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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857 views

Complex Analysis- Research

I was interested in doing some research in complex analysis. I already have a basic understanding of the subject. i.e. I read Saff and Snider's book "Fundamentals of Complex Analysis". But now I would ...
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1answer
133 views

Analytic function in open connected set that is bounded by another analytic function

Let $G$ be an open connected set and $f, g$ analytic functions on $G$. If $|f|\le |g|$ then there exists an analytic function $h$ such that $f(z)=h(z)g(z)$. We know $|f/g|\le 1$ everywhere in ...
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2answers
219 views

Interpolation of analytic function on unit disk

Been thinking about this problem for a long time without any progress, can someone help? Consider a bounded function $f: \mathbb{D} \rightarrow \mathbb{D}$ with the following property : for every ...
6
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1answer
130 views

$f:\mathbb{C}\rightarrow\mathbb{C}$ is entire function such that $g(z)=f(1/z)$ has a pole at $z=0$, then is $f$ surjective?

$f:\mathbb{C}\rightarrow\mathbb{C}$ is entire function such that $g(z)=f(1/z)$ has a pole at $z=0$, then is $f$ surjective? I can prove that $f$ will be a polynomial. and hence $f$ is surjective. am ...
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2answers
84 views

$z_{0}$ is a zero of order $m$. Prove that $|z_{0}|^m\geq|f(0)|$ where $f$ is analytic in the unit disc

$f$ is an analytic function in the unit disc, so that $|f(z)|\leq1$. Let $z_{0}$ be a zero of order $m$. Prove that $|z_{0}|^m\geq|f(0)|$ My approach: We can write: $$(1) \ \ \ ...
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1answer
247 views

any two simply connected open set in the plane R^2 are diffeomorphic

Prove that any two simply connected open set in the plane R^2 are diffeomorphic. I know that in the complex plane any simply connected open set is diffeomorphic to either complex plane or open unit ...
3
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1answer
111 views

Domain of bijectivity of function $f:\mathbb{C}\rightarrow\mathbb{C}$

There is a type of problems in my course in Complex analysis that I don't fully understand them. Given function $f:\mathbb{C}\rightarrow\mathbb{C}$, $f(z)=z^2$. You must specify the analytic and ...
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1answer
127 views

Holomorphic function with uncountable set of zeros?

I am aware that on a region, this is only possible if the function is identical to zero. If the domain is not a region, is it possible to have a non-trivial holomorphic function with uncountable zero ...
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2answers
119 views

Computing $\lim_{s \to 1} \Gamma \left(\frac{1-s}{2}\right) (s-1)$

I want to evaluate the following limit: $$\lim_{s \to 1}\; \Gamma \left( \frac{1-s}{2} \right) (s-1).$$ I know that the gamma function has simple poles at $-n$ for $n \in \mathbb{N}_0$ with residue ...
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2answers
179 views

$f^2+2f+1$ is a polynomial implies that $f$ is a polynomial

This is a complex analysis problem. Let $f$ be an entire function and $f^2+2f+1$ be a polynomial. Prove that $f$ is a polynomial.
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0answers
88 views

Modular form weight 0

Why is an entire modular form of weight 0 must be a constant? In particular, does a function defined on the upper half plane that is analytic everywhere, including i/infty, imply boundedness? Can we ...
2
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3answers
160 views

Prove inequality in complex numbers in an unit circle

Given $|\omega| < 1$, $\omega \neq 0$ and $|z| < 1$. Prove inequality: $$\frac{|\frac{|\omega|}{\omega}z+1|}{|1-z \bar \omega|} \le \frac{2}{1-|z|}$$ It is simple but i have problems with it. ...
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3answers
154 views

