Benutzer:Carlos-X/Dump

$f$ $U$ $U$ $U$ $f\colon U\to \mathbb {C}$ $\mathbb {C}$ $z\mapsto f(z)$ $w$ $U$ $z\mapsto f(w)+f'(w)(z-w)$ $f'(w)$ $f$ $w$ $x^{2}+1=0$ $i$ $i^{2}=-1$ $z$ $z=x+iy$ $x$ $y$ $x$ $y$ $x+iy$ $(x,y)$ $x$ $y$ $z$ $i^{2}=-1$ $(-3+2i)+(4+10i)=1+12i$ $(a+bi)(c+di)=(ac-bd)+(ad+bc)i.$ $-bd$ $bidi$ ${\frac {1+i}{1-2i}}={\frac {(1+i)(1+2i)}{(1-2i)(1+2i)}}={\frac {-1+3i}{5}}=-{\frac {1}{5}}+{\frac {3}{5}}i.$ $\mathbb {C}$ $\mathbb {R}$ $M$ $N$ $f\colon M\rightarrow N$ $M$ $N$ $f(z)=z^{2}$ $x\mapsto x^{2}$ $(x,f(x))$ $x$ $f$ $x$ $f(x)$ $(x,f(x))$ $(x,y)$ $z\mapsto z$ $x_{0}$ $x_{0}$ $f\colon \mathbb {R} \to \mathbb {R}$ $a$ $a$ $x\mapsto m(x-a)+c$ $h$ $f(a+h)\approx mh+c$ $h=0$ $c=f(a)$ $f$ $a$ $\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}=f'(a)$ $f$ $a$ $\lim _{h\to 0}{\frac {f(c+h)-f(c)}{h}}=:f'(c)\,\,\,$ $h$ $h>0$ $c$ $c$ $f(c+h)\approx f'(c)h+f(c),$ $h$ $f$ $\textstyle z\mapsto {\frac {1}{3}}(2z+1)^{2}$ $z+2$ $z^{3}-4z^{2}+3z-1$ $a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots +a_{0}.$ $z^{n}$ $\pi$ $f(z):=3+z+4z^{2}+z^{3}+5z^{4}+9z^{5}+2z^{6}+6z^{7}+\cdots ,$ $f\left({\frac {1}{10}}\right)=3+{\frac {1}{10}}+{\frac {4}{100}}+{\frac {1}{1000}}+{\frac {5}{10000}}+\cdots =3{,}1415926\dotsc =\pi .$ $f(z)$ $|z|$ $z$ $\mathbb {C}$ $f(z)={\tfrac {1}{1-z}}$ $z=1$ $z$ $1+z+z^{2}+z^{3}+z^{4}+z^{5}+\cdots ={\frac {1}{1-z}}.$ $2\pi$ $z\mapsto \sin(z)$ $z$ $\sin(z)\approx z-{\frac {z^{3}}{6}}.$ $\textstyle z\mapsto z-{\frac {z^{3}}{6}}$ $\sin(0)=0$ $\sin(0{,}2)=0{,}1986693308\dots$ $\textstyle 0{,}2-{\frac {0{,}2^{3}}{6}}=0{,}1986666\dots$ $\sin(z)=z-{\frac {z^{3}}{6}}+{\frac {z^{5}}{120}}-{\frac {z^{7}}{5040}}+{\frac {z^{9}}{362880}}-{\frac {z^{11}}{39916800}}+{\frac {z^{13}}{6227020800}}-{\frac {z^{15}}{1307674368000}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{(2n+1)!}},$ $!$ $\Sigma$ $z^{n}$ $f(z)=f(c)+f'(c)(z-c)+{\frac {f''(c)}{2}}(z-c)^{2}+{\frac {f'''(c)}{6}}(z-c)^{3}+\cdots$ $z$ $c$ $z\mapsto {\sqrt {z}}$ $c=25$ $\textstyle ({\sqrt {z}})'={\frac {1}{2{\sqrt {z}}}}$ $\textstyle ({\sqrt {z}})''=-{\frac {1}{4{\sqrt {z}}^{3}}}$ ${\sqrt {z}}\approx {\sqrt {25}}+{\frac {1}{2{\sqrt {25}}}}(z-25)-{\frac {1}{8\cdot {\sqrt {25}}^{3}}}(z-25)^{2}=5+{\frac {1}{10}}(z-25)-{\frac {1}{1000}}(z-25)^{2}$ $z$ $25$ $\textstyle 5+{\tfrac {1}{10}}(z-25)-{\tfrac {1}{1000}}(z-25)^{2}$ $z=25$ $5$ $25$ ${\sqrt {26}}\approx 5+{\frac {1}{10}}-{\frac {1}{1000}}=5{,}099$ ${\sqrt {26}}=5{,}0990195\dots$ $25+0{,}4i$ $5{,}00016+0{,}04i$ ${\sqrt {25+0{,}4i}}$ $(5{,}00016+0{,}04i)^{2}=25{,}0000000256+0{,}4000128i$ $z^{n}$ $z^{n}$ $nz^{n-1}$ $z^{n}$ $1+z+z^{2}+z^{3}+\cdots ={\frac {1}{1-z}}$ $1+2z+3z^{2}+4z^{3}+\cdots ={\frac {1}{(1-z)^{2}}}$ $f$ $0{,}001$ $p$ $z$ $\textstyle p_{k}(z):=\sum _{n=0}^{k}{\frac {(-1)^{n}z^{2n+1}}{(2n+1)!}}.$ $p_{0}(z)$ $\sin(z)$ $p_{0}(z)=z$ $p_{1}(z)=z-{\tfrac {z^{3}}{6}}$ $p_{2}(z)=z-{\tfrac {z^{3}}{6}}+{\tfrac {z^{5}}{120}}$ $p_{3}(z)$ $p_{4}(z)$ $p_{5}(z)$ $p_{6}(z)$ ${\tfrac {z^{n+1}}{n+1}}$ $z^{n}$ $\textstyle -\log(1-z)=\int \!{\frac {1}{1-z}}\,\mathrm {d} z$ $\int \!\left(1+z+z^{2}+z^{3}+z^{4}+\cdots \right)\,\mathrm {d} z=\int \!{\frac {1}{1-z}}\,\mathrm {d} z{\overset {(*)}{\implies }}z+{\frac {z^{2}}{2}}+{\frac {z^{3}}{3}}+{\frac {z^{4}}{4}}+\cdots =-\log(1-z).$ $(*)$ $\textstyle \int (1+z+z^{2}+\cdots )=\int 1+\int z+\int z^{2}+\cdots$ $z\mapsto az+b$ $z_{1},z_{2},z_{3},\dots ,z_{n},\dots$ $z_{\infty }$ $z_{\infty }$ $z_{\infty }$ $1,2,3,4,\dots$ $z_{1},z_{2},\dotsc \to z_{\infty }$ $\int _{-\infty }^{\infty }\!\mathrm {e} ^{-t^{2}}\,\mathrm {d} t={\sqrt {\pi }}$ $t\mapsto \mathrm {e} ^{-t^{2}}$ $a_{1}+a_{2}+a_{3}+\cdots$ $a_{n}$ $1+{\frac {1}{10}}+{\frac {1}{10^{2}}}+{\frac {1}{10^{3}}}+{\frac {1}{10^{4}}}+\cdots =1+0{,}1+0{,}01+0{,}001+0{,}0001+\cdots =1{,}11111\ldots ={\frac {10}{9}}.$ $1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+{\frac {1}{5^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}$ $1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots ={\frac {7\pi ^{3}}{180}}-2\left({\frac {1}{\mathrm {e} ^{2\pi }-1}}+{\frac {1/8}{\mathrm {e} ^{4\pi }-1}}+{\frac {1/27}{\mathrm {e} ^{6\pi }-1}}+\cdots \right)$ $x>0$ $\sum _{n=-\infty }^{\infty }\mathrm {e} ^{-\pi xn^{2}}=1+2\mathrm {e} ^{-\pi x}+2\mathrm {e} ^{-4\pi x}+2\mathrm {e} ^{-9\pi x}+\cdots ={\frac {1}{\sqrt {x}}}\left(1+2\mathrm {e} ^{-\pi /x}+2\mathrm {e} ^{-4\pi /x}+2\mathrm {e} ^{-9\pi /x}+\cdots \right)={\frac {1}{\sqrt {x}}}\sum _{n=-\infty }^{\infty }\mathrm {e} ^{-\pi n^{2}/x}.$ $e$ $\pi$ $78=8^{2}+3^{2}+2^{2}+1^{2}$ $0,1,2,3,4,\dots ,n,\dots$ $a_{0},a_{1},a_{2},a_{3},a_{4},\dots ,a_{n},\dots$ $0,1,4,9,16,25,\dots$ $2,3,5,7,\dots$ $1,1,2,3,5,8,\dots$ $a_{0},a_{1},a_{2},\dots$ $f(z)=a_{0}+a_{1}z+a_{2}z^{2}+a_{3}z^{3}+\cdots +a_{n}z^{n}+\cdots$ $a_{n}$ $f$ $a_{n}$ $f$ $a_{n}$ $a_{n}$ $n$ $p$ ${\begin{aligned}4&=4\\&=3+1\\&=2+2\\&=2+1+1\\&=1+1+1+1\end{aligned}}$ $p(4)=5$ $p(100)=190\ 569\ 292$ $p(1000)=24\ 061\ 467\ 864\ 032\ 622\ 473\ 692\ 149\ 727\ 991.$ $p(0)=1$ $P(z)=\sum _{n=0}^{\infty }p(n)z^{n}=1+z+2z^{2}+3z^{3}+5z^{4}+{\color {red}{7}}z^{\color {blue}{5}}+11z^{6}+15z^{7}+22z^{8}+\cdots$ $P.$ $P$ $p(n)$ $z$ $|z|>1$ $p(n)\approx {\frac {1}{4{\sqrt {3}}n}}\mathrm {e} ^{\pi {\sqrt {\frac {2n}{3}}}},$ $n$ $e^{x}$ $\pi =3{,}14159\dots$ ${\sqrt {3}}$ ${\tfrac {1}{f(z)}}$ $z\to \infty$ $x\mapsto {\tfrac {1}{x^{2}+1}}$ $\mathbb {R}$ $x\mapsto c$ $\sin(2+10i)=10014{,}30435528\ldots +4583{,}12202096\dots i.