z) t^n \]

   D: Rodovna funkcija

2. \[ \frac{z - t}{\left( 1- 2zt + t ^2 \right)^{\frac{3}{2}}} = \sum\limits_{n = 1}^{\infty} n P_n (z) t^{n - 1} \]

   D: odvod po \( t \)

3. \[ \frac{z - t}{\sqrt{1 - 2zt + t ^2}} = \sum\limits_{n = 0}^{\infty} P_n (z) t^n (z - t) \]

   D: <1>1 pomnožena z \( z -t \)

4. \[ \frac{z - t}{\sqrt{1 - 2zt + t ^2}} = \sum\limits_{n = 0}^{\infty} n P_n (t) t^{n - 1} (1 -2tz + z ^2) \]

   D: <1>2 pomnožena z \( (1 - 2zt + t ^2) \)

5. \[ zP_n (z) - P_{n - 1} (z) = (n + 1) P_{n + 1} (z) -2nz P_n (t) = (n - 1) P_{n - 1} (z) \]

   D: Koeficienti pri \( t^n \)

6. QED

Dokazujemo

\[ \rho = \left. \left[\bar{v} u' - \bar{v} ' u \right]\right|_a ^b \]

1. A1: \( \alpha_1 \ne 0 \)

   A2: \( \beta_2 \ne 0 \)

   \begin{align*} \alpha_1 u (a) + \alpha_2 u'(a) &= 0\\ \beta_1 u(b) + \beta_2 u'(b) &= 0 \end{align*}

   D: definicija ločenih robnih pogojev za \( u \)

2. \begin{align*} \alpha_1 v (a) + \alpha_2 v'(a) &= 0\\ \beta_1 v(b) + \beta_2 v'(b) &= 0 \end{align*}

   D: Definicija ločenih robnih pogojev za \( v \)

3. \[ \rho = \bar{v}(b) u'(b) - \bar{v}'(b) u(b) - \bar{v}(a) u'(a) + \bar{v}' (a) u(a) = 0 \]

   D: Greenova identiteta upoštevajoč meje

4. \[ \rho = 0 \]

   Dokaz:

   1. \[ u(a) = - \frac{\alpha_2}{\alpha_1} u'(a) \]

      D: <1>1

   2. \[ u'(b) = - \frac{\beta_1}{\beta_2} u(b) \]

      D: <1>1

   3. \[ \bar{v}(a) = -\frac{\alpha_1}{\alpha_2} \bar{v}'(a) \]

      D: <1>2

   4. \[ \bar{v}'(b) = - \frac{\beta_1}{\beta_2} \bar{v}(b) \]

      D: <1>2

   5. QED
5. QED

1. A1: \( L \) je formalno sebi-adjungiran, torej velja

   \[ \left\langle Lu, v \right\rangle = \left\langle u, L v \right\rangle \]

   A2: \( u \) lastna funkcija uteženega lastnega problema

   A3: \( \lambda \) pripadajoča lastna vrednost \( u \)

   Lastne vrednosti so realne.

   1. \[ \left\langle Lu, u \right\rangle = - \lambda \left\lVert u \right\rVert_{\omega} ^2 \]

      1. \[ Lu = - \lambda \omega u \]

         D: P5

      2. \[ \left\langle Lu, u \right\rangle = - \lambda \left\langle \omega u, u \right\rangle \]

         D: skalarni produkt z \( u \)

      3. \[ = -\lambda \int\limits_a^b \omega u \bar{u} \, \mathrm{d} x \]

         D: definicija skalarnega produkta z utežjo

      4. \[ = - \lambda \left\langle u, u \right\rangle_{\omega} = - \lambda \left\lVert u \right\rVert_{\omega} ^ \]

      5. QED

         D: Definicija norme

   2. \[ \left\langle u, Lu \right\rangle = - \bar{\lambda} \left\lVert u \right\rVert_{\omega} ^2 \]

      1. \[ \left\langle u, Lu \right\rangle = \left\langle u, - \lambda \omega u \right\rangle \]

         D: P5 in skalarni produkt z leve

      2. \[ = - \bar{\lambda} \left\langle u, \omega u \right\rangle \]

         D: lastnost skalarnega produkta \( \left\langle u, \lambda v \right\rangle = \bar{\lambda} \overline{\left\langle v, u \right\rangle} \)

      3. \[ = - \bar{\lambda} \int\limits_a^b u \overline{\omega u} \, \mathrm{d} x \]

         D: Definicija skalarnega produkta z utežjo

      4. \[ = - \bar{\lambda} \int\limits_a^b u \omega \bar{u} \, \mathrm{d} x \]

         D: pozitivnost funkcije??

