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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23 views

Show that $f(V_0)\cap f(V_1)\neq\emptyset$

Let $U$ be a connected subset of $\mathbb C$ and $z_0,z_1\in U$ and if $f$ is holomorphic on $U\setminus\{z_0\}$, with essential singularity in $z_0$, prove that for any subsets $V_0,V_1$ of $U$ ...
5
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2answers
34 views

let $f$ be holomorphic on the unit sphere and $|f(z)| = 1$ for $|z| = 1$ and $f(-1) = 1$. Furthermore $f$ has no zero's, determine $f$

let $f$ be holomorphic on the unit sphere and continous on the closure, suppose $|f(z)| = 1$ for $|z| = 1$ and $f(-1) = 1$. furthermore $f$ has no zero's, determine $f$. So far i know with the ...
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1answer
24 views

Find value of complex function at a point

Let $f(z)$ be analytic in $ D = \{z \in \Bbb C : |z| < 1\}$, and $f(z) = 1$ when $Im(z) = 0$ and $-\frac{1}{2} \leq Re(z) \leq \frac{1}{2}$. What is the value of $f(\frac{1}{2}+i\frac{1}{2})$? I'm ...
3
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1answer
109 views

A UCLA Qualifying Complex Analyis Problem , possibly related to Phragmén-Lindelöf Theorem

Let $f$ be a bounded analytic function on the open right half plane such that $f(x) \to 0, x\to 0$ along the positive real axis. Suppose $0<\phi<\pi/2$. Prove that $f(z) \to 0, z \to 0$ ...
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0answers
36 views

Inequality on complex polynomial

For every $a\geq 0$, let $p_a(z)=1-z+az^3$. What is the maximal value of $a$ such that $$ p_a(|z|)\leq |p_a(z)| $$ for all $z\in \mathbb C$? EDIT: I claim that $a=\frac{4}{27}$ is the maximal value. ...
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1answer
125 views

$F$ entire with $\lim_{k \rightarrow \infty} F(z + N_{k}) = h(z)$ for every $h$ entire

Universal Entire Functions. Prove that there exists an entire function with the following "universal" property: Given any entire function $h$, there is an increasing sequence $\{ ...
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2answers
28 views

Taylor expansion of a complex function

Trying to find Taylor series of $$\frac{z^2}{(1+z)^2}$$ I write it in the form $1- \frac{2}{1+z} + \frac{1}{(1+z)^2}$ and I can find Taylor expansion for each factor, is there another method without ...
3
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1answer
172 views

Finite subgroups of $\operatorname{Aut}(\mathbb{P}^1)$

I would like to know all finite subgroups of $\operatorname{Aut}(\mathbb{P}^1)$. I am aware that any automorphism of $\mathbb{P}^1$ is given by a Möbius transformation $$ z\mapsto\frac{az+b}{cz+d} ...
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1answer
42 views

Show that $f$ is constant in $D(0,1)$. [on hold]

Considere this, $f: D(0,1) \to D(0,1)$ analytic. Suposse that $|f(z^2)|>|f(z)|$, for all $z \in D(0,1)$. Show that $f$ is constant in $D(0,1)$. Any help.
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2answers
40 views

Taylor series of $1\over z^2$

How to find the Taylor series of $1\over z^2$ near $2$ ( in the power of $z-2$) I have tried to write it in the form: $1\over ((z-2)^2+4z-4)$ But I reached nothing, any help please
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0answers
23 views

calculate complex integral $\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$

I don't know how to calculate this complex integral: $$\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$$
2
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1answer
45 views

Gaussian integral with a shift in the complex plane

$$ \int_{-\infty}^\infty e^{-(x+ia)^2} \text{d}x $$ where $a\in \mathbb{R}$. I don't know where to start but have reasons to believe the answer is $\sqrt{\pi}$. Namely $\int_{-\infty}^\infty ...
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0answers
15 views

