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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9
votes
1answer
300 views

Help in calculating the following integral $\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x. $

I was asked to calculate this: $$\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x. $$ My idea was to change the integration limits to $|z|=1$ in the complex plane and to ...
2
votes
1answer
87 views

How to find the all elements of $\text{Aut}(\Bbb{D})$?

Use Schwarz lemma, we can verify that $$f(z)=e^{i\theta}\frac{\alpha-z}{1-\overline{\alpha}z}$$exhaust all automorphisms of the disc($\theta \in \Bbb{R},\alpha \in \Bbb{D}$). When I first see the ...
0
votes
2answers
66 views

for $z\in\mathbb C-\{0\},~\dfrac{1}{1+nz}\to0.$

How to show that for $z\in\mathbb C-\{0\},~\dfrac{1}{1+nz}\to0.$ I've tried triangle inequality couldn't arrive at any conclusion. Please help me.
2
votes
1answer
141 views

Paley-Wiener Theorem

In Proposition 5.3.11 of Bratteli-Robinson Vol II the Paley-Wiener Theorem is stated as follows: Is this the correct statement? Aren't $f$ and $\hat{f}$ mixed up? I know the following version of ...
0
votes
1answer
41 views

Expansion of $z^3 \log ( (z-a)/(z-b))$ in $\infty$

I need some hints to evaluate the expansion of $z^3 \log((z-a)/(z-b))$ in $\infty$. I thougt that evaluating $\log(\frac 1 z -a)$ in $z = 0$ may be helpful. How can I proceed ?
18
votes
0answers
443 views

Is there an exposition of complex analysis firmly separating the algebra, analysis, and topology?

Complex analysis seems to work because of the interplay between algebraic geometry over $\mathbb{C}$, and analysis and topology exploiting the fact that $\mathbb{C}/\mathbb{R}$ happens to be a ...
1
vote
2answers
294 views

Residue Theorem to Compute Integrals of Rational Functions

Any help would be very much appreciated. Thanks. $$\int_{-\infty}^{\infty}\frac{x^2}{x^4-4x^2+5}dx$$ Integral for the above using Residue Theorem.
0
votes
1answer
64 views

Calculate the Radius of convergence of $\sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$

I need your help: Calculate the Radius of convergence of the following: $$ \sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$$ Im new to this subject, so I'd appreciate it if you can add explanations to ...
4
votes
1answer
141 views

Integral formula for $\frac{1}{\Gamma(z)}$

Let $c>0$. How to prove that for any complex number $z$, $$\frac{1}{\Gamma(z)}=\frac{1}{2\pi}\int_{-\infty}^\infty (c+it)^{-z}e^{c+it}\,dt?$$ where $\Gamma(z)$ is the Gamma function.
0
votes
1answer
39 views

Expansion of $(z-1)^2 / (z-2)(z^2+1)$ in $z = 2$

I have to expand $$ f(z) := \frac { (z-1)^2 }{(z-2)(z^2+1)} $$ around $z = 2$. I wanted to write $f(z) = \frac 1 {z-2} g(z)$ and then expand $g(z)$. Some hints would help a lot :)
2
votes
0answers
46 views

using Paley-Wiener to get support and then estimate inf sup

Define the function $$ \tilde{f}_n(\omega)=\frac1{\sqrt{2\pi}} \frac{\sin R\omega/2}{R\omega/2} s_n(R\omega/2\pi),$$ where (using the Weierstrass product representation for $\sin$) $$ s_n(w) = ...
0
votes
1answer
27 views

Validity of residue outside the domain

Using the identity theorem I can see that $f(z)=\dfrac{2}{3+z}$ and hence 1 is true and 4 is false. This far is easy. But for 2 and 3 I can see that $f$ is not defined at $z=3$ and $-3$ is not a ...
0
votes
1answer
247 views

Entire function and odd/even function [duplicate]

f is an entire function.If $$f(R)\subset R,f(iR)\subset iR,$$then f(z) is an odd function;Similarly,if$$f(R)\subset R,f(iR)\subset R,$$then f is an even function.
2
votes
1answer
69 views

Description of elements of $\text{Aut}(\Bbb{H})$

The automorphism of the disc is $\varphi(z)=\exp(i\theta)\frac{\alpha-z}{1-\overline{\alpha}z}$,where $\theta \in \Bbb{R}$ and $\alpha \in \Bbb{D}$ (we denote the open unit disc centered at the origin ...
1
vote
1answer
59 views

Is this the Laurent series expansion of $f$ in $D?$

Let a complex valued function $f$ be analytic on $D=\{z\in\mathbb C:R_1<|z-z_0|<R_2\}$ where $0<R_1,R_2<\infty.$ Moreover let in $D,~f$ can be represented as ...
2
votes
0answers
82 views

How to obtain the infimum of this inequalities?