$ $a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots +a_{1}z+a_{0}=0$ $a_{n}\not =0$ $n\in \mathbb {N}$ $z^{4}-z^{3}+2z^{2}+17z+20=0$ $f(z)=z^{4}-z^{3}+2z^{2}+17z+20$ $z\mapsto {\tfrac {1}{f(z)}}$ $z$ $f(z)\not =0$ $z^{4}-z^{3}+2z^{2}+17z+20=0$ $z\mapsto {\frac {1}{f(z)}}={\frac {1}{z^{4}-z^{3}+2z^{2}+17z+20}}$ $\mathbb {C}$ $f(z)$ $z$ ${\tfrac {1}{f(z)}}$ $x+iy$ $U$ $U$ $\mathbb {N}$ $\mathbb {Z}$ $\mathbb {Q}$ $\mathbb {R}$ $\mathbb {C}$ $U\subset \mathbb {C}$ $D\subset \mathbb {C}$ $c$ $r>0$ $\partial M$ $M$ $\textstyle \oint$ $\mathbb {R} ^{2}$ $|x+iy|={\sqrt {x^{2}+y^{2}}}$ $U\subset \mathbb {C}$ $z_{0}\in U$ $z_{0}\in U$ $\varepsilon >0$ $B_{\varepsilon }(z)$ $U$ $U\subseteq \mathbb {C}$ $z_{0}\in U$ $f\colon U\to \mathbb {C}$ $z_{0}$ $\lim _{h\to 0}{\frac {f(z_{0}+h)-f(z_{0})}{h}}$ $f'(z_{0})$ $\textstyle \lim _{h\to 0}$ $h_{n}$ $h_{n}\not =0$ $n$ $\lim _{n\to \infty }{\frac {f(z_{0}+h_{n})-f(z_{0})}{h_{n}}}$ $h_{n}$ $U$ $U$ $U$ $h$ $z+h$ $U$ $h$ $f$ $f(z)=\operatorname {Re} (f(z))+i\operatorname {Im} (f(z))=u(x,y)+i\ v(x,y)$ $u,v\colon \mathbb {R} ^{2}\rightarrow \mathbb {R}$ $f$ $z_{0}=x_{0}+i\ y_{0}$ $u(x,y)=u(x_{0},y_{0})+a\Delta x+b\Delta y+o(\Delta x,\Delta y),$ $v(x,y)=v(x_{0},y_{0})+c\Delta x+d\Delta y+o(\Delta x,\Delta y),$ $o$ $\Delta x,\Delta y$ ${\frac {o(\Delta x,\Delta y)}{\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}}\rightarrow 0$ $(\Delta x)^{2}+(\Delta y)^{2}\rightarrow 0.$ $a,b,c,d$ $u$ $v$ $f$ $\mathbb {R} ^{2}$ ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{pmatrix}{\frac {\partial u}{\partial x}}&{\frac {\partial u}{\partial y}}\\{\frac {\partial v}{\partial x}}&{\frac {\partial v}{\partial y}}\end{pmatrix}}.$ $\Delta x$ $\Delta y$ $f(x_{0}+iy_{0}+i\Delta y)$ $f(x_{0}+\Delta x+iy_{0})$ $\Delta x$ $\Delta y$ $\Delta x$ $\Delta y$ $\Delta z$ $z_{0}$ $f(z)=f(z_{0})+\omega \Delta z+o(\Delta z),$ $\Delta z:=z-z_{0}$ $\mathbb {C}$ ${\frac {f(z)-f(z_{0})}{\Delta z}}=\omega +{\frac {o(\Delta z)}{\Delta z}}$ $\lim _{\Delta z\to 0}{\frac {o(\Delta z)}{\Delta z}}=0$ $\Delta z=\Delta x+i\Delta y$ $\omega =\operatorname {Re} (\omega )+i\operatorname {Im} (\omega )$ $\omega \Delta z=(\operatorname {Re} (\omega )+i\operatorname {Im} (\omega ))(\Delta x+i\Delta y)=(\operatorname {Re} (\omega )\Delta x-\operatorname {Im} (\omega )\Delta y)+i(\operatorname {Im} (\omega )\Delta x+\operatorname {Re} (\omega )\Delta y)$ ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{pmatrix}\operatorname {Re} (\omega )&-\operatorname {Im} (\omega )\\\operatorname {Im} (\omega )&\operatorname {Re} (\omega )\end{pmatrix}}={\begin{pmatrix}{\frac {\partial u}{\partial x}}&{\frac {\partial u}{\partial y}}\\{\frac {\partial v}{\partial x}}&{\frac {\partial v}{\partial y}}\end{pmatrix}}.$ ${\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}$ ${\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}$ $f\colon U\rightarrow \mathbb {C}$ $z_{0}\in U$ $f=u+iv$ $U$ $u$ $v$ $U$ ${\mathcal {O}}$ $f$ $U$ $z\in U$ $U=\mathbb {C}$ $f$ $U$ $U$ $U$ ${\mathcal {O}}(U)$ ${\mathcal {O}}$ $f,g\colon U\to \mathbb {C}$ $z\in U$ $f+g$ $f-g$ $f\cdot g$ ${\tfrac {f}{g}}$ $z$ $g$ $f=u+iv$ $\operatorname {det} {\begin{pmatrix}u_{x}&u_{y}\\v_{x}&v_{y}\end{pmatrix}}$ $f$ $\Phi \colon \mathbb {C} \to \mathbb {R} ^{2\times 2}$ $\Phi (a+ib):={\begin{pmatrix}a&-b\\b&a\end{pmatrix}}$ $r>0$ $\Phi (r\mathrm {e} ^{i\theta })=r{\begin{pmatrix}\cos(\theta )&-\sin(\theta )\\\sin(\theta )&\cos(\theta )\end{pmatrix}}.$ $Df(x,y)=\Phi (f'(z))$ $z=x+iy$ $f(z)=z^{n}$ $n\geq 2$ $n$ $f\colon D\to \mathbb {C}$ $D$ $a,b\in D$ $a\not =b$ $z_{1},z_{2}\in (a,b):=\{z=at+b(1-t)\colon t\in (0,1)\}\subset D$ $\operatorname {Re} \left({\frac {f(a)-f(b)}{a-b}}\right)=\operatorname {Re} (f'(z_{1}))$ $\operatorname {Im} \left({\frac {f(a)-f(b)}{a-b}}\right)=\operatorname {Im} (f'(z_{2})).$ $f\colon D\to \mathbb {C}$ $D$ $a,b\in D$ $a\not =b$ $f(a)=f(b)=0$ $z_{1},z_{2}\in (a,b):=\{z=at+b(1-t)\colon t\in (0,1)\}\subset D$ $\operatorname {Re} (f'(z_{1}))=\operatorname {Im} (f'(z_{2}))=0.$ $f,g\colon U\to \mathbb {C}$ $a\in U$ $f(a)=f'(a)=\cdots =f^{(n-1)}(a)=0$ $g(a)=g'(a)=\cdots =g^{(n-1)}(a)=0$ $f^{(n)}(a)\not =0\not =g^{(n)}(a)$ $n\in \mathbb {N}$ $\lim _{z\to a}{\frac {f(z)}{g(z)}}={\frac {f^{(n)}(a)}{g^{(n)}(a)}}.$ $a$ $b$ $a$ $b$ $\textstyle \oint$ $D\subset \mathbb {C}$ $f\colon D\to \mathbb {C}$ $\gamma \colon [0,1]\to D$ $\int _{\gamma }\!f(z)\,\mathrm {d} z:=\int _{0}^{1}\!f(\gamma (t))\gamma '(t)\,\mathrm {d} t.$ $\operatorname {Re} (f)+i\operatorname {Im} (f)$ $\gamma '(t)\mathrm {d} t$ $\mathrm {d} \gamma (t)$ $\gamma ({\tfrac {1}{n}})-\gamma (0),\gamma ({\tfrac {2}{n}})-\gamma ({\tfrac {1}{n}}),\ldots ,\gamma (1)-\gamma ({\tfrac {n-1}{n}})$ $n\to \infty$ $\int _{\gamma }\!f(z)\,\mathrm {d} z$ $\gamma (0)=a$ $\gamma (1)=b$ $f$ $\gamma$ $f$ $F$ $\int _{\gamma }\!f(z)\,\mathrm {d} z=F(\gamma (1))-F(\gamma (0))=F(b)-F(a),$ $x_{0}$ $\textstyle x\mapsto \int _{x_{0}}^{x}$ $\gamma$ $D$ $F$ $D$ $F$ $f$ $D$ $\int _{\gamma }\!f(z)\,\mathrm {d} z=\int _{a}^{b}\!f(z)\,\mathrm {d} z$ $f$ $D=\mathbb {C} \setminus \{0\}$ $f(z)={\tfrac {1}{z}}$ $f$ $D$ $f$ $D$ $D$ $f\colon D\to \mathbb {C}$ $z_{0}\in D$ $z\in D$ $D$ $F(z):=\int _{z_{0}}^{z}\!f(w)\,\mathrm {d} w$ $z_{0}$ $z$ $D\subseteq \mathbb {C}$ $\gamma$ $D$ $\int _{\gamma }\!f(z)\,\mathrm {d} z=0.$ $D$ $\gamma$ $D\subset \mathbb {C}$ $f\colon D\to \mathbb {C}$ $\langle z_{1},z_{2},z_{3}\rangle$ $\Delta$ $D$ $\oint _{\langle z_{1},z_{2},z_{3}\rangle }\!f(z)\,\mathrm {d} z=0,$ $f$ $f\colon D\to \mathbb {C}$ $D$ $g\colon D\to \mathbb {C}$ $f=e^{g}$ $g$ $f$ $f$ $n$ $n\in \mathbb {N}$ $D$ $h\colon D\to \mathbb {C}$ $f=h^{n}$ $U$ $B_{r}(a)$ $r$ $a\in U$ $f\colon U\to \mathbb {C}$ $B_{r}(a)$ $U$ $z\in D$ $f(z)={\frac {1}{2\pi i}}\oint _{\partial B_{r}(a)}\!{\frac {f(w)}{w-z}}\,\mathrm {d} w.$ $k\in \mathbb {N} _{0}$ $f^{(k)}(z)={\frac {k!}{2\pi i}}\oint _{\partial B_{r}(a)}\!{\frac {f(w)}{(w-z)^{k+1}}}\,\mathrm {d} w.$ $k!$ $k$ $z_{0}$ $z_{0}$ $z_{0}$ $z_{0}$ $f(a)={\frac {1}{2\pi }}\int _{0}^{2\pi }\!f(a+re^{i\varphi })\,\mathrm {d} \varphi ,$ $|f(a)|\leq \sup _{|z-a|=r}|f(z)|$ $f$ ${\overline {B_{r}(0)}}$ $z\in B_{r}(0)$ $f(z)={\frac {1}{2\pi i}}\oint _{\partial B_{r}(0)}\!{\frac {f(w)}{w}}{\frac {r^{2}-|z|^{2}}{|w-z|^{2}}}\,\mathrm {d} w,$ $f(z)={\frac {1}{2\pi i}}\oint _{\partial B_{r}(0)}\!{\frac {\operatorname {Re} (f(w))}{w}}{\frac {w+z}{w-z}}\,\mathrm {d} w+i\operatorname {Im} (f(0))$ $f$ ${\overline {B_{R}(0)}}$ $f(re^{i\theta })={\frac {1}{2\pi }}\int _{0}^{2\pi }f(Re^{i\Theta }){\frac {R^{2}-r^{2}}{R^{2}-2Rr\cos(\Theta -\theta )+r^{2}}}\mathrm {d} \Theta .