The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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Condition to be polynomial.

Let $f$ be entire function on complex plane. If for any sequence $z_n$ in $\mathbb{C}$ satisfying $z_n \to \infty, f(z_n) \to \infty$ , then $f$ is nonconstant polynomial. I think so $f \to \...
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3answers
73 views

$e^{a 2\pi i} = (e^{2\pi i})^a$.

When $a$ is any real number , Is it true $e^{a 2\pi i} = (e^{2\pi i})^a$ ? The reason why I ask this question is that I met this situation wheter this equality hold in Calculating Integral in Complex ...
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3answers
74 views

Trying to solve improper integral

I've been trying to solve this $$ \int_{-\infty}^\infty {\sin(x)\over x+1-i }dx $$ using residue theorem. I've tried using a square contour pi, pi+pii, -pi+pii, pi and half a circle but with the ...
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0answers
78 views

Normal family theorem regarding meromorphic functions (Schiff, Joel)

I have a question regarding Theorem 3.3.1 from pages 76-77 in Joel Shiff's book Normal Families. The theorem is stated as such: $\textit{Let} \, \, \mathcal{F}$ $\textit{be a family of meromorphic ...
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26 views

Maximum Principle variation.

Let $f$ be a non-constant holomorphic function in a bounded open connected set $\Omega$ in $\mathbb{C}$. Let $M:=\lim \sup_{n \to \infty} |f(z_n)|$ for every sequence ${z_n}$ in $\Omega$ which ...
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58 views

Real integral using a contour integral

I am going to calculate $\int_{-\infty}^{\infty}\dfrac{x \sin \pi x}{x^2+2x+5}dx$ So I have to compute the following limit $\lim_{R \to \infty}\int_{C_1}\dfrac{z \sin \pi z}{z^2+2z+5}dz$ where $C_1$ ...
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1answer
104 views

Invariance of subharmonicity under a holomorphic map

If $f:U_1\rightarrow U_2$ is holomorphic and $u$ is subharmonic on $U_2$, then prove that $u\circ f $ is subharmonic on $U_1$. I know how to prove the same argument with $f$ being conformal. In that ...
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39 views

Complex curve integral $\frac{1}{1+z^2}$

I want to calculate $\int_\gamma\frac{1}{1+z^2}\,\mathrm dz$ where $\gamma = \delta B(i,1)$, circle with radius $1$ around $i$. So i have $\gamma(t) = i+\exp(it),\,t \in [0,2\pi]$ with $$\int_\gamma \...
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2answers
47 views

Inverting a fraction

I'm doing a complex analysis question and it is as follows $$\int\limits_{c}\frac{1}{z(e^z-1)}dz$$ Using Maclaurin Series I simplified this expression upto this point: $$f(z) = \frac{1}{z(e^...
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1answer
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Calculate residues in all singularities

I need to calculate residues in all singularities of $f(z) = e^{z^2 + \frac{1}{z^2}}$. I found that point $z_0 = 0$ is a pole, but i can`t find an order of it. I tried to calculate the derivatives of $...
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1answer
46 views

There is not an holomorphic function on a bounded domain such that $\lim_{z \to w} f(z)= \infty$ for every $w \in Fr(\Omega)$

I had to solve the following question in an exam: Prove that there is not a holomorphic function on a bounded domain $\Omega$, $f:\Omega \to \mathbb{C}$ such that $\lim_{z \to w} f(z)= \infty$ ...
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34 views

Isolated singularites of $f(z)=\frac{\log(1+z)}{z^2}$?

Let $f:\mathbb{C}\setminus\{0\}\to\mathbb{C}$ be the holomorphic function given by $$f(z)=\frac{\log(1+z)}{z^2}.$$ I know that $z_0=0$ is a pole of degree $1$. My question: What about those points ...
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29 views

Pdes -theoretical answer

Question.Let $Ω$ be a bounded Connected on $R^3$ with smooth boundary $\partial{Ω}$.Let $u$ be a harmonic function on $Ω$ with continuous derivatives on $Ω\cup \partial{Ω}$ prove that. $$\iint_V \ {\...
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0answers
8 views

Hints to write an example of the Implicit function theorem involving $\Re\zeta(s)$ and $\Im\zeta(s)$ for $0<\Re s<1$

I would like to explore the following function, and refresh my knowledges of analysis in several real variables. I know the statement of the Implicit function theorem (previous is the Wikipedia Page), ...
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15 views

An example of a bounded domain $\Omega\subset \left\{ 0<\Re s< 1\right\} $ for which $\Re \zeta(s)$ is non-negative

Denoting the complex variable $s=\sigma+it$ (and we know that $\mathbb{C}$ and $\mathbb{R}^2$ are isomorphic, thus $s\equiv(\sigma,t)\in\mathbb{R}^2$) one has for $0<\Re s=\sigma<1$ that $$\zeta(...
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2answers
48 views