Let $A$ be the family of functions $f(z)=z+a_2z^2+\cdots$ that are analytic in unit disk $D:\{z:|z|<1\}$ and $S$ is the subfamily of functions that are univalent in $D$. $R(a)$ is the subfamily of ...
1
vote
1answer
519 views

Bijective conformal map from half disc to upper half plane

I'm trying to find a bijective conformal map from the half disc $\{z: |z| < 1, \Re(z)>0\}$ to the upper half plane $\{z: \Re(z) > 0\}$. Any help is appreciated. Thanks!
8
votes
1answer
276 views

Proof of the product formula for sine function

I am looking for a simple way to prove $$\frac{\sin \pi z}{\pi z}=\prod_{n=1}^\infty \left(1-\frac{z^2}{n^2}\right)$$ using mainly on the fact that the entire function has simple zeros at $n=\pm 1, ...
1
vote
1answer
181 views

Schwarz-Christoffel transformation: point at infinity

In this paper how can one simply write $\displaystyle A = A' \frac{1}{x_n^{\phi_n/\pi - 1}}$ on $(21.6)$. Can anyone hint me what is meant by "with it then being dropped from the list of points in the ...
3
votes
2answers
276 views

Type of singularities of $\frac{z}{e^z-1}$

I don't really understand how one can find the type of singularities for a given function. Say if $$f(z) = \frac{z}{e^z-1}$$ then I know that the singularities are at $z = 2n\pi i$ However, how do I ...
1
vote
1answer
61 views

$\displaystyle\lim_{z\to 0} |zf(z)| = 0$ implies $z = 0$ is a removable singularity.

Given that $f$ is analytic around $z = 0$ and that $\displaystyle\lim_{z\to 0} |zf(z)| = 0$, we can conclude that $f(z)$ has a removable singularity at $z = 0$. I think that we should deduce that ...
3
votes
1answer
101 views

Showing an analytic function takes certain values exactly once

Let $D$ be the open unit disk in the complex plane, and let $f(z)$ be a map from $D$ to $D$ with $f(0)=0$. Denoting $|f'(0)|=\delta$, we further require $\delta>0$. Fix $\eta>0$ with ...
1
vote
1answer
108 views

At $z=0$ the function $f(z)=\exp({z\over 1-\cos z})$ has

At $z=0$ the function $f(z)=\exp({z\over 1-\cos z})$ has $1$. A removable singualrity $2$. A pole $3$. An essential singularity $4$. Laurent series around $z=0$ has infinitely man positive and ...
2
votes
0answers
144 views

Complex conjugate of contour integral

Suppose I have a closed-loop counter-clockwise contour integral over a function $f(z,t)$: $$F(t)=\oint_C dz\enspace f(z,t)$$ Then suppose I want to know the complex conjugate of $F(t)$. What ...
1
vote
1answer
69 views

If I have a holomorphic function on the unit disc, do I know anything about the radius of convergence of its series expansion about zero?

I'm looking at a proof that assumes only that $f : \mathbb{D} \rightarrow \mathbb{D}$ is holomorphic with $f(0) = 0$ The first step in the proof is to "expand $f$ in a power series centered at $0$ ...
0
votes
1answer
188 views

Prove Fibonacci sequence and find partial fraction decomposition

We have a function $z/(1-z-z^2)$ which has simple poles at $z=(-1\pm \sqrt5)/2$. Furthermore we have the powerseries $\frac{z}{1-z-z^2}=\sum_{n=0}^\infty{F_nz^n}$. I have calculated it's radius of ...
2
votes
2answers
76 views