$ $c\in U$ $U$ $B_{r}(c)$ $c$ $U$ $f\colon U\to \mathbb {C}$ $f$ $c$ $\textstyle \sum _{n=0}^{\infty }a_{n}(z-c)^{n}$ $B_{r}(c)$ $a_{n}={\frac {f^{(n)}(c)}{n!}}={\frac {1}{2\pi i}}\oint _{\partial B_{d}(c)}{\frac {f(w)}{(w-c)^{n+1}}}\mathrm {d} w$ $z\mapsto (z-c)^{-k-1}$ $k=0$ $\textstyle f(z)=\sum _{n=0}^{\infty }a_{n}(z-c)^{n}$ $R>0$ $f(z)$ $|z-c|>R$ $R$ $R={\frac {1}{\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}|}}}}.$ $a_{n}\not =0$ $n\in \mathbb {N}$ $\liminf _{n\to \infty }\left|{\frac {a_{n}}{a_{n+1}}}\right|\leq R\leq \limsup _{n\to \infty }\left|{\frac {a_{n}}{a_{n+1}}}\right|.$ $+\infty$ $R=\infty$ $\infty \leq R$ $f$ ${\overline {B_{R}(0)}}$ $f(z)=\sum _{k=0}^{N-1}{\frac {f^{(k)}(0)}{k!}}z^{k}+{\frac {z^{N}}{2\pi i}}\oint _{\partial B_{R}(0)}{\frac {f(w)}{w^{N}(w-z)}}\mathrm {d} w.$ $\left|\sum _{k=N}^{\infty }{\frac {f^{(k)}(0)}{k!}}z^{k}\right|\leq {\frac {|z|^{N}}{2\pi }}\oint _{\partial B_{R}(0)}\left|{\frac {f(w)}{w^{N}(w-z)}}\mathrm {d} w\right|\leq \left({\frac {|z|}{R}}\right)^{N}R\max _{|w|=R}\left|{\frac {f(w)}{w-z}}\right|$ $z$ $\left|\sum _{k=N}^{\infty }{\frac {f^{(k)}(0)}{k!}}z^{k}\right|\ll _{R,N}z^{N}\max _{|w|=R}|f(w)|$ $R$ $N$ $z$ $f$ $2\pi i$ $f(z):={\frac {1}{1-z}}=\sum _{n=0}^{\infty }z^{n},$ $z=1$ $\mathbb {C} \setminus \{1\}$ $R=1$ $f(2)=-1$ $z=2$ $z$ $f\colon B_{1}(0)\cup B_{1}(3)\to \mathbb {C}$ $f(z)={\begin{cases}e^{z},&\quad z\in B_{1}(0)\\0,&\quad z\in B_{1}(3)\end{cases}}$ $\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}$ $f$ $B_{1}(0)$ $z=3$ $f\colon \mathbb {C} \setminus \mathbb {R} _{\leq 0}\to \mathbb {C}$ $f(z):=\mathrm {Log} (z)$ $z_{0}=-5+i$ $R={\sqrt {26}}$ $-5-i$ $f(z)=\sum _{n=0}^{\infty }a_{n}(z-c)^{n}$ $w\in \partial B_{R}(c)$ $W$ $w$ ${\widehat {f}}\colon W\to \mathbb {C}$ $f|_{B_{R}(c)\cap W}={\widehat {f}}|_{B_{R}(c)\cap W}$ $\partial B_{R}(c)$ $f$ $\partial B_{R}(c)$ $f$ $B_{R}(c)$ $f$ $f(z)=\sum _{\nu =0}^{\infty }b_{\nu }z^{\lambda _{\nu }}$ $\delta >0$ $\lambda _{\nu +1}-\lambda _{\nu }\geq \delta \lambda _{\nu }$ $\nu \in \mathbb {N} _{0}$ $B_{R}(0)$ $f$ $\sum _{n=0}^{\infty }a_{n}z^{n}$ $R=1$ $\mathbb {E}$ $\varepsilon _{n}\in \{-1,1\}$ $z\mapsto \sum _{n=0}^{\infty }\varepsilon _{n}a_{n}z^{n}$ $\mathbb {E}$ $f$ $f(z)=g(z)+h\left({\frac {1}{z}}\right)$ $g\colon B_{R}(0)\to \mathbb {C}$ $\textstyle h\colon B_{\frac {1}{r}}(0)\to \mathbb {C}$ $h(0)=0$ $c$ $f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-c)^{n}.$ $f$ $f(z+1)=f(z)$ $f$ $f(z)=\sum _{n=-\infty }^{\infty }a_{n}e^{2\pi izn}.$ $D$ $a_{n}=\int _{0}^{1}f(x+iy)e^{-2\pi in(x+iy)}\mathrm {d} x$ $z\mapsto e^{2\pi iz}$ $f(z)=z^{2}$ $\lim _{h\to 0}{\frac {f(z+h)-f(z)}{h}}=\lim _{h\to 0}{\frac {(z+h)^{2}-z^{2}}{h}}=\lim _{h\to 0}{\frac {2zh+h^{2}}{h}}=\lim _{h\to 0}(2z+h)=2z=f'(z).$ $\lim _{h\to 0 \atop h\in \mathbb {R} }{\frac {|1+h|-1}{h}}\not =\lim _{h\to 0 \atop h\in \mathbb {R} }{\frac {|1+ih|-1}{ih}},$ $p(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots +a_{1}z+a_{0}$ $\mathbb {C}$ $x\mapsto e^{x}$ $\mathbb {C}$ $e^{z}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}=1+z+{\frac {z^{2}}{2}}+{\frac {z^{3}}{6}}+\cdots .$ $z,w\in \mathbb {C}$ $e^{z+w}=e^{z}e^{w}$ $(e^{z})'=e^{z}$ $e^{iw}=\cos(w)+i\sin(w),$ $w$ $2\pi i$ $z\in \mathbb {C}$ $e^{z+2\pi i}=e^{z}$ $z\mapsto \exp(z)$ $z=\pi i$ $\exp(S)=\mathbb {C} \setminus \mathbb {R} _{\leq 0}=:\mathbb {C} _{-},$ $\mathrm {Log} \colon \mathbb {C} _{-}\rightarrow S$ $\exp(\mathrm {Log} (z))=z$ $\mathbb {C} _{-}$ $S$ $2\pi i$ $\mathrm {Arg} (z)$ $\mathrm {Log} (z)=\log |z|+i\mathrm {Arg} (z)$ $\log(x)$ $\lim _{z\to a \atop \operatorname {sgn}(\operatorname {Im} (z))=\pm 1}\mathrm {Log} (z)=\log |a|\pm \pi i,$ $\operatorname {sgn}(x)$ $\mathbb {C} _{-}$ $\mathrm {Log} '(z)={\frac {1}{z}}.$ $\sin(z)={\frac {e^{iz}-e^{-iz}}{2i}}$ $\mathbb {C}$ $\{{\tfrac {2k+1}{2}}\pi i\mid k\in \mathbb {Z} \}$ $z=0$ $s\in \mathbb {C}$ $z\in \mathbb {C} \setminus \{0\}$ $z^{s}:=e^{s\operatorname {Log} (z)}.$ $\mathbb {C} \setminus \mathbb {R} _{\leq 0}$ $z^{s}=e^{s(\pi i+\operatorname {Log} (-z))}.$ $z\in \mathbb {C}$ $z\mapsto |z|$ $z\mapsto \operatorname {Re} (z)$ $z\mapsto \operatorname {Im} (z)$ $z\mapsto {\overline {z}}$ $z\mapsto |z|^{2}$ $z=0$ $0$ $U\subseteq \mathbb {C}$ $f\colon U\to \mathbb {C}$ $U$ $U$ $a\in U$ $U(a)\subset U$ $f|_{U(a)}$ $U$ $f$ $\quad {\frac {\partial f}{\partial {\bar {z}}}}=0,$ ${\tfrac {\partial }{\partial {\bar {z}}}}$ ${\tfrac {\partial }{\partial {\bar {z}}}}:={\tfrac {1}{2}}\left({\tfrac {\partial }{\partial x}}+i{\tfrac {\partial }{\partial y}}\right)$ $D$ $a\in \mathbb {C}$ $f\colon D\to \mathbb {C}$ $f^{-1}(a):=\{z\in D\mid f(z)=a\}$ $D$ $f$ $a$ $a=0$ $f$ $f,g$ $D$ $\gamma$ $D$ $z\in \gamma ([0,1])$ $f$ $f+g$ $\gamma ([0,1])$ $f\not =0$ $D$ $D$ ${\overline {B_{r}(0)}}$ $z_{1},z_{2},\dots ,z_{n}$ $f$ $B_{r}(0)$ $f(0)\not =0$ $\log |f(0)|=-\sum _{k=1}^{n}\log \left({\frac {r}{|z_{k}|}}\right)+{\frac {1}{2\pi }}\int _{0}^{2\pi }\log |f(re^{i\theta })|\mathrm {d} \theta .$ $f(z)\not =0$ $\log |f(z)|=-\sum _{k=1}^{n}\log \left|{\frac {r^{2}-{\overline {z_{k}}}z}{r(z-z_{k})}}\right|+{\frac {1}{2\pi }}\int _{0}^{2\pi }\operatorname {Re} \left({\frac {re^{i\theta }+z}{re^{i\theta }-z}}\right)\log |f(re^{i\theta })|\mathrm {d} \theta .$ $a\in U$ $f\colon U\setminus \{a\}\rightarrow \mathbb {C}$ $a$ $U$ $f$ $a$ $f\colon U\setminus \{a\}\rightarrow \mathbb {C}$ $a$ $z\mapsto {\frac {\sin(z)}{z}}$ $z\mapsto {\frac {1}{z}}$ $z\mapsto \exp \left({\frac {1}{z}}\right)$ $\mathbb {C} \setminus \{0\}$ $f$ $a$ $|f(z)|\leq M$ $z$ $a$ $U$ $f$ $U$ ${\widehat {f}}\colon U\rightarrow \mathbb {C}$ $U\setminus \{a\}$ $f$ $f(z)={\frac {z^{2}-1}{z-1}}$ $a=1$ ${\widehat {f}}(z)=z+1$ $f(z)={\frac {\sin(z)}{z}}$ $a=0$ $1-{\tfrac {z^{2}}{6}}+{\tfrac {z^{4}}{120}}-\cdots$ $f\colon U\setminus \{c\}\rightarrow \mathbb {C}$ $m\in \mathbb {N}$ $c$ $c$ $f(z)={\frac {h(z)}{(z-c)^{m}}}$ $h$ $h(c)\not =0$ $f$ $f$ $a$ $\lim _{z\to a}|f(z)|=\infty$ $c$ $m$ $f(z)$ $f$ $f(z)={\frac {a_{-m}}{(z-c)^{m}}}+{\frac {a_{-m+1}}{(z-c)^{m-1}}}+\cdots =\sum _{n=-m}^{\infty }a_{n}(z-c)^{n},\qquad a_{-m}\not =0.