Meromorphic complex function [on hold]

I want to find all function $f$ which are meromorphic in $C$ and satisfy $|f(z) - tan(z)| < 2$ for all $z$ which are neither poles of $f$ nor poles of $tan(z)$
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1answer
28 views

Taylor series in complex analysis [on hold]

I am working on finding the Taylor series of $$\frac1{az+b}$$ in powers of $z.$ How to start with it Any help in details...
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1answer
38 views

Convergence of complex power series $z^{n!}$ at boundary

I'm revising for an exam at the moment and I'm struggling with part of a question. I'm asked to find the radius of convergence of the series $\sum_{n=0}^{\infty }z^{n!}$ and then find where it ...
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1answer
21 views

Residue Calculus - Showing that the quotient of polynomials have integral $0$ in a simple closed contour in a special case.

I'm having difficulty understanding the solution to the following problem. In the solution below, I can't understand why since $b_m\neq 0$, the quotient of these polynomials is represented by a ...
3
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1answer
42 views

Does a weaker form of the mean value property already imply harmonicity for continuous functions?

If $u:\mathbb{C}\to \mathbb{R}$ is continuous and satisfies $u(z)=\frac{1}{2\pi}\int_0 ^{2\pi}u(z+\frac{e^{i\theta}}{n})d\theta$ for all $n\in \mathbb{N}$ and $z\in \mathbb{C}$, is $u$ harmonic? What ...
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0answers
22 views

Exactly one supporting line for a $C^1$ Jordan curve [on hold]

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a convex Jordan curve (closed, simple, continuous) that has $C^1$ regularity, with $\gamma '(t)\neq 0,\ \forall t\in [a,b]$. Prove that there is exactly one ...
2
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1answer
43 views

integral of harmonic function

I'm having trouble with this one: Let $u$ be a real-valued harmonic function on $D(0,1)$, and let $\gamma$ be a closed curve in that disk. Then $\int_\gamma u=0.$ I'm supposed to prove or disprove ...
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1answer
26 views

Antiholomorphic function

Let f be an antiholomorphic function in C. $z_0 \in C - C(0,1). $ Show that $\frac{1}{2 \pi i}\oint \frac {f(z)}{z-z_0} = \begin{cases}f(0) &\text{for } |z_0| < 1\\f(0) - f(\frac{1}{z_0}) ...
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2answers
25 views

Is any simply connected domain conformally equivalent to Cartesian product of unit disks?

By Riemann mapping theorem, any simply connected domain is conformally equivalent to the unit disk. Is any simply connected domain in the complex plane conformally equivalent to the Cartesian product ...
2
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0answers
37 views

Does convergence of power series on radius of convergence imply absolute convergence?

Let $R$ be radius of convergence of power seires $\displaystyle\sum_{k}a_kz^k$. If the power series converges for all $|z|=R$, can we say that it converges absolutely on the radius of convergence? I ...
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1answer
47 views

Why is $|e^{i \lambda z}| |e^{- \lambda y}|= |e^{- \lambda y}|$ here?

Let $z \in \Gamma (R)$ where this is the upper semi circle centred at the origin with radius $R>1$. Let $z=x+iy$ with $x \in \mathbb{R}$ and $y \geq 0$. So $$|e^{i \lambda z}|=|e^{i \lambda z}| ...
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1answer
33 views

With the aid of series, show that if $f(z)=\frac{\operatorname{cos}z}{z^2-(\pi/2)^2}$, then $f$ is an entire function.

Prove that if $$f(z)= \begin{cases} \frac{\operatorname{cos}z}{z^2-(\pi/2)^2}, & \text{when} \; z\neq \pm \pi/2, \\ -\frac{1}{\pi}, & \text{when} \; z=\pm \pi/2, \end{cases} $$ then $f$ is ...
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1answer
23 views

Show that if $f(z)=\frac{\operatorname{Log}z}{z-1}$ when $z\neq 1$ and $f(1)=1$, then $f$ is analytic throughout the domain.