      5. \[ = - \bar{\lambda} \left\lVert u \right\rVert_{\omega} ^2 \]

      6. QED

         D: Definicija norme

   3. \[ \lambda = \bar{\lambda} \]

      D: <2>1 in <2>2 za \( u \ne 0 \), kar povzroči \( \left\lVert u \right\rVert _{\omega} \ne 0 \)

   4. QED

      D: \( \lambda \in \mathbb{R} \)

2. A4: \( \eta \) različna lastna vrednost od \( \lambda \)

   A5: \( \eta \) pripada različni od \( u \) lastni funkciji \( v \)

   \[ \left\langle u, v \right\rangle_{\omega} = 0 \]

   1. \[ Lu = - \lambda \omega u \]

      D: P5 za \( u \)

   2. \[ Lu = - \eta \omega u \]

      D: P5 za \( v \)

   3. \[ \left\langle Lu, v \right\rangle = - \lambda \left\langle u, v \right\rangle_{\omega} \]

      Dokaz:

      \[ \left\langle Lu, v \right\rangle = - \lambda \left\langle \omega u, v \right\rangle = - \lambda \left\langle u, v \right\rangle_{\omega} \]

      ter definicija skalarnega produkta z utežjo.

   4. \[ \left\langle u, Lv \right\rangle = - \eta \left\langle u, v \right\rangle_{\omega} \]

      D: <1>1 ter <2>3

   5. \[ \left\langle L u, v \right\rangle = \left\langle u, L v \right\rangle \]

      D: A1

   6. QED

      D:

      \[ \left( \lambda - \eta \right) \left\langle u, v \right\rangle_{\omega} = 0, \ \lambda - \eta \ne 0 \]

3. A6: \( u \) je lastna funkcija za \( \lambda \)

   A7: \( v \) je lastna funkcija za \( \lambda \)

   \( u \) in \( v \) linearno odvisni.

   1. \( u \)

      \begin{align*} \alpha_1 u(a) + \alpha_2 u'(a) &= \begin{bmatrix} \alpha_1 \\ \alpha_2 \end{bmatrix} \begin{bmatrix} u(a) \\ u'(a) \end{bmatrix} = 0 \\ \beta_1 u(b) + \beta_2 u'(b) &= \begin{bmatrix} \beta_1 \\ \beta_2 \end{bmatrix} \begin{bmatrix} u(b) \\ u'(b) \end{bmatrix} = 0 \end{align*}

      D: Ločena robna pogoja za \( u \)

   2. \( v \)

      \begin{align*} \alpha_1 v(a) + \alpha_2 v'(a) &= \begin{bmatrix} \alpha_1 \\ \alpha_2 \end{bmatrix} \begin{bmatrix} v(a) \\ v'(a) \end{bmatrix} = 0 \\ \beta_1 v(b) + \beta_2 v'(b) &= \begin{bmatrix} \beta_1 \\ \beta_2 \end{bmatrix} \begin{bmatrix} v(b) \\ v'(b) \end{bmatrix} = 0 \end{align*}

   3. A8: \( \begin{bmatrix} \alpha_1 \\ \alpha_2 \end{bmatrix} \) je neničeln

      Vektor \( \begin{bmatrix} \alpha_1 \\ \alpha_2 \end{bmatrix} \) je pravokoten na \( \begin{bmatrix} u(a) \\ u'(a) \end{bmatrix} \) ter \( \begin{bmatrix} v(a) \\ v'(a) \end{bmatrix} \)

      D: <2>1 in <2>2

   4. A9: \( C_1, C_2 \in \mathbb{R} \), ki nista hkrati \( 0 \)

      Linearna odvisnost

      \[ C_1 \begin{bmatrix} u(a) \\ u'(a) \end{bmatrix} + C_2 \begin{bmatrix} v(a) \\ v'(a) \end{bmatrix} = 0 \]

      oziroma

      \[ y = C_1 u(a) + C_2 v(a) \]

      D: Po <2>3 sta oba vektorja pravokotna na \( \begin{bmatrix} \alpha_1 \\ \alpha_2 \end{bmatrix} \), kar je v \( \mathbb{R} ^2 \) mogoče samo, če sta linearno odvisna v točki \( a \) (in \( b \)).

   5. Imamo Cauchyjev problem za \( y \)

      D: Rešujemo problem 2. reda

      \[ Ly = - \lambda \omega y \]

      z začetnima pogojema

      \begin{align*} y(a) &= C_1 u(a) + C_2 v(a) = 0 \\ y'(a) &= 0 \end{align*}

   6. \( y = 0 \) je unikatna rešitev.

      D: Uganili rešitev ter eksistenčni izrek za diferencialno enačbo 2. reda, ki nam pove, da je to unikatna rešitev.

   7. QED

      \[ C_1 u = - C_2 v \]

      Funkciji \( u \) in \( v \) sta linearno odvisni na celotne intervalu.

4. Izpustimo.