Winding number in complex plane homework

If $\gamma_1,\gamma_2$ are closed paths s.t. $|\gamma_1(t)|>|\gamma_2(t)|$ for all $t$, and $\Gamma(t) := \gamma_1(t) + \gamma_2(t)$, show that $n(\Gamma,0) = n(\gamma_1,0)$, where $n$ is the ...
2
votes
2answers
219 views

Integrating $\frac{z}{1-\cos(z)}$ over the unit circle

I want to evaluate $ \int_C f(z)\,dz $ where $f(z)$ is : $$f(z)=\frac{z}{1-\cos(z)}$$ and $C$ is the unit circle, counterclockwise. I kept having problems with it. If someone can help, it would ...
1
vote
1answer
32 views

Show that $w(\gamma ,0)=w(\gamma _1 ,0)+w(\gamma _2 ,0)$

$\gamma _1$ and $\gamma _2$ are two closed curves in $\Bbb C\setminus\{0\}$ both with parameter interval $[a,b]$. We define the curve $\gamma(t)=\gamma_1(t)\gamma_2(t)$. I've just shown that $\gamma$ ...
4
votes
2answers
142 views

Show that $f(z)=0$.

Suppose that $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic on the open unit disc and continuous on the closed unit disc. Assume that $f(z)=0$ on an arc of the circle $\{z\in\mathbb{C}:|z|=1\}$. Show ...
12
votes
2answers
221 views

Zero of a complex polynomial satisfying one of three assertions.

Let $n$ be a positive integer greater than $1$. Prove that if $x$ is a zero of $ X^n+1+(-1)^n(X+1)^n$ then $|x|=1$ or $|x+1|=1$ or $|x+1|=|x|$. My initial thought was to study the cases $n=2,3,4$ ...
1
vote
1answer
68 views

Show that analytic $f_n$ on $D_1(0)$ have an uniformly convergent subsequence, given that $\int_0^{2\pi} |f_n(e^{i\theta})|d\theta< 1$

I am preparing for my qualifying exam in complex analysis, and I'm working on the following problem. Let $f_n = u_n + iv_n$ be a family of functions that are analytic on $D_1(0)$ and continuous up ...
1
vote
1answer
164 views

Check my work: $\lim a_n = 0 \Rightarrow \lim \sqrt{a_n} = 0 $? (for $a_n$ positive)

I'm trying to prove, as "properly" as possible the following:$$\left[ \lim z_n = z \right] \iff \left[ \lim x_n = x \quad \wedge \quad \lim y_n = y \right]$$ where $z_n = x_n + i y_n$ and $z=x+iy$. ...
1
vote
0answers
93 views

What happens to small squares in Riemann mapping?

I have a square $S$, and I want to convert it to the unit disc $D$. The Riemann mapping theorem says that I can to it with a conformal bijective map. But, any such mapping will cause some distortion. ...
2
votes
0answers
47 views

Representing series $f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$ as a Dirac comb function.

Consider the function $$f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$$ where $\omega_n= \sqrt{(\frac{n \pi c}{l})^2-(\frac{r_0}{2})^2}.$ If we neglect the term ...
2
votes
2answers
49 views

$f(z)={2z+1\over 5z+3}$ maps

Define $H^{+}=\{z:y>0\}$ $H^{-}=\{z:y<0\}$ $L^{+}=\{z:x>0\}$ $L^{-}=\{z:x<0\}$ $f(z)={2z+1\over 5z+3}$ maps $1.$ $H^+\to H^+$ and $H^-\to H^-$ $2$. $H^+\to H^-$ and $H^-\to H^+$ ...
4
votes
1answer
198 views

A branch cut problem

In Ahlfors' Complex Analysis text, chapter 3, section 4 the transformation $z=\zeta+\frac{1}{\zeta}$ is discussed. The author notes that for every $z$, there exists 2 solutions for $\zeta$ and they ...
2
votes
1answer
260 views

Showing that if $u$ is a real-valued harmonic function then for any real $c$ we have that $u^{-1}(c)$ is unbounded

I have the following homework question: Let $u$ be a non-constant real-valued harmonic function in $\mathbb{C}$. Prove that $u^{-1}(c)$ is unbounded for every real number $c$ There is a hint ...
3
votes
2answers
72 views

Is there an image to the point at infinity through this map?

