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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4
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2answers
183 views

Evaluate these infinite products $\prod_{n\geq 2}(1-\frac{1}{n^3})$ and $\prod_{n\geq 1}(1+\frac{1}{n^3})$

What is $\prod\limits_{n\geq 2}(1-\frac{1}{n^3})=?$ $\prod\limits_{n\geq 1}(1+\frac{1}{n^3})=?$ I am sure about their convergence. But don't know about exact values. Know some bounds as well. For ...
4
votes
2answers
612 views

Proving surjectivity of $\cos(z)$ and $\sin(z)$ and find all $z : \cos(z) \in \mathbb R$ and all $z: \sin(z) \in \mathbb R$

I am trying to solve the following two problems: 1) Prove that the functions $\cos(z)$, $\sin(z)$ are surjective over the complex numbers. 2) Find all $z \in \mathbb C$: $cos(z) \in \mathbb R$ and ...
4
votes
2answers
157 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 ...
3
votes
3answers
611 views

How can I show this statement.

Show that there is no holomorphic fuction $f$ in the unit disc $D$ that extends continuously to boundary of $D$ such that $f(z)=\frac{1}{z} ~for~ z\in \partial( D) $. I tried to apply maximum ...
3
votes
3answers
1k views

Proof of Cauchy-Schwarz inequality - Why select s so that so that $||x-sy||$ would be minimized?

I was looking at a number of different proofs of the cauchy schwarz inequality in an inner product space ($\mathbb{R}^n$ or $\mathbb{C}^n$). All of them used the idea of $||x-sy||$ where $s$ was ...
3
votes
2answers
546 views

if a complex function $f$ is real-differentiable, then $f$ or $\overline{f}$ are complex-differentiable

This is an exercise from Remmert's Theory of Complex functions. Let $D\subset \mathbb{C}$ be a domain and $f:D\rightarrow \mathbb{C}$ a real-differentiable function. Assume that the following limit ...
2
votes
1answer
265 views

Why a holomorphic function satisfying these conditions has to be linear?

Let $\Omega$ be a bounded open subset of $\mathbb{C}$ and $f:\Omega\rightarrow\Omega$ be holomorphic in $\Omega$. Prove that if there exists a point $z_0$ in $\Omega$ such that ...
2
votes
2answers
606 views

Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$ [closed]

Show that $$\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)},$$ where $0\leq r <1$. Using this, prove that $\sum_{n=0}^\infty r^n ...
2
votes
2answers
1k views

Find conformal mapping from sector to unit disc

Find a conformal mapping between the sector $\{z\in\mathbb{C} : -\pi/4<\arg(z) <\pi/4\}$ and the open unit disc $D$. I know that it should be a Möbius transformation, but other than that I am ...
2
votes
1answer
162 views

Prove the following equation of complex power series.

Show that for $|z| \lt 1$ with $z \in \Bbb C$, we have $$ \sum_0^\infty \frac{{z^2}^k}{1-{z^2}^{k+1}} = \frac{z}{1-z} $$ $$ \sum_0^\infty \frac{2^k{z^2}^k}{1+{z^2}^{k}} = \frac{z}{1-z} $$ My guess ...
2
votes
4answers
872 views

Improper integral of $\sin^2(x)/x^2$ evaluated via residues

I have come across another improper integral I wish to evaluate via residues. The integral is: $$\int_{-\infty}^\infty{\frac{\sin(x)^2}{x^2}}dx$$ $\sin(z)$ behaves in an uneasy way so I tried ...
1
vote
1answer
154 views

Finding the residue of function with Laurent series $\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{y^n(A+By+Cy^{-1})^k}{\beta (\beta i)^n \ k!}$

I have been trying to find the residue of $f(\omega) = \frac{e^{i \omega a} e^{\frac{-b \omega}{\omega + ib}}}{i \omega}$ at the essential singularity $\omega = -ib$ for a while, but it is giving me ...
1
vote
1answer
453 views

Integrating squared absolute value of a complex sequence

I was reading through my book in complex analysis and i encountered this problem. Given, $F=\sum_{n=0}^{\infty} a_nX^n$ is a convergent power series with radius of convergence R. We are asked to show ...
1
vote
2answers
688 views

How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?

How do I prove the error function $$ \mbox{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} dt. $$ is entire? Could you