$ $m$ $\mathbb {C} \setminus \{0\}$ $f(z)={\tfrac {e^{z}}{z^{4}}}$ $z=0$ $z\mapsto \exp({\tfrac {1}{z}})$ $V\setminus \{c\}$ $f$ $\varepsilon >0$ $v\in \mathbb {C}$ $w\in V\setminus \{c\}$ $c$ $f$ $c$ $f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-c)^{n}$ $a_{n}\not =0$ $c$ $\infty :={\tfrac {1}{0}}$ $f\colon D\to \mathbb {C} \cup \{\infty \}$ $S(f):=f^{-1}(\{\infty \})$ $D$ $f_{0}\colon D\setminus S(f)\to \mathbb {C}$ $S(f)$ $f$ $\mathbb {C}$ $f$ ${\dot {U}}_{\delta }(c)$ $c$ $f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-c)^{n}$ ${\dot {U}}_{\delta }(c)$ ${\tfrac {a_{-1}}{z-c}}.$ $\mathrm {Res} (f;c):=a_{-1}.$ $U$ $a_{k}$ $f$ $D$ $\{a_{1},\dots ,a_{n}\}\subset D$ $n$ $f\colon D\setminus \{a_{1},\dots ,a_{n}\}\to \mathbb {C}$ $\gamma \colon [0,1]\to D\setminus \{a_{1},\dots ,a_{n}\}$ ${\frac {1}{2\pi i}}\oint _{\gamma }f(z)\mathrm {d} z=\sum _{j=1}^{n}\chi (\gamma ;a_{j})\operatorname {Res} (f;a_{j}),$ $\chi (\gamma ;a_{j})$ $\gamma$ $a_{j}$ $f$ $f$ $D$ $\gamma$ $f$ $N(0)$ $N(\infty )$ ${\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f'(z)}{f(z)}}\mathrm {d} z=N(0)-N(\infty ).$ $\int _{-\infty }^{\infty }{\frac {t^{2k}}{1+t^{2n}}}\mathrm {d} t={\frac {\pi }{n\sin \left({\frac {(2k+1)\pi }{2n}}\right)}}.$ $\int _{-\infty }^{\infty }e^{-t^{2}}\mathrm {d} t={\sqrt {\pi }}$ $D$ $f$ $g$ $D$ $f=g$ $z\in D$ $M\subset D$ $z$ $M$ $f=g$ $\{z\in D\mid f(z)=g(z)\}$ $D$ $z_{0}\in D$ $n\geq 0$ $f^{(n)}(z_{0})=g^{(n)}(z_{0})$ $D$ $f(z)={\begin{cases}0,&\quad z\in B_{1}(0)\\1,&\quad z\in B_{1}(42)\end{cases}}$ $g(z)={\begin{cases}0,&\quad z\in B_{1}(0)\\2,&\quad z\in B_{1}(42)\end{cases}}$ $B_{1}(0)$ $f(42)\not =g(42)$ $D=B_{1}(0)\cup B_{1}(42)$ $z\mapsto \sin \left({\tfrac {\pi }{z}}\right)$ $\mathbb {C} \setminus \{0\}$ $z=1,{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},\dots ,{\tfrac {1}{n}},\dots$ $n\mapsto {\tfrac {1}{n}}$ $\mathbb {C} \setminus \{0\}$ $D\subseteq \mathbb {C}$ $f\colon D\rightarrow \mathbb {C}$ $f(D)\subseteq \mathbb {C}$ $\sin(\mathbb {R} )=[-1,1]$ $z\mapsto z^{n}$ $x^{2}\geq 0$ $f$ $z\mapsto \operatorname {Re} (f(z))$ $z\mapsto \operatorname {Im} (f(z))$ $z\mapsto |f(z)|$ $f$ $f$ $f(0)=0$ $n$ $U$ $\varphi \colon U\to V$ $\varphi (0)=0$ $f(z)=\varphi (z)^{n}$ $z\in U$ $z=\varphi ^{-1}(w)$ $f(\varphi ^{-1}(w))=w^{n}$ $w\in V$ $n$ $f$ $n=1$ $D$ $f$ $D$ $z$ $z\in D$ $U$ $|f(a)|\leq |f(z)|$ $a\in U$ $f$ $D$ $\partial D$ $D$ $g(z)=e^{e^{z}}$ $\left|e^{e^{z}}\right|=e^{\operatorname {Re} (e^{x+iy})}=e^{e^{x}\cdot \cos(y)},$ $z=x+iy$ $g(z)$ $\operatorname {Re} (z)\to \infty$ $f(z)$ $f(D)$ $|f(z)|$ $f$ $z$ $D$ $z$ $f$ $D$ $f\colon D\rightarrow \mathbb {C}$ $\varphi \colon D\rightarrow \mathbb {C}$ $\infty$ $\partial _{\infty }D=A\cup B$ $M>0$ $a\in A$ $\limsup _{z\to a}|f(z)|\leq M$ $b\in B$ $\mu >0$ $\limsup _{z\to b}|f(z)||\varphi (z)|^{\mu }\leq M$ $|f(z)|\leq M$ $z\in D$ $\limsup$ $f$ $-{\tfrac {\pi }{2}}\leq \operatorname {Re} (z)\leq {\tfrac {\pi }{2}}$ $|f(z)|\leq 1$ $z$ $\operatorname {Re} (z)={\tfrac {\pi }{2}}$ $\operatorname {Re} (z)=-{\tfrac {\pi }{2}}$ $C>0$ $|f(z)|\leq \exp \left(Ce^{\alpha |z|}\right).$ $|f(z)|\leq 1$ $\alpha =1$ $z\mapsto e^{e^{z}}$ $f$ ${\overline {S}}$ $|f(z)|\leq 1$ $S$ $c,C>0$ $|f(z)|\leq Ce^{c|z|}$ $z\in S$ $|f(z)|\leq 1$ $z\in S$ $f$ $M(r):=\sup _{|z|=r}|f(z)|,$ $\log \left({\frac {\beta }{\alpha }}\right)\log M(r)\leq \log \left({\frac {\beta }{r}}\right)\log M(\alpha )+\log \left({\frac {r}{\alpha }}\right)\log M(\beta ).$ $f$ $a\leq \operatorname {Re} (z)\leq b$ $M(\sigma ):=\sup _{t\in \mathbb {R} }|f(\sigma +it)|$ $f$ $\psi (\sigma )$ $a\leq \sigma \leq b$ $|f(\sigma +it)|\ll |t|^{\psi (\sigma )+\varepsilon }$ $\varepsilon >0$ $\psi (\sigma )$ $[a,b]$ $f$ $|f|$ $z\mapsto \cos(z)$ $\mathbb {C}$ $\textstyle \sup _{z\in \mathbb {C} }|f(z)|\leq C$ $z\in \mathbb {C}$ $r>|z|$ $|f'(z)|=\left|{\frac {1}{2\pi i}}\oint _{|w-z|=r}{\frac {f(w)}{(w-z)^{2}}}\mathrm {d} w\right|\leq {\frac {2\pi r}{2\pi }}{\frac {C}{r^{2}}}={\frac {C}{r}}.$ $2\pi r$ $r>0$ $f'=0$ $\mathbb {C}$ $f$ $x\mapsto x^{2}+1$ $P$ $z\mapsto {\tfrac {1}{P(z)}}$ $f\colon {\overline {\mathbb {C} }}\rightarrow \mathbb {C}$ ${\overline {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}$ $|f(z)|\leq 1+|z|^{\frac {1}{2}}$ $f$ $f$ $f$ $R,B$ $K$ $\operatorname {Re} (f(z))\leq B|z|^{K}$ $|z|\geq R>0$ $f$ $K$ $f(\mathbb {C} )$ $f$ $\mathbb {C}$ $f\colon \mathbb {C} \rightarrow \mathbb {C}$ $f(\mathbb {C} )=\mathbb {C}$ $f(\mathbb {C} )=\mathbb {C} \setminus \{a\}$ $a\in \mathbb {C}$ $z\mapsto e^{z}$ $f\colon \mathbb {E} \rightarrow \mathbb {E}$ $\mathbb {E}$ $f(0)=0$ $|f(z)|\leq |z|$ $z\in \mathbb {E}$ $|f'(0)|\leq 1$ $f\colon \mathbb {E} \to \mathbb {C}$ $f(0)=0$ $f'(0)=1$ $\mathbb {E}$ $f$ $f\colon \mathbb {E} \to \mathbb {C}$ $z\mapsto zf'(z)$ $z\in \mathbb {E}$ $\operatorname {Re} \left(1+{\frac {zf''(z)}{f'(z)}}\right)>0$ $\textstyle f(z)=\sum _{n\geq 1}a_{n}z^{n}$ $\textstyle g(z)=\sum _{n\geq 1}b_{n}z^{n}$ $\textstyle (f*g)(z):=\sum _{n\geq 1}a_{n}b_{n}z^{n}$ $\operatorname {Li} _{\alpha }(z)$ $\alpha \geq 0$ $f$ $\gamma \subset \mathbb {E}$ $f$ $\operatorname {Re} \left({\frac {f(z)-f(\zeta )}{(z-\zeta )f'(z)}}\right)\geq 0$ $(z,\zeta )\in \mathbb {E} \times \mathbb {E}$ $f\colon \mathbb {E} \to D$ $D$ $D_{r}:=f(B_{r}(0))$ $D$ $f(0)$ $D_{r}$ $f(0)$ $f\colon \mathbb {E} \to \mathbb {C}$ $f(0)=0$ $|f'(0)|\geq 1$ $f(\mathbb {E} )$ $B_{\frac {1}{2}}(0)\subset f(\mathbb {E} )$ $U\subset \mathbb {C}$ $\{z\in \mathbb {C} \mid |z|\leq 1\}$ $f\colon U\rightarrow \mathbb {C}$ $f(0)=0$ $f'(0)=1$ $S\subset U$ $f|_{S}\colon S\rightarrow \mathbb {C}$ $f(S)$ ${\tfrac {1}{72}}$ $\beta (f)$ $r$ $f|_{S}\colon S\rightarrow \mathbb {C}$ $S\subset U$ $f(S)$ $r$ $\beta (f)$ $f$ $B:=\inf\{\beta (f)\mid f\in {\mathcal {O}}(V),\{z\in \mathbb {C} \mid |z|\leq 1\}\subset V\}.