Integrate $\int_{C} \frac{1}{r-\bar{z}}dz$ - conflicting answers

In an homework exercise, we're asked to integrate $\int_{C} \frac{1}{k-\bar{z}}dz$ where C is some circle that doesn't pass through $k$. I tried solving this question through two different ...
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1answer
166 views

on the convergence exponent of zeros of entire functions

Let $\{z_j\}$ be the sequence of zeros on an entire function $f$. We define the convergence exponent of $\{z_j\}$ as $$b=\inf\left\{\lambda>0\ \text{s.t.}\ ...
5
votes
1answer
279 views

Complex infinite sum convergence problem

Suppose that the complex infinite sum $ \sum_{n=1}^{\infty}(-1)^{n}Z_n$ converges. Define $A \subset \mathbb{C}$ by $A=\{{z\in\mathbb{C}\mid\exists ...
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votes
1answer
62 views

Prove, that $f: S_{k} \rightarrow \mathbb{C}-\{0\}$ is a surjection.

For $k>0$ define $S_{k} := \{z=x+iy\in\mathbb{C}\mid |z|<k,\ \ \ \ k\cdot y>|x|\}\subset\mathbb{C}$ Let $f(z)=\exp(1/z)\ \ \ \text{for}\ \ \ z\neq 0$ Prove, that $f: S_{k} \rightarrow ...
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1answer
241 views

Average value of a complex valued function on a circle.

The following is an exercise from Complex Analysis by Stephen Fisher. Fix a complex number $a$ and a positive real number $R$. Suppose $u$ is a function defined on the circle of radius $R$ centered ...
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1answer
60 views

A problem on complex integration where $\gamma$ is a closed and continuously differentiable path in the upper half plane

Let $\gamma$ be a closed and continuously differentiable path in the upper half plane $\{z \in \mathbb{C} : z = x + iy,\; x, y \in\mathbb{ R}, \;y > 0 \}$ not passing through the point $i$. ...
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1answer
164 views

complex integration along closed contour

Let $I_r= \int dz/(z(z-1)(z-2))$ along $C_r$, where $C_r = \{z\in\mathbb C : |z|=r\}$, $r>0$. Then a. $I_r= 2\pi i$ if $r\in (2,3)$ b. $I_r= 1/2$ if $r\in (0,1)$ c. ...
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1answer
190 views

if $f$ is entire then show that $f(z)f(1/z)$ is also entire

This is again for an old exam. Let $f$ be an entire function, show that f(z)f(1/z) is entire. How do I go about showing the above. Do I use the definition of analyticity?., Call g: f(z)f(1/z) and ...
2
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1answer
209 views

An application of Rouche's theorem involving $e^z$

Put $p_n(z)=\sum_{k=0}^n\frac{z^k}{k!}$. Show that for any $r>0$ and any $n\ge 0$, there exists a point $z_0$ with $|z_0|=r$ such that $|p_n(z_0)|=|e^{z_0}|$. This is actually the second part ...
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1answer
78 views

Characteristic function as limit of integrals.

I've seen the following claim in a book without proof and don't know why it holds. Let $a<b \in\mathbb{R}$. Then the integral $$\frac{1}{2\pi ...
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0answers
91 views

Integration of sine^2 w.r.t. some norm

Let $||x||$ be any norm over $\mathbb R^n$. Let $B_T$ the open ball with radius $T$ w.r.t. to our norm, i.e. all $x\in\mathbb R^n$ such that $||x||<T$. Let $n\in\mathbb N$. How much ...
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1answer
251 views

Is this proof valid

This is a clear version of an earlier closed question. Let: $x, u, y, v : \mathbb{C} \to \mathbb{R}$ be functions in the complex variable $s=a+ib$ defined by convergente series. I have this ...
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votes
2answers
67 views

Why function $j(\tau)$ has degree 1?

We have $$ j(\tau)=\frac{1}{q}+\sum_{n=0}^{\infty}a_nq^n, a_n\in\mathbb{Z},q=e^{2\pi i\tau} $$ Then it is said that because $j$'s only pole is simple, $j$ has degree 1 as a map ...
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1answer
146 views

Complex Variable, properties.