$\operatorname{Log}z=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(z-1)^n \; (|z-1|\lt 1).$ Use this fact to show that if $$f(z)=\frac{\operatorname{Log}z}{z-1} \; \text{when} z\neq 1$$ and $f(1)=1$, ...
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2answers
41 views

If $f(z)= \frac 1z $ be defined and analytic on region $ |z| \gt 1 $ in $ \Bbb C $ then can we find an entire function $g$ such that :

$g$ should be such that $f(z)=g(z)$ on $ |z| \gt 1$ in $\Bbb C $. Now,Can we plainly apply uniqueness theorem and say that such a function $g$ can not exist?
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0answers
43 views

Laurent series for $\frac{2}{(z)(z-1)(z-2)}$

! So I think I am getting the hang of Laurent Series, but having a bit of trouble with one of the fractions for part a). So I split this up in to partial fractions: $\frac{1}{z} - ...
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1answer
39 views

What's an importance of multi-valued functions?

For example, let's define $\log z= \{w\in \mathbb{C} : e^w=z\}$. Then, we call this set $\log z$ a "multi-valued function". Formally saying, this $\log z$ is merely a set but not a function to ...
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1answer
45 views

A necessary condition for a multi-complex-variable holomorphic function. [on hold]

Let $\Omega\subset \mathbb{C}^n$ be an open unit ball, $f:\Omega \to\mathbb{C}$ is a bounded function. For $a \in \mathbb{C}^n$, define $$ ...
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1answer
580 views

All values of z for which cos(z) has real values

Find all values of z for which $\cos(z)$ has real values. My approach: By definition $\cos(z)= 0.5(e^{iz}+e^{-iz})$ Subbing in $z=x+iy$, $\cos(z)=0.5(e^{ix}e^{-y}+e^{-ix}e^{y})$ Therefore ...
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0answers
28 views

Contour integral from first principles

What does it mean by 'evaluate from first principles? Does it mean use ? For part (a) do I parametrise as $\gamma(t)=a+2e^{it}$ with $t$ between $0$ and $2\pi$? Doing this I end up with the ...
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1answer
43 views

Showing $f$ is meromorphic or that $f(a_j)$ converges to some point in the plane.

Let $S$ be a sequence of points in $\mathbb{C}$ that converges to $0$. Let $f$ be analytic and defined on some disc centered at $0$ except possibly at the points of $S$ and at $0$. Show that either ...
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0answers
24 views

Counting poles that are shared between $f$ and $g$

Suppose I have a meromorphic function $f(z)$ with poles at $f_i$ and $\mathcal{Res}(f,f_i)=1$, and $g(z)$ with poles at $g_i$ and $\mathcal{Res}(g,g_i)=1$. I would like to construct a function ...
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1answer
476 views

Most general linear transformation which leaves the origin fixed and preserves all distances?

I'm working on the following problem from Ahlfors (complex analysis): Prove that the most general (linear) transformation which leaves the origin fixed and preserves all distances is either a ...
3
votes
2answers
105 views

Evaluating $\int_0^\infty \frac{x\sin x}{1+x^2}$ using contour integration?

I'd like to Evaluate $$\int_0^\infty \frac{x\sin x}{1+x^2}$$ The sine function makes the obvious choice $\dfrac{z \sin z}{1+z^2}$ useless since if we integrate over a semicircle sine can become ...
4
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3answers
66 views

Finding local max of analytic function

Given a function $f=z^2+iz+3-i$. I need to find the the maximum of $|f(z)|$ in the domain $|z|\leq 1$ I know that the maximimum should be on $|z|=1$ but when I tried to put $z=e^{i\theta} $ in the ...
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3answers
1k views

definition of winding number, have doubt in definition.