$ $B\geq {\tfrac {1}{72}}$ $f(z)=z$ $B\leq 1$ $0{,}43\leq B\leq {\frac {\Gamma \left({\frac {1}{3}}\right)\Gamma \left({\frac {11}{12}}\right)}{{\sqrt {1+{\sqrt {3}}}}\Gamma \left({\frac {1}{4}}\right)}}=0{,}47186165\ldots$ $\Gamma (s)$ $B$ $0\leq \beta \leq 1$ $C(\alpha ,\beta )$ $D\subset \mathbb {C}$ $\{z\in \mathbb {C} \mid |z|\leq 1\}$ $f\colon D\rightarrow \mathbb {C}$ $|f(0)|\leq \alpha$ $|f(z)|\leq C(\alpha ,\beta )$ $|z|\leq \beta$ ${\overline {B_{R}(0)}}=\{z\in \mathbb {C} \mid |z|\leq R\}$ $R>0$ $D$ ${\overline {B_{R}(0)}}$ $f\colon D\rightarrow \mathbb {C}$ $|f(0)|\leq \alpha$ $C(\alpha ,\beta )$ $|f(z)|\leq C(\alpha ,\beta )$ $|z|\leq \beta R$ $f\colon \mathbb {E} \to \mathbb {C}$ $f(0)=0$ $f'(0)=1$ $B_{\frac {1}{4}}(0)\subset f(\mathbb {E} )$ $\textstyle f(z)=\sum _{n=-\infty }^{\infty }a_{n}z^{n}$ $|z|=r$ $L$ $L$ $\pi \sum _{n=-\infty }^{\infty }n|a_{n}|^{2}r^{2n}.$ $r\leq |z|\leq R$ $\pi \sum _{n=-\infty }^{\infty }n|a_{n}|^{2}(R^{2n}-r^{2n}).$ $a_{1},a_{2},...$ $\Delta ^{1}(a_{n}):=a_{n}-a_{n+1}$ $\Delta ^{2}(a_{n}):=\Delta ^{1}(\Delta ^{1}(a_{n}))$ $\Delta ^{r}(a_{n})=\sum _{j=0}^{r}{\binom {r}{j}}(-1)^{j}a_{n+j}.$ $b_{n}$ $k$ $\Delta ^{r}(b_{n})\geq 0$ $n$ $1\leq r\leq k$ $\textstyle f(z)=\sum _{n=0}^{\infty }b_{n}z^{n}$ $\mathbb {E}$ $|z|=1$ $b_{n}$ $g(\theta )=|f(e^{i\theta })|^{2}$ $[0,\pi ]$ $\textstyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}$ $\mathbb {E}$ $\mathbb {E}$ $\sum _{n=0}^{\infty }{\frac {|a_{n}|}{6^{n}}}\leq 1$ $\textstyle \sum _{n=0}^{\infty }{\frac {|a_{n}|}{3^{n}}}\leq 1$ $r={\tfrac {1}{3}}$ $z_{0}$ $D$ $z_{0}$ $D\setminus \{z_{0}\}$ $f$ $D$ $U$ $z_{0}$ $f(U\setminus \{z_{0}\})$ $\mathbb {C}$ $z_{0}$ $z_{0}$ $f$ $U$ $c\in U$ $f\colon U\setminus \{c\}\rightarrow \mathbb {C}$ ${\dot {V}}\subset U$ $c$ $f({\dot {V}})=\mathbb {C}$ ${\dot {V}}\subset U$ $c$ $f({\dot {V}})=\mathbb {C} \setminus \{a\}$ $a\in \mathbb {C}$ $U\subset \mathbb {C}$ $(f_{n})_{n\in \mathbb {N} }$ $U$ $(f_{n})_{n\in \mathbb {N} }$ $K\subset U$ $f$ $f$ $(f'_{n})_{n\in \mathbb {N} }$ $f'$ $(f'_{n})_{n\in \mathbb {N} }$ $D$ $(f_{n})_{n\in \mathbb {N} }$ $D$ $\partial D$ $(f_{n}|_{\partial D})_{n\in \mathbb {N} }$ $\partial D$ $(f_{n})_{n\in \mathbb {N} }$ $D$ $\partial D$ $f\colon [a,b]\rightarrow \mathbb {R}$ $(f'_{n})_{n\in \mathbb {N} }$ $f_{n}\colon U\to \mathbb {C}$ $\textstyle f=\sum _{n=1}^{\infty }f_{n}$ $f\colon U\to \mathbb {C}$ $U$ $z\in U$ $z\in V\subset U$ $(z,t)\mapsto F(z,t)$ $U\times [a,b]$ $U\subset \mathbb {C}$ $z\mapsto F(z,t)$ $t$ $f(z)=\int _{a}^{b}F(z,t)\mathrm {d} t$ $U$ $\Gamma _{n}(z):=\int _{\frac {1}{n}}^{n}e^{-t}t^{z-1}\mathrm {d} t$ $\{z\in \mathbb {C} \mid \operatorname {Re} (z)>0\}$ $n\to \infty$ $\Gamma (z)=\int _{0}^{\infty }e^{-t}t^{z-1}\mathrm {d} t$ $\{z\in \mathbb {C} \mid \operatorname {Re} (z)>0\}$ $D$ $(f_{n})_{n\in \mathbb {N} }$ $f_{n}\colon D\rightarrow \mathbb {C}$ $f$ $f(z_{0})=0$ $z_{0}\in D$ $B_{r}(z_{0})\subset D$ $N$ $f_{n}$ $n\geq N$ $B_{r}(z_{0})$ $f_{n}\colon D\rightarrow \mathbb {C}$ $f_{n}\colon D\to \mathbb {C}$ $f\colon D\to \mathbb {C}$ $K\subset D$ $N$ $n\geq N$ $f_{n}|_{K}$ $(f_{n})_{n\in \mathbb {N} }$ $U$ $(f_{n})_{n\in \mathbb {N} }$ $U$ $S\subset U$ $U$ $D$ $(f_{n})_{n\in \mathbb {N} }$ $D$ $c\in D$ $k\in \mathbb {N} _{0}$ $f_{1}^{(k)}(c),f_{2}^{(k)}(c),f_{3}^{(k)}(c),\dots$ $A:=\{z\in D\mid \lim _{n\to \infty }f_{n}(z)\in \mathbb {C} \}$ $D$ $a,b\in \mathbb {C}$ $a\not =b$ $(f_{n})_{n\in \mathbb {N} }$ $f_{n}\colon D\to \mathbb {C} \setminus \{a,b\}$ $\lim _{n\to \infty }f_{n}(w)$ $D$ $D$ $(f_{n})_{n\in \mathbb {N} }$ $D$ $(f_{n})_{n\in \mathbb {N} }$ $D$ $f$ $(f_{n})_{n\in \mathbb {N} }$ $D'\subset D$ $f$ $D'$ $f_{n}$ $D$ $u(x,y)$ ${\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0$ $D$ $\Delta ={\tfrac {\partial ^{2}}{\partial x^{2}}}+{\tfrac {\partial ^{2}}{\partial y^{2}}}$ $\Delta u=0$ $u$ $f$ $g(z):={\tfrac {\partial u}{\partial x}}-i{\tfrac {\partial u}{\partial y}}$ $g$ $u$ $f=u+iv$ $u$ $v$ $u$ $z\mapsto \sin(\pi z)$ $\mathbb {Z}$ $S\subset \mathbb {C}$ $m\colon S\to \mathbb {N}$ $s\mapsto m_{s}$ $f$ $S=N(f):=\{z\in \mathbb {C} \mid f(z)=0\}$ $m_{s}=\operatorname {ord} (f;s)$ $s\in S$ $S$ $s\in S$ $f$ $s\in S$ $\sin(\pi z)=\pi z\prod _{k=1}^{\infty }\left(1-{\frac {z^{2}}{k^{2}}}\right),$ $\mathbb {C}$ $D$ $z_{1},z_{2},...$ $D$ $m_{1},m_{2},...$ $f\colon D\to \mathbb {C}$ $z_{n}$ $z_{n}$ $m_{n}$ $h\colon D\to {\overline {\mathbb {C} }}$ $D$ $S\subset \mathbb {C}$ $\mathbb {C} \setminus S$ $s\in S$ $s\in S$ $h_{s}\colon \mathbb {C} \to \mathbb {C}$ $h_{s}(0)=0$ $f\colon \mathbb {C} \setminus S\to \mathbb {C}$ $s\in S$ $h_{s}$ $z\mapsto f(z)-h_{s}\left({\frac {1}{z-s}}\right)$ $z=s$ $S$ $f(z)=\sum _{s\in S}h_{s}\left({\frac {1}{z-s}}\right)$ $S$ $K\subset \mathbb {C}$ $f$ $K$ $f$ $K$ $(r_{n})_{n\in \mathbb {N} }$ $r_{n}$ $K$ $U\subset \mathbb {C}$ ${\overline {\mathbb {C} }}\setminus U$ ${\overline {\mathbb {C} }}$ $U$ $f$ $\varepsilon >0$ $K\subset U$ $p$ $z\in K$ $f$ $C,\varrho >0$ ${\frac {\mathrm {d} }{\mathrm {d} z}}\left({\frac {f(z)}{\sin(\varrho z)}}\right)=\sum _{n=-\infty }^{\infty }{\frac {\varrho (-1)^{n+1}f\left({\frac {\pi n}{\varrho }}\right)}{(\varrho z-n\pi )^{2}}}.$ $f$ $f(-z)=-f(z)$ ${\frac {f(z)}{2\varrho z\cos(\varrho z)}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}f\left({\frac {\pi (n+{\frac {1}{2}})}{\varrho }}\right)}{(\pi (n+{\frac {1}{2}}))^{2}-\varrho ^{2}z^{2}}}.$ $f(z)=\sin(\varrho z)$ $z_{1},z_{2},...,z_{n}$ $L$ $z_{1},z_{2},...,z_{n}$ $f$ $L$ $L$ $\textstyle \omega (z)=\prod _{j=1}^{n}(z-z_{j})$ $P(z)={\frac {1}{2\pi i}}\oint _{L}{\frac {f(\zeta )}{\omega (\zeta )}}{\frac {\omega (\zeta )-\omega (z)}{\zeta -z}}\mathrm {d} \zeta$ $n-1$ $z=z_{1},...,z_{n}$ $f(z)$ $\textstyle f(z)=\sum _{n=0}^{\infty }a_{n}(z-c)^{n}$ $R$ $r$ $\textstyle M(r):=\max _{|z-c|=r}|f(z)|$ $|a_{n}|=\left|{\frac {f^{(n)}(c)}{n!}}\right|\leq {\frac {M(r)}{r^{n}}}.$ $U$ $K\subset D$ $K\subsetneq L\subset U$ $z\in \partial K$ $L$ $k\in \mathbb {N}$ $M_{k,L}>0$ $||f^{(k)}||_{K}\leq M_{k,L}||f||_{L},$ $f\in {\mathcal {O}}(U)$ $||.||$ $K=L$ $K=L={\overline {\mathbb {E} }}$ $f_{n}(z):=z^{n}$ $f\colon B_{R}(c)\rightarrow \mathbb {C}$ $M:=\sup _{z\in B_{R}(c)}|f(z)|$ $z\in B_{R}(c)$ $k\in \mathbb {N}$ $|f^{(k)}(z)|\leq {\frac {k!MR}{(R-|z-c|)^{k+1}}}.