Let $ f $ be a non-constant entire function. Justify that $ f $ satisfies one of the following two statements: (a) For all $ w \in \mathbb{C} $, the equation $ f(z) = w $ has a solution. (b) For all ...
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1answer
165 views

How to calculate $\oint_{C}\frac{dz}{z(z-1)(z-2)}$ when $C$ is a circle around the origin with radius $1.5$?

I wish to calculate $\oint_{C}\frac{dz}{z(z-1)(z-2)}$ when $C$ is a circle around the origin with radius $1.5$. I guess that I should somehow apply Cauchy's integral formula here, but ...
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1answer
120 views

Value of Arg(f(z)) after rotation of z

An elementary problem asks for Arg($f(z)$) after a single complete counter-clockwise rotation of the point z about the origin, beginning at the point z = 2 and taking the angle there to be 0, with ...
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1answer
254 views

Holomorphic function $\varphi$ with fixed point $z_0$ such that $\varphi'(z_o)=1$ is linear?

This is an exercise in complex analysis: Let $\Omega\subset{\Bbb C}$ be open and bounded, and $\varphi:\Omega\to\Omega$ a holomorphic function. Prove that if there exists a point $z_0\in\Omega$ ...
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4answers
90 views

Minimum value of $|z-w|$ where $z,w \in \mathbb C$ such that $|z|=11$, and $|w+4+3i|=5$?

I was thinking about the problem: What is the minimum value of $|z-w|$ where $z,w \in \mathbb C$ such that $|z|=11$, and $|w+4+3i|=5$? My attempts: I notice that $|z-w| \geq |z|-|w|=11-|w|$. Also if ...
2
votes
1answer
64 views

What can you say about $f(z)$ with finitely many poles?

Assume that $f$ is analytic with finitely many poles {$z_1,z_2,...,z_n$}. At $z=z_i$, $f$ has a pole with multiplicy $m_j>0$. Suppose that $|f(z)\le C(1+|z|)^m$ for $|z|>R$. What can you say ...
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2answers
314 views

Finding $\frac{1}{2\pi}\int_{0}^{2\pi} \cos^{2n} x dx$

I have a question that asks me to find the value of $\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi} \cos^{2n} x dx$ $\ $ by considering the integral $$ \displaystyle \oint_{\gamma} \frac{1}{z}\left ( ...
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votes
1answer
104 views

sign of roots of a quadratic equation with complex coefficients.

Consider $x^2+ax+b=0$, where $x$ is the variable and $a,b$ are complex coefficients. Is there any condition on $a$ and $b$ which makes sure the roots of the equation have negative real parts?
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1answer
120 views

Let $f (z) $ be an entire function such that $|f (z)|≤K|z|$, $∀z∈\mathbb{C}$, for some $K>0$. If $f (1) =i$, then$f (i) $ is

Let $f (z) $ be an entire function such that $|f (z)|≤K|z|$, $∀z∈\mathbb{C}$, for some $K>0$. If $f (1) =i$, the value of $f (i) $ is (A) $ 1 $ (B)$-1$ (C) $i$ (D) $-i$ how can I able to ...
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2answers
189 views

Complex integral revision, this is just Cauchy's Theorem right?

(a) Give the definition of $e^z$ for a complex number $z = x+iy$ (2 marks) (b) Use the Cauchy-Riemann equations to prove that $f\colon \mathbb C \to \mathbb C$, $f(z) = e^{2z+i}$ is ...
0
votes
1answer
47 views

A Triangular Domain with Different Bounds on the Sides

I have the following problem: Let $K$ be the equilateral triangle centered at 0. Assume that $f$ is continuous on $K$ and analytic inside of $K$, and assume that $|f(z)|\leq 8$ on one of the sides ...
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2answers
269 views

Entire function with prescribed zeros

I want to construct an entire function which vanishes at points $n+in$ for all $n$ integers. I'm looking for the most simple entire function which satisfies this condition, in the sense that the ...
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1answer
154 views

complex zeros of an equation inside the unit ball

I found this exercise and I don't know where i do wrong: Let $a > e$ be a real number. Prove that the equation $a z^4 e^{−z} = 1$ has a single solution in $D(0, 1)$, which is real and positive. ...
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1answer
160 views

Exercise complex variable, series.