I encountered a conformal mapping on the complex plane:$$z\rightarrow e^{i\pi z}$$ and I am not sure about where it does send the point at infinity. If I could say something along the lines: ...
2
votes
1answer
181 views

Wirtinger derivative of composition of functions

So I have a very basic question : let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a $C^1$ function, and let $g : \mathbb{C} \rightarrow \mathbb{R}$ be defined by $g(z)=h(z \overline{z})$. I want to ...
1
vote
0answers
38 views

Calculate $\lim\limits_{n\to \infty} \frac{\prod\limits_{z^n=-1}(f(z)^n-1)}{\prod\limits_{z^n=1}(f(z)^n-1)}$

I am trying to calculate the following limit $$\lim_{n\to \infty} \frac{\prod_{z^n=-1}(f(z)^n-1)}{\prod_{z^n=1}(f(z)^n-1)},$$ where $f$ is a holomorphic function of $z$ such that $|f(z)|<1$ for ...
2
votes
2answers
97 views

Principal value of $\int_0^\infty \frac{x^{-p}}{x-1}dx$ for $|p|<1$

If $|p|<1$, how to find the Cauchy Principal Value of $$\int_0^\infty \frac{x^{-p}}{x-1}dx$$ I tried spliting the integration from $0\to 1$ and $1 \to \infty$ and switching $x = 1/u$, but no luck ...
0
votes
2answers
78 views

How find the number of $z$,such that$ |a^2-b^2-b+1|\le 10$

let $f:C\longrightarrow C$ and $f(z)=z^2+zi+1$,find the number of $z$,such $Im(z)>0$ and $|Im(f(z))|\le 10,|Re(f(z))|\le 10$,and $ Im(f(z)) ,Re(f(z)) $are integer numbers. where $i=\sqrt{-1}$, ...
8
votes
2answers
298 views

Problem $13.7$ in Apostol's Mathematical Analysis.

I am looking at problem $13.7$ in Apostol's Mathematical Analysis. Let $D=\{|z|<1\}$ and let $f=u+iv$ be such that $(\rm i)$ $u,v\in{\mathcal C}^1(D)$ $(\rm ii)$ $f$ is continuous ...
2
votes
1answer
109 views

Square Root and Winding Number

Let $f\in C(S^{1},S^{1})$ such that $w(f)=0$ where $w$ denotes winding number. Can we conclude that there exists a function $g\in C(S^{1},S^{1})$ such that $f=g^{2}$?
2
votes
2answers
120 views

Supposed counterexample to Liouville's theorem

I'm trying to understand Liouville's theorem, and I don't see why $f(z)=e^{-|z|^2}$ isn't a counterexample. It's bounded ($0 < f(z) \leq 1$), so it must be somehow that it's not holomorphic. ...
0
votes
2answers
32 views

The function $w(z)=-(\frac1z+bz),-1<b<1 ,$ maps $|z|<1$ onto …

I am stuck on the following problem that says: The function $w(z)=-(\frac1z+bz),-1<b<1 ,$ maps $|z|<1$ onto (A) A half plane (B) Exterior of the circle (C) Exterior of an Ellipse ...
1
vote
1answer
47 views

$f(z)={z\over 3z+1}$ maps

Define $H^{+}=\{z:y>0\}$ $H^{-}=\{z:y<0\}$ $L^{+}=\{z:x>0\}$ $L^{-}=\{z:x<0\}$ $f(z)={z\over 3z+1}$ maps $1.$ $H^+\to H^+$ and $H^-\to H^-$ $2$. $H^+\to H^-$ and $H^-\to H^+$ $3.$ ...
1
vote
1answer
45 views

For all $f: D_1(0) \to D_1(0)$ analytic with $f(\frac{i}{3}) = 0$, find $\displaystyle \sup_f\{\operatorname{Im} f(0) \}$

Let $\mathcal{F}$ denote the family of all analytic functions $f$ that map the unit disc onto itself with $f(\frac{i}{3}) = 0$. Find $M \equiv\sup\{\operatorname{Im} f(0) : f \in \mathcal{F}\}$. I am ...
6
votes
4answers
307 views

Holomorphic vs differentiable (in the real sense).

Why a holomorphic function is infinitely differentiable just because of satisfying the Cauchy Riemann equations, but on the other side, a two variable real function that is twice differentiable is not ...