$ $f^{(n)}(0)=n!^{2}$ $r_{0},r_{1},r_{2},\dots$ $f\colon \mathbb {R} \to \mathbb {R}$ $f^{(n)}(0)=r_{n}$ $n\in \mathbb {N} _{0}$ $f$ $D$ ${\overline {B_{R}(a)}}\subset \mathbb {C}$ $A(r):=\max _{|z-a|=r}\operatorname {Re} (f(z)).$ $|f(z)|\leq {\frac {2|z-a|}{R-|z-a|}}A(R)+{\frac {R+|z-a|}{R-|z-a|}}|f(a)|$ $|f^{(n)}(z)|\leq {\frac {2^{n+2}n!R}{(R-|z-a|)^{n+1}}}(A(R)+|f(a)|)$ $n\in \mathbb {N}$ $f$ $U$ $0$ $\textstyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}$ $r>0$ $B_{\varrho }(0)\subset U$ $\sum _{n=0}^{\infty }|a_{n}|^{2}\varrho ^{2n}\leq \left(\max _{|z|=\varrho }|f(z)|\right)^{2}.$ $f$ $U$ ${\overline {B_{r}(c)}}\subset U$ $r>0$ $z_{1}\not =z_{2}$ $B_{r}(c)$ $\left|{\frac {f(z_{1})-f(z_{2})}{z_{1}-z_{2}}}\right|\leq \max _{|z-c|=r}|f'(z)|.$ $|f(z_{1})-f(z_{2})|=\left|\int _{z_{2}}^{z_{1}}f'(z)\mathrm {d} z\right|\leq |z_{1}-z_{2}|\max _{z\in [z_{1},z_{2}]}|f'(z)|\leq |z_{1}-z_{2}|\max _{|z-c|=r}|f'(z)|.$ $\max _{|z-c|=r}|f'(z)|$ $f(z)=az+b$ $[z_{1},z_{2}]$ $\left|{\frac {f(z_{1})-f(z_{2})}{z_{1}-z_{2}}}\right|\leq \max _{z\in [z_{1},z_{2}]}|f'(z)|.$ $\textstyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}$ ${\overline {\mathbb {E} }}$ $\int _{-1}^{1}|f(x)|\mathrm {d} x\leq {\frac {1}{2}}\int _{0}^{2\pi }|f(e^{it})|\mathrm {d} t.$ $f$ $H^{1}(\mathbb {E} )$ $\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {|a_{m}||a_{n}|}{m+n+1}}\leq \pi \sum _{n=0}^{\infty }|a_{n}|^{2}.$ $f$ $f(0)=0$ $f'(0)=1$ $g(z)={\frac {1}{f\left({\frac {1}{z}}\right)}}.$ $z=\infty$ $g(z)=z+\sum _{n=0}^{\infty }b_{n}z^{-n},$ $|z|>1$ $c_{m,n}$ $\mathrm {Log} \left({\frac {g(\zeta )-g(z)}{\zeta -z}}\right)=-\sum _{m,n>0}{\frac {c_{n,m}}{n}}z^{-m}\zeta ^{-n},\qquad |z|,|\zeta |\gg 1.$ $\lambda _{1},...,\lambda _{N}$ $\sum _{n=1}^{\infty }n\left|\sum _{m=1}^{N}c_{m,n}\lambda _{m}\right|^{2}\leq \sum _{n=1}^{N}n|\lambda _{n}|^{2}.$ $\left|\sum _{1\leq m,n\leq N}nc_{m,n}\lambda _{m}\lambda _{n}\right|\leq \sum _{n=1}^{N}n|\lambda _{n}|^{2}$ $\textstyle \varphi (z)=\sum _{n=1}^{\infty }\alpha _{n}z^{n}$ $B_{r}(0)$ $\varphi (0)=0$ $\psi (z)=\exp(\varphi (z))$ $B_{r}(0)$ $\psi (z)=\sum _{k=0}^{\infty }\beta _{k}z^{k}.$ $\sum _{k=0}^{\infty }|\beta _{k}|^{2}\leq \exp \left(\sum _{k=1}^{\infty }k|\alpha _{k}|^{2}\right),$ $\alpha _{k}={\tfrac {\omega ^{k}}{k}}$ $k\in \mathbb {N}$ $n=1,2,...$ $\sum _{k=0}^{n}|\beta _{k}|^{2}\leq (n+1)\exp \left({\frac {1}{n+1}}\sum _{m=1}^{n}\sum _{k=1}^{m}\left(k|\alpha _{k}|^{2}-{\frac {1}{k}}\right)\right),$ $\alpha _{k}={\tfrac {\omega ^{k}}{k}}$ $1\leq k\leq n$ $|\omega |=1$ $n=1,2,...$ $|\beta _{n}|^{2}\leq \exp \left(\sum _{k=1}^{n}\left(k|\alpha _{k}|^{2}-{\frac {1}{k}}\right)\right),$ $\alpha _{k}={\tfrac {\omega ^{k}}{k}}$ $1\leq k\leq n$ $|\omega |=1$ $\operatorname {Re} (f)$ $\operatorname {Im} (f)$ $|f|$ $f$ $f$ $\{z\in \mathbb {C} \colon \operatorname {Re} (z)\geq 0\}$ $A,B>0$ $z$ $\operatorname {Re} (z)\geq 0$ $C,\gamma >0$ $|f(\pm ir)|\leq Ce^{(\pi -\gamma )r}$ $r\geq 0$ $f(n)=0$ $n\in \mathbb {N} _{0}$ $f\equiv 0$ $f(z)=\sin(\pi z)$ $|f(\pm ir)|\leq Ce^{-\gamma r}$ $r\geq 0$ $f\equiv 0$ $f$ $\mathbb {E}$ $k\in \mathbb {N} _{0}$ $\lim _{t\to 1^{-} \atop t\in \mathbb {R} }f^{(k)}(t)=0.$ $z=1$ $f$ $f\equiv 0$ $z=1$ $f$ $\mathbb {H} :=\{z\in \mathbb {C} \colon \operatorname {Im} (z)>0\}$ $f(z+2)=f(z)$ $z\in \mathbb {H}$ $f(-{\tfrac {1}{z}})=f(z)$ $z\in \mathbb {H}$ $f$ $\mathbb {H}$ $f$ $\operatorname {SL} _{2}(\mathbb {Z} )$ $f$ $\mathbb {H} :=\{z\in \mathbb {C} \colon \operatorname {Im} (z)>0\}$ $f(z_{k})=0$ $z_{k}=x_{k}+iy_{k}$ $|x_{k}|\leq 1$ $\textstyle \sum _{k=1}^{\infty }y_{k}=\infty$ $f$ $f\equiv 0$ $\alpha$ $\alpha =\operatorname {inf} \{c\in \mathbb {R} \mid |f(z)|\ll _{c}\exp(|z|^{c})\}.$ $e^{z},e^{-3z^{2}},\dots$ $f$ $\log(f)$ $\operatorname {Re} (\log(f(z)))\ll C|z|^{\alpha +\varepsilon }$ $\varepsilon >0$ $e^{g}$ $g$ $\textstyle f(z)=\sum _{k=0}^{\infty }c_{k}z^{k}$ $\limsup _{n\to \infty }n|c_{n}|^{\frac {1}{n}}\leq ce$ $f(z)^{-(c+\varepsilon )|z|}$ $f$ $\varepsilon >0$ $f\not =0$ $M(r):=\sup\{|f(z)|\colon |z|=r\}$ $r\geq 0$ $\limsup _{r\to \infty }{\frac {\log(M(r))}{r}}=\limsup _{n\to \infty }\left({\frac {n}{e}}\right)|c_{n}|^{\frac {1}{n}}=\limsup _{n\to \infty }\left|f^{(n)}(0)\right|^{\frac {1}{n}}$ $\alpha$ $(z_{n})_{n\in \mathbb {N} }$ $f$ $z_{n}\not =0$ $P_{1}\equiv 0$ $k>1$ $P_{k}(z)=z+{\frac {z^{2}}{2}}+\cdots +{\frac {z^{k-1}}{k-1}}.$ $k$ $k>\alpha$ $h$ $\alpha$ $f(z)=z^{m}e^{h(z)}\prod \left(1-{\frac {z}{z_{n}}}\right)e^{P_{k}({\tfrac {z}{z_{n}}})}.$ $m$ $f$ ${\mathcal {O}}({\mathcal {\mathbb {C} }})$ $\mathbb {C} [z]\subset {\mathcal {O}}({\mathcal {\mathbb {C} }})$ $\varepsilon \colon \mathbb {R} \to (0,\infty )$ $f\colon \mathbb {R} \to \mathbb {C}$ $g$ $x\in \mathbb {R}$ $x\to \sin(x)$ $\mathbb {R}$ $f_{1},\dots ,f_{n}$ $|f(z)|\leq C^{|z|^{\alpha }}$ $C>1$ $\alpha$ $\mathbb {Q} [f_{1},\dots ,f_{n}]$ $\partial :={\tfrac {\mathrm {d} }{\mathrm {d} z}}$ $P_{j}$ $\partial f_{j}=P_{j}(f_{1},\dots ,f_{n}).$ $w_{1},\dots ,w_{N}$ $f_{j}(w_{\ell })\in \mathbb {Q}$ $j=1,\dots ,n$ $\ell =1,\dots ,N$ $N\leq 4\alpha$ $\mathbb {Q} [z,e^{z}]$ $\partial$ $e^{w}$ $w\not =0$ $e^{w},e^{2w},e^{3w},\dots$ $f$ $D$ $f$ $c\in D$ $R>0$ $c$ $D$ $f$ $D$ $f\colon I\to \mathbb {R}$ $I=(a,b)$ $f$ $D\supset I$ $f$ $D\subset \mathbb {C}$ $M\subset D$ $D$ $f\colon M\to \mathbb {C}$ ${\widetilde {f}}\colon D\to \mathbb {C}$ $f$ ${\widetilde {f}}|_{M}=f$ $M=\mathbb {R}$ $D=\mathbb {C}$ $f(x)=e^{x}$ $\mathbb {C}$ ${\widetilde {(e^{z})}}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.