Get the power series expansion centered at the origin of the function f, and calculate the radius of convergence of the corresponding series in each of the following cases: ...
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3answers
129 views

$e^{i\theta_n}\to e^{i\theta}\implies \theta_n\to\theta$

How to show $e^{i\theta_n}\to e^{i\theta}\implies \theta_n\to\theta$ for $-\pi<\theta_n,\theta<\pi.$ I'm completely stuck in it. Please help.
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1answer
200 views

Sheaf of meromorphic functions on non-compact Riemann surfaces

Why does the first cohomology group $H^1(X, K)$ of the sheaf of meromorphic functions on a non-compact Riemann surface $X$ vanish?
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359 views

Prove that $\sum z^n/(1-z^n)$ is holomorphic on the unit disc.

Prove that the series $$f(z)=\sum_{n>1}\frac{z^n}{1-z^n}$$ converges in the unit disc $D=\{z:|z|<1\}$ and defines there a holomorphic function.
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2answers
36 views

Complex Variable. Linearly dependent. [duplicate]

Possible Duplicate: Holomorphic functions and limits of a sequence Let $\Omega$ a domain and $f,g$ holomorphic function in $\Omega$. Suposse that $\exists$ a sequence $\{a_n\}$ in $\Omega$ ...
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43 views

Proving that a holomorphic $f$ such that $\lim_{z\rightarrow0}f(z)=\lim_{z\rightarrow\infty}f(z)=\infty$ has a zero.

Let $f\in H(C\backslash\{0\})$ a function such that $$\lim_{z\rightarrow0}f(z)=\lim_{z\rightarrow\infty}f(z)=\infty.$$ Prove that $f$ have some zero in $C\backslash\{0\}$.
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1answer
106 views

Entire function. Prove that $f(\bar{z})=\overline{f(z)}, \forall z\in C$

Let $f$ a entire function: $f(R)\subset R.\;$ Prove that $f(\bar{z})=\overline{f(z)}, \forall z\in C$
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votes
1answer
365 views

order of growth of entire function

Show that $$f(z)=\frac{\sin\sqrt z}{\sqrt z}$$ is an entire function of finite order $\rho$ and determine $\rho$. I observed that the two determinations of the square root differ only for the signum. ...
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votes
0answers
129 views

winding number question

This is part of a proof from Banach algebra techniques in Operator theory by Ronald Douglas on page 170. Let $\epsilon>0$. Let $T$ be the unit circle and $\phi\in H^\infty+C(T)$. Choose $\psi\in ...
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votes
2answers
223 views

The minimum value of $|z+1|+|z-1|+|z-i|$ for $z \in \mathbb C?$

I was thinking about the following problem: How can i find the minimum value of $|z+1|+|z-1|+|z-i|$ for $z \in \mathbb C?$ There are four options which are $(a)2,(b)2\sqrt 2,(c)1+ \sqrt 3,(d)\sqrt ...
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1answer
121 views

About the problem related to first category (meager)

We have $X$ an $F$-space, $Y$ a subspace of $X$ whose complement is of the first category. Prove that $Y=X$. This problem have a hint: need to show that $Y $ intersect $x+Y$ $\forall x \in X$, but i ...
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1answer
381 views

Must a meromorphic function on a compact set have same number of zeros and poles?

Let $f:X\rightarrow\mathbb{C}\cup\{\infty\}$ be a meromorphic function while $X$ is compact. Must $f$ have same number of zeros and poles?