Uniform convergence of n-fold composition using Schwarz lemma

Let $f$ be an analytic function mapping the unit disk $\mathbb D$ to itself with $f(0) = 0$ and $|f'(0)| < 1$. Let $f^{n} = f \circ f \circ \dots \circ f$ be the function obtained by composing $f$ ...
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4answers
51 views

Help understanding the transform $f(z)=\frac{2z+1}{5z+3}$

Consider the Mobius transformation $f(z)=\frac{2z+1}{5z+3}$. Show that this function maps the upper half plane and the lower half planes onto themselves. What can you say about the left and right half ...
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1answer
33 views

To prove ring properties of analytic functions

Let $R$ be the ring of entire functions $f: \mathbb{C} \rightarrow \mathbb{C}$ that are analytic at every point of $\mathbb{C}$ with respect to point-wise addition and multiplication. Then show that ...
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46 views

Entire function $f$ and a function $g = \overline{f(\bar z)}$

Let $f$ be an entire function $f$ and a function $g = \overline{f(\bar z)}$. Then which of the followings are true. a) if $f(z) \in \mathbb R$ for all $z\in \mathbb R$, then $f =g$. b) if $f(z) \...
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48 views

To solve multivariate polynomial equations

For a system of multivariate polynomial equations like this: $$ \left( {\begin{array}{*{20}c} {\frac{{124}} {3}} & { - 24} & {\frac{{ - 68}} {3}} & {\frac{{68}} {3}} \\ {32} & {...
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How to solve $\int_{-\infty}^{\infty} \frac {sin(t)}{t^2+1} dt$?

I'm considering here the fact that $$\lim\limits_{R\to\infty} \int_{\Gamma_R} \frac {e^{iz}}{z^2+1} dz=0$$ , where $\Gamma$ is a contour defined as a semicircle centred about the origin, of radius $...
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Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$ \theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) , $$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
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Upper bounds for the modulus of $f(s)=\prod_{n=1}^\infty \left( 1-\frac{\sigma(n)}{n^3}s\right)$

Let the complex variable $s=x+iy$, and $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisors function (is a known multiplicative function in number theory, for example $\sigma(1)=1$ and $\sigma(6)=1+2+3+6=...
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1answer
19 views

Edge of convergence radius behaviour

what do i have to do if the excercise is "examine the behaviour at the edge of the convergence radius". I don't even know if that's the correct translation, please fix if not. For my actual ...
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0answers
151 views

How did Hecke come up with Hecke-operators?

I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d,...
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1answer
79 views

Is $|z^2|$ entire function?

I want to know whether $|z^2|$ is entire function or not. If I am not wrong then $z^2$ is entire but $|z|$ is not entire (Am I wrong here?) So, now how to say anything whether $|z^2|$ is entire or ...
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1answer
164 views

Is there a way to prove that i²=-1? [duplicate]

I have 4 questions regarding the imaginary and complex numbers. (And some ideas) My questions are about the way that I’m trying to come up with a proof to the equation i²=-1 (and from there maybe ...
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0answers
25 views

Basics on Schwarz -Christoffel Integral

I've just began to study the Schwarz-Christoffel integral, but I'm having trouble to understand some very basic points. For example, take $S:\mathbb{H}\to \mathbb{C}$ (where $\mathbb{H}:=\{z\in \...
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1answer
26 views

Is there a meaningful measure on analytic functions?

Let $\mathcal{B}$ be the functions analytic on the unit disk and continuous on its boundary. With the supremum norm this becomes a Banach space. Is there any way to define a meaningful measure on ...
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1answer
47 views

Flip integral boundaries of Delta function to get contradiction

Look at this equation: $\int_{-\infty}^{+\infty}dx\int_{-\infty}^{+\infty}dy\delta\left(x-y\right)f(y)=\int_{-\infty}^{+\infty}dxf(x)$ If I flip integration boundaries of both integrals, minus ...
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1answer
33 views

Seeming Contradiction With Complex Exponents

I was fooling around while waiting for a page to load and came across the following "contradiction". Let $x=(-1)^{i}$. Then $x^{i}=(-1)^{i\cdot i}=(-1)^{-1}=-1$. Thus, $x=\left(x^{i}\right)^{-i}=(-1)^...
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1answer
48 views

Improper integral complex analysis $\int_{-\infty}^\infty \frac{e^{ax} \, dx}{\cosh(x)}$

I tried the following problem but I don't think I got the right answer. I checked it by substituting $a=\frac{1}{2}$ into the integral and putting that through Wolfram Alpha but it didn't match the ...
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0answers
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Holomorphic function restricted to curves

Supppose $F:B_{r}(z_0) \to \mathbb{C}$ holomorphic with $F(z_0) = F'(z_0) = 0$ and $F''(z_0) \neq 0$. Show that there exist two curves $\gamma_1, \gamma_2: [0, 1] \to B_{r}(z_0)$ such that: i) $\...
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2answers
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Complex function $f$ is either constant or unbounded, but maximum value still does exist even if $f$ is not constant?