$ $f$ $z_{0}$ $f$ $f$ $z_{1}$ $z_{0}$ $z_{0}$ $z_{1}$ $\gamma _{0}$ $\gamma _{1}$ $t\mapsto \gamma _{t}$ $0\leq t\leq 1$ $f$ $\gamma _{t}$ $f_{0}$ $f_{1}$ $f$ $\gamma _{0}$ $z_{1}$ $D\not =\emptyset$ $\mathbb {R}$ $z\in D\implies {\overline {z}}\in D$ $D_{\pm }:=\{z\in D\mid \pm \operatorname {Im} (z)>0\}$ $D_{0}:=D\cap \mathbb {R}$ $f\colon D_{+}\cup D_{0}\to \mathbb {C}$ $f|_{D_{+}}\colon D_{+}\to \mathbb {C}$ $f(D_{0})\subset \mathbb {R}$ ${\widetilde {f}}(z):={\begin{cases}f(z),&\qquad z\in D_{+}\cup D_{0},\\{\overline {f({\overline {z}})}},&\qquad z\in D_{-}\end{cases}}$ ${\widetilde {f}}\colon D\to \mathbb {C}$ $z\mapsto {\overline {z}}$ $D$ $D$ $f$ $c\in D$ $f$ $c$ $D$ $D$ $f$ $D$ $f$ $D'\supseteq D$ ${\widehat {f}}\in {\mathcal {O}}(D')$ ${\widehat {f}}|_{D}=f$ $D$ $\mathbb {C} _{-}$ $z\mapsto {\sqrt {z}}$ $z\mapsto \log(z)$ $z\mapsto {\sqrt {z}}$ $z\mapsto \log(z)$ $c\in \mathbb {C} _{-}$ $B_{|c|}(c)$ $B_{|c|}(c)\not \subset \mathbb {C} _{-}$ $z\mapsto {\sqrt {z}}$ $z\mapsto \log(z)$ $\mathbb {C} _{-}$ $z\mapsto {\sqrt {z}}$ $z\to \log(z)$ $U$ $f\in {\mathcal {O}}(U)$ $U\cap \mathbb {C} _{-}$ $z\mapsto {\sqrt {z}}$ $z\to \log(z)$ $\mathbb {C}$ $\mathbb {C} ^{\times }$ $\mathbb {E}$ $z\mapsto z$ $z\mapsto {\tfrac {1}{z}}$ $z\mapsto \sum _{n=0}^{\infty }z^{2^{n}}$ $D$ $f\colon U\rightarrow \mathbb {C}$ $f'(a)\not =0$ $a\in U$ $f$ $a\in V\subset U$ $f|_{V}\colon V\rightarrow f(V)$ $e^{0}=e^{2\pi i}=1$ $f$ $\mathbb {E}$ $f(0)=0$ $f'(0)=1$ $f$ $B_{\frac {1}{M}}(0)$ $B_{\frac {1}{2M}}(0)$ $D$ $f\colon D\rightarrow f(D)$ ${\overline {B_{r}(a)}}\subset D$ $w\in f(B_{r}(a))$ $f^{-1}(w)={\frac {1}{2\pi i}}\oint _{\partial B_{r}(a)}{\frac {\zeta f'(\zeta )}{f(\zeta )-w}}\mathrm {d} \zeta .$ $f$ $f(0)=0$ $0$ $f(z)=\sum _{n=1}^{\infty }a_{n}z^{n},$ $a_{1}=f'(0)\not =0$ $f^{-1}(w)=\sum _{k=1}^{\infty }b_{k}w^{k}$ $b_{k}={\frac {1}{ka_{1}^{k}}}\sum _{\ell _{1},\ell _{2},\ell _{3},\dots \geq 0 \atop \ell _{1}+2\ell _{2}+3\ell _{3}+\cdots =k-1}(-1)^{\ell _{1}+\ell _{2}+\ell _{3}+\cdots }{\frac {k(k+1)\cdots (k-1+\ell _{1}+\ell _{2}+\cdots )}{\ell _{1}!\ell _{2}!\ell _{3}!\cdots }}\left({\frac {a_{2}}{a_{1}}}\right)^{\ell _{1}}\left({\frac {a_{3}}{a_{1}}}\right)^{\ell _{2}}\cdots .$ $!$ $b_{1}={\frac {1}{a_{1}}}$ $b_{2}=-{\frac {a_{2}}{a_{1}^{3}}}$ $b_{3}={\frac {2a_{2}^{2}-a_{1}a_{3}}{a_{1}^{5}}}$ $b_{4}={\frac {5a_{1}a_{2}a_{3}-a_{1}^{2}a_{4}-5a_{2}^{3}}{a_{1}^{7}}}$ $\mathbb {C}$ $\mathbb {C}$ $\mathbb {C}$ $\mathbb {E}$ $\mathbb {C}$ $\mathbb {E}$ $h\colon \mathbb {C} \rightarrow \mathbb {E}$ $h(z)={\frac {z}{1+|z|}}.$ ${\overline {\mathbb {C} }}$ $\mathbb {E}$ $\mathbb {C}$ ${\overline {\mathbb {C} }}$ $f\colon \mathbb {E} \to D$ $D$ $f$ ${\overline {\mathbb {E} }}$ ${\overline {D}}$ $D$ $\varphi \colon \partial \mathbb {E} \to \mathbb {C}$ $\varphi (\partial E)=\partial D$ $f$ ${\overline {\mathbb {E} }}$ ${\overline {D}}$ $\partial D$ $\varphi$ $\mathbb {E}$ $D$ $U\subset \mathbb {C}$ $\operatorname {Aut} (U)$ $f\colon U\rightarrow U$ $\operatorname {Aut} (\mathbb {C} )$ $D$ $\mathbb {C}$ $f(z)=az+b$ $a\not =0$ $\operatorname {Aut} (\mathbb {C} )=\{f(z)=az+b\mid a,b\in \mathbb {C} ,a\not =0\}.$ $f$ $\infty$ $f^{-1}$ $0$ $f$ $f$ $\infty$ $\mathbb {C} ^{\times }:=\mathbb {C} \setminus \{0\}$ $\operatorname {Aut} (\mathbb {C} ^{\times })=\{f(z)=az\mid a\in \mathbb {C} ^{\times }\}\cup \{f(z)=az^{-1}\mid a\in \mathbb {C} ^{\times }\}.$ $\operatorname {Aut} (\mathbb {C} ^{\times })$ $\mathbb {C} ^{\times }$ $f\in \operatorname {Aut} (\mathbb {E} )$ $f(0)=0$ $f(z)=\zeta z$ $|\zeta |=1$ $\operatorname {Aut} (\mathbb {E} )$ $\mathbb {E} ^{\times }:=\mathbb {E} \setminus \{0\}$ $\operatorname {Aut} (\mathbb {E} ^{\times })=\{f(z)=az\ \colon |a|=1\}$ $\operatorname {Aut} (\mathbb {E} ^{\times })$ $\mathbb {H} \rightarrow \mathbb {E} ,z\mapsto {\tfrac {z-i}{z+i}}$ $\operatorname {Aut} (\mathbb {H} )=\left\{f(z)={\frac {az+b}{cz+d}}\ \colon a,b,c,d\in \mathbb {R} ,ad-bc=1\right\}.$ $f\in \operatorname {Aut} (\mathbb {H} )$ $M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \operatorname {SL} _{2}(\mathbb {R} )$ $f,g\in \operatorname {Aut} (\mathbb {H} )$ $f=g$ $M_{f}$ $M_{g}$ $M_{f}=\pm M_{g}$ $\operatorname {SL} _{2}(\mathbb {R} )$ $2\times 2$ $\operatorname {Aut} (\mathbb {H} )$ $\operatorname {SL} _{2}(\mathbb {R} )/\{\pm E_{2}\}$ $f\mapsto [M_{f}]$ $E_{2}$ $2\times 2$ $D$ $\operatorname {Aut} (D)=\{\mathrm {id} \colon D\to D\}$ $D$ $\mathbb {E} \setminus \{0,{\tfrac {1}{2}},{\tfrac {3}{4}}\}$ $\Sigma \ a_{n}z^{n}$ $f\colon D\to \mathbb {C}$ $0\in \partial D$ $k\in \mathbb {N} _{0}$ $\lim _{z\to 0 \atop z\in D}z^{-k}\left(f(z)-\sum _{n=0}^{k}a_{n}z^{n}\right)=0.$ $f$ $f(z)=\exp({\tfrac {1}{z}})$ $D=\mathbb {C} ^{\times }$ $f\in {\mathcal {O}}(D)$ ${\widehat {f}}\in {\mathcal {O}}({\widehat {D}})$ ${\widehat {D}}$ $0\in {\widehat {D}}\supset D$ $f$ ${\widehat {f}}$ $f$ $\textstyle f(z)\sim \sum _{n=0}^{\infty }a_{n}z^{n}$ $z\to 0$ $f'(z)\sim \sum _{n=1}^{\infty }na_{n}z^{n-1}$ $S$ $f(z)-\sum _{n=0}^{N}a_{n}z^{n}=O_{N}(z^{N+1})$ $z\to 0$ $S$ $N\in \mathbb {N}$ $O$ $z$ $D$ $0\in \partial D$ $z\in D$ $c_{\nu }$ $[c_{\nu },z]$ $D$ $f$ $D$ $f^{(n)}(0):=\lim _{z\to 0}f^{(n)}(z)$ $f$ $\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}z^{n}.$ $D$ $S$ $\Sigma \ a_{n}z^{n}$ $S$ $f$ $f\sim _{S}\sum a_{\nu }z^{\nu }.$ $\textstyle g(z)=\sum _{n=1}^{\infty }f(nz)$ $z\to 0^{+}$ $f$ $D_{\theta }:=\{re^{i\alpha }\colon r\geq 0,|\alpha |\leq \theta \}$ $f\colon \mathbb {C} \to \mathbb {C}$ $D_{\theta }$ $k\in \mathbb {N} _{0}$ $\varepsilon >0$ $f(w)=O(w^{-1-\varepsilon })$ $|w|\to \infty$ $D_{\theta }$ $a\in \mathbb {R}$ $N\in \mathbb {N} _{0}$ $\sum _{m=0}^{\infty }f(w(m+a))={\frac {1}{w}}\int _{0}^{\infty }f(x)\mathrm {d} x-\sum _{n=0}^{N-1}{\frac {B_{n+1}(a)f^{(n)}(0)}{(n+1)!}}w^{n}+O_{N}(w^{N})$ $w\to 0$ $D_{\theta }$ $B_{n}(x)$ $f$ $b_{-1}$ $a\in \mathbb {R} \setminus (-\mathbb {N} _{0})$ $\sum _{m=0}^{\infty }f(w(m+a))={\frac {b_{-1}\operatorname {Log} ({\frac {1}{w}})}{w}}+{\frac {b_{-1}(1-a)}{w}}\sum _{m=0}^{\infty }{\frac {1}{(m+a)(m+1)}}+{\frac {1}{w}}\int _{0}^{\infty }\left(f(x)-{\frac {b_{-1}e^{-x}}{x}}\right)\mathrm {d} x-\sum _{n=0}^{N-1}{\frac {B_{n+1}(a)f^{(n)}(0)}{(n+1)!}}w^{n}+O_{N}(w^{N})$ $w\to 0$ $D_{\theta }$ $F(s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}},$ $\operatorname {Re} (s)>1$ $|a_{n}|\leq C$ $n\in \mathbb {N}$ $F(s)$ $\operatorname {Re} (s)=1$ $\sum _{n=1}^{\infty }{\frac {a_{n}}{n}}=F(1).