could any one tell me why in the definition of index number or winding number of a curve $\gamma(t)$ around some point $a$ we take this integral : $$\frac{1}{2\pi i}\int_{\gamma}\frac{1}{z-a} $$ why ...
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1answer
44 views

Contour integral $|z-i|=1/9$

Calculate \begin{equation*} \int_{\Gamma}\frac{1}{z^4+16}dz, \end{equation*} where $\Gamma :|z-i|=\frac{1}{9}$. I have asked I similar question to this but I still do not understand.... when I find ...
2
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1answer
18 views

If an entire function $f$ satisfies $|f(z)| \le |\log z|,$ what can we say about $f$?

Let $f$ be an entire function. Define $\Omega=\mathbb{C}-(-\infty,0]$, the complex plane with the ray $(-\infty,0]$ removed. Suppose that for all $z \in \Omega$ , $|f(z)| \le |\log z|$, where $\log z$ ...
2
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0answers
25 views

Conformal Mapping from Equilateral triangle to Isosceles Right Triangle

This is an exercise problem. Does there exist a conformal mapping from an equilateral triangle onto an isosceles right triangle such that, under correspondence of boundary, vertices are mapped to ...
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7answers
3k views

Picard's Little Theorem Proofs

Picard's little theorem says that If there exist two complex numbers $a,b$ such that $f: \Bbb{C} \to \Bbb{C}\setminus \{a,b\}$ is holomorphic then $f$ is constant. I am interested in proofs for ...
3
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1answer
16 views

$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$ onto upper half space $\{\,z\in \mathbb C : \operatorname{Im}(z)>0 \,\}$

I am search an one to one mapping that maps the domain $$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$$ onto upper half space $$\{\,z\in \mathbb C : \operatorname{Im}(z)>0\, ...
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1answer
39 views

Complex number, power series

Develop $\sinh z$ in powers of $z-\pi i$ to show that $$\lim_{z\to \pi i}\frac{\sinh z}{z-\pi i}=-1$$ I know that $\sinh z=\sum_{n=1}^\infty \frac{z^{2n-1}}{(2n-1)!}$. Edit: Following the hint ...
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2answers
23 views

equation of a line into the complex form

So if i am given an equation of a line in complex form for example $Re|(1+i)z| = 0$, I could turn this into its real counter part on the x-y plane and graph it. Is there a way to go in the other ...
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1answer
32 views

Write $1/z$ as a power series

Show that the function $f(z)=1/z$ can be represented as a power series in a ball $B(z_0,r)$, where $z_0 \neq 0$. Find the radius of convergence of this power series. $$f(z)=\frac1z = ...
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0answers
21 views

Inverse of a constant function on an open set

I was working on holomorphic functions and Riemann surfaces, and I was wondering about the inverse of a constant function: Let $f:U\rightarrow V$ be a holomorphic function between two Riemann ...
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1answer
27 views

Understanding a Wermer's counterexample.

I am reading some lecture notes on holomorphic functions of several complex variables, see page 105. The part I am struggling with is a proof by Wermer I have asked about runge domains, and ...
0
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0answers
27 views

complex variable inequality

Let $B$ and $C$ be nonegative real numbers and $A$ a complex number. Suppose that $$ 0\leq B-2Re(\overline{\lambda}A) + |\lambda|^2 C \ \forall \ \lambda \in \mathbb{C} $$ Conclude that $|A|^2 \leq ...
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1answer
51 views

How can I expand this

How can I expand $\dfrac{\pi \csc(z\pi)}{(2z+1)^3}$? so then I can find the residue ? thanks
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0answers
24 views

Existence of analytic function on a unit disc $\triangle$ [on hold]

Let $\triangle$ be the open unit disc. Then can there be analytic functions with the property (1) $f(\frac{3}{4})=\frac{3}{4}$ and $f'(\frac{2}{3})=3/4$ 2) $f(\frac{3}{4})= -\frac{3}{4}$ and ...