In Complex Variables and Applications, Brown & Churchill (9th edition), I stumbled upon a chapter which got me somewhat confused. On page 175 of the book, there is the theorem, which states the ...
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2answers
73 views

How to read that sum?

My exercise is to show that $\sum_{n=0}^{\infty}{N+n \choose n}z^{n} =1/(1-z)^{N+1}$ where $N\in\{0,1,2,\ldots\}$ and $z\in\mathbb{C}$ with $\lvert z \rvert <1$. Now, this doesn't even work with $...
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1answer
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Modulus of function values on circle is necessarily less than or equal to modulus at centre?

Consider a complex function $f(z)$, a circle $C_\rho$ with radius $\rho$, centred at $z_0$, which can be defined as $z=z_0+\rho e^{i\theta}$ $(\theta \in [0, 2\pi]$). In the book on complex variables ...
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1answer
44 views

Power Series Extensions into the Complex Plane

I'm working through Complex Analysis by Serge Lang, and I came across a part that I can't figure out on my own, and I was hoping that I can get help here. This is the theorem in question: Theorem ...
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2answers
65 views

Proof that $\overline{P(z)} = P(\overline{z})$ for polynomial $P$ with real coefficients

Let $$ a_0, a_1, a_2, a_3, \ldots , a_n \quad (n \ge 1)$$ denote real numbers, and let $z$ be any complex number. With the aid of $$ \overline {z_1 +z_2+ \ldots +z_n} = \overline z_1 +\overline z_2+ \...
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2answers
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Prove $\sum \frac{\cos nz}{n!}$ converges on compact sets. [closed]

Prove that $$\sum \displaystyle\frac{\cos nz}{n!}$$ converges on compact subsets of complex plane.
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Inequality in proof of thm. 10.30 Rudin's Real and Complex Analysis.

I'm trying to understand the proof of theorem 10.30 and I'm missing something at the very beginning. Theorem 10.30 Suppose $\varphi\in H(\Omega)$, $z_0\in \Omega$, and $\varphi'(z_0)\neq 0$. ...
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Specific fucnction has 11 different zeros

Let $f : \mathbb{C} \to \mathbb{C}$ be given by $$ f(z) = z^{11} + 4 e^{z + 1} - 2 $$ Show that $f$ has 11 different zeros in the annulus $\{z \in \mathbb{C} : 1 < |z| < 3\}$. This is an old ...
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2answers
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Show complex equation of closed curve integral

I need to show this equation: $$\frac{1}{2ia} \cdot \oint _{\gamma } \frac{e^{iz}}{z-ia}dz = \frac{e^{-a}}{2ia} \cdot \oint _{\gamma } \frac{1}{z-ia}dz$$ I have an hint to using Taylor. I have no ...
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1answer
34 views

An entire function of irrational order

It's easy to construct an entire function of order $\frac{p}{q}$ for any positive integers $p,q$. But is there an example of an entire function of irrational order ($\sqrt2$ for example)?
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Complex Analysis Problem (Argument principle or Rouché's Theorem ?)

My problem: Let f be analytic in $\overline{B(0;R)}$ with $f(0) = 0$, $f'(0)\ne0$ and $f(z)\ne0$ for $0 < |z| \le R$. Put $\rho = \min\limits_{|z|=R} |f(z)| > 0$. Define $N: B(0; \rho) \...
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Fourier transform spherically symmetric function with complex constant

In Gradshteyn's section 17.24 on Fourier transform pairs for spherically symmetric functions, the third entry relates $\frac{e^{-ar}}{r}$ and $\sqrt{\frac{2}{\pi}}\frac{1}{(a^2 + k^2)^2}$. I think ...
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1answer
26 views

biholomorphic on unit disk

Let $D$ be the unit disk and $f: D\rightarrow G$, $\; p_1$ the maximum value of $dist(f(z),f(0))\;$ and $p_2$ the minimum value of $dist(f(z),f(0))$ for $z\in \partial \bar G$ Prove that : $|f(z)-...
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1answer
51 views

Pullback of euclidean metric on the disc.

$\newcommand{\Im}{\operatorname{Im}}\newcommand{Re}{\operatorname{Re}}$Consider the biolomorphism $$f : D \to H$$ where $H$ is the complex upper hyperplane $\{\Im(z) > 0\}$ and $D$ is the unic disc,...