$ $\textstyle z\mapsto \sum \ a_{n}z^{n}$ $\partial \mathbb {E}$ $\textstyle \sum \ a_{n}$ $f(z)=\int _{C}k(w,z)g(w)\mathrm {d} w,$ $C$ $g$ $k$ $a\in \mathbb {C}$ $U$ $f\colon [a,\infty )\times U\to \mathbb {C}$ $F(z):=\int _{a}^{\infty }f(t,z)\mathrm {d} t$ $U$ $\textstyle \lim _{b\to \infty }\int _{a}^{b}f(t,z)\mathrm {d} t$ $U$ $z\mapsto f(t,z)$ $t\in (a,\infty )$ $F$ $U$ $F^{(k)}(z)=\int _{a}^{\infty }{\frac {\partial ^{k}}{\partial z^{k}}}f(t,z)\mathrm {d} t$ $k\in \mathbb {N}$ $f\colon \mathbb {R} \to \mathbb {C}$ $|x|\to \infty$ ${\hat {f}}\colon \mathbb {R} \to \mathbb {C}$ ${\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }\mathrm {d} x$ $f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )e^{2\pi i\xi x}\mathrm {d} \xi .$ ${\hat {f}}$ $|{\hat {f}}(\xi )|\leq Ae^{-2\pi a|\xi |}$ $a,A>0$ $f$ $\mathbb {R}$ $f$ $|f(x)|\leq {\tfrac {B}{1+x^{1+\varepsilon }}}$ $B,\varepsilon >0$ $x\in \mathbb {R}$ $f$ $|f(z)|\leq Ae^{2\pi M|z|}$ $A,M>0$ $z\in \mathbb {C}$ ${\hat {f}}$ $[-M,M]$ $f$ ${\hat {f}}$ ${\hat {f}}(\xi )=0$ $f$ $\{z\in \mathbb {C} \colon \operatorname {Im} (z)\geq 0\}$ $x\mapsto e^{-\pi x^{2}}$ $f$ $a,b>0$ ${\hat {f}}$ $|{\hat {f}}(\xi +i\eta )|\leq c'e^{-a'\xi ^{2}+b'\eta ^{2}}$ $a',b',c'>0$ ${\mathcal {M}}(f)(s):=\int _{0}^{\infty }f(t)t^{s-1}\mathrm {d} t.$ $f$ $(0,\infty )$ $f(t)=O(t^{-\alpha })$ $t\to 0^{+}$ $f(t)=O(t^{-\beta })$ $t\to \infty$ $s\mapsto {\mathcal {M}}(f)(s)$ $f$ $t\to \infty$ $t\to 0^{+}$ ${\mathcal {M}}(f)$ $\mathbb {C} \setminus \{-\alpha _{1},-\alpha _{2},-\alpha _{3},...\}$ $m_{n}+1$ $s=-\alpha _{n}$ ${\frac {(-1)^{m_{n}}m_{n}!}{(s+\alpha _{n})^{m_{n}}}}.$ ${\mathcal {M}}(f)$ $f(t)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\mathcal {M}}(f)(s)t^{-s}\mathrm {d} s,$ $f\colon [0,\infty )\to \mathbb {C}$ ${\mathcal {L}}(f)(z):=\int _{0}^{\infty }f(t)e^{-zt}\mathrm {d} t.$ $t\geq 0$ $|f(t)|\leq Ae^{Bt}$ $A>0$ $B\in \mathbb {R}$ $z\mapsto {\mathcal {L}}(f)(z)$ $\{z\in \mathbb {C} \colon \operatorname {Re} (z)>B\}$ $f(t)=b_{0}+b_{1}t+{\frac {b_{2}}{2!}}t^{2}+\cdots +{\frac {b_{N-1}}{(N-1)!}}t^{N-1}+O(t^{N})$ $t\to 0^{+}$ ${\mathcal {L}}(f)(z)={\frac {b_{0}}{z}}+{\frac {b_{1}}{z^{2}}}+{\frac {b_{2}}{z^{3}}}+\cdots +{\frac {b_{N-1}}{z^{N}}}+O\left({\frac {1}{\operatorname {Re} (z)^{N+1}}}\right)$ $\operatorname {Re} (z)\to \infty$ $f\colon [0,\infty )\to \mathbb {C}$ ${\mathcal {L}}(f)$ $\{z\in \mathbb {C} \colon \operatorname {Re} (z)\geq 0\}$ $\int _{0}^{\infty }f(t)\mathrm {d} t={\mathcal {L}}(f)(0).$ $f\colon [0,\infty )\to \mathbb {R}$ ${\mathcal {L}}(f)$ $\{z\in \mathbb {C} \colon \operatorname {Re} (z)>1\}$ $C$ $z\mapsto {\mathcal {L}}(f)(z)-{\frac {C}{z-1}}$ $\{z\in \mathbb {C} \colon \operatorname {Re} (z)\geq 1\}$ $f(t)e^{-t}\,\,\,{\overset {t\to \infty }{\longrightarrow }}\,\,\,C.$ ${\mathcal {O}}(D)$ $\mathbb {C}$ ${\mathcal {O}}(D)$ ${\mathcal {M}}(D)$ ${\mathcal {O}}(D)$ ${\mathcal {O}}(D)$ $\mathbb {Z}$ $\mathbb {Z} [i]$ $A\subset D$ ${\mathfrak {a}}:=\{f\in {\mathcal {O}}(D)\mid f\ {\text{verschwindet fast überall auf}}\ A\},$ ${\mathcal {O}}(D)$ ${\mathcal {O}}(D)$ $u,v$ $\omega \in {\mathcal {O}}(D)$ $u\omega ^{-1},v\omega ^{-1}\in {\mathcal {O}}(D)$ $a,b\in {\mathcal {O}}(D)$ $au+bv=1.$ ${\mathfrak {b}}\subset {\mathcal {O}}(D)$ ${\mathcal {O}}(D)$ ${\mathfrak {b}}\subset {\mathcal {O}}(D)$ ${\mathfrak {b}}$ ${\mathfrak {b}}$ ${\mathfrak {b}}$ $f_{n}\in {\mathfrak {b}}$ ${\mathfrak {b}}$ $\mathbb {C}$ $\mathbb {C}$ $D,{\widehat {D}}$ $\mathbb {C}$ $\varphi \colon {\mathcal {O}}(D)\to {\mathcal {O}}({\widehat {D}})$ $h\colon {\widehat {D}}\to D$ $\varphi (f)=f\circ h$ $f\in {\mathcal {O}}(D)$ $h=\varphi (\mathrm {id} _{D})\in {\mathcal {O}}({\widehat {D}})$ $\varphi$ $h$ $D$ ${\widehat {D}}$ ${\mathcal {O}}(D)\cong {\mathcal {O}}({\widehat {D}})$ $\mathbb {C}$ $\mathbb {C}$ $\varphi \colon {\mathcal {O}}(D)\to {\mathcal {O}}({\widehat {D}})$ $(f_{n})_{n\in \mathbb {N} }\in {\mathcal {O}}(D)$ $\varphi (f_{n})_{n\in \mathbb {N} }\in {\mathcal {O}}({\widehat {D}})$ $D$ ${\widehat {D}}$ $\varphi \colon {\mathcal {M}}(D)\to {\mathcal {M}}({\widehat {D}})$ $\mathbb {C}$ $h\in {\mathcal {O}}({\widehat {D}})$ $\varphi (f)=f\circ h$ $n$ $a_{n}$ $a_{n}=O(n^{A})$ $A$ $s$ $\operatorname {Re} (s)>A+1$ $F(s):=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}$ $\sigma _{0}$ $\operatorname {Re} (s)>\sigma _{0}$ $\operatorname {Re} (s)>\sigma _{0}$ $\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}},\qquad \operatorname {Re} (s)>1,$ $\mathbb {C} \setminus \{1\}$ $\{z\in \mathbb {C} \mid \operatorname {Im} (z)>0\}$ $f$ $\Gamma \subset \operatorname {SL} _{2}(\mathbb {Z} )$ $\mathbb {Q} \cup \{i\infty \}$ $f(z)=\sum _{n=0}^{\infty }a_{n}q^{\frac {n}{N}},\qquad q:=e^{2\pi iz}$ $N$ $k\in \mathbb {Z}$ $a_{n}$ $F(q)=\sum _{n=0}^{\infty }a_{n}q^{n},$ $|q|>1$ $a(n)$ $e^{n^{\delta }}$ $a_{n}={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {F(q)}{q^{n+1}}}\mathrm {d} q,$ $n$ $r=r_{n}$ $r_{n}\to 1$ $n\to \infty$ $a_{n}$ $F(q)$ $q=1$ $q=1$ $a_{n}$ $F(q)$ $q=1$ $a_{n}$ $p$ $p(n)\sim {\frac {1}{4{\sqrt {3}}n}}\exp \left(\pi {\sqrt {\frac {2n}{3}}}\right),\qquad n\to \infty .$ $\sum _{n=0}^{\infty }p(n)q^{n}=\prod _{n=1}^{\infty }{\frac {1}{1-q^{n}}}.$ $p(n)$ $D\subseteq \mathbb {C} ^{n}$ $f\colon D\to \mathbb {C}$ $w=(w_{1},\dotsc ,w_{n})\in D$ $\Delta (w;r_{1},\dotsc ,r_{n})\subset D$ $f(z)=\sum _{k_{1},\dotsc ,k_{n}=0}^{\infty }a_{k_{1},\dotsc ,k_{n}}(z_{1}-w_{1})^{k_{1}}\cdots (z_{n}-w_{n})^{k_{n}}$ $z\in \Delta (w;r_{1},\dotsc ,r_{n})$ $z$ $a_{k_{1},\dotsc ,k_{n}}\in \mathbb {C}$ $f\colon D\to \mathbb {C}$ $j$ $z_{j}$ $\textstyle {\frac {\partial }{\partial z^{j}}}$ $\textstyle {\frac {\partial }{\partial {\overline {z}}^{j}}}$ $f\colon D\to \mathbb {C}$ $D\subseteq \mathbb {C} ^{n}$ $f$ $f$ $f$ $f$ $\textstyle {\frac {\partial }{\partial {\overline {z}}^{j}}}f=0$ $j=1,\dotsc ,n$ $f=(f_{1},\dotsc ,f_{m})\colon D\to \mathbb {C} ^{m}$ $D\subseteq \mathbb {C} ^{n}$ $f_{j}\colon D\to \mathbb {C}$ $f\colon D\to \mathbb {C} ^{m}$ $f\colon U\to \mathbb {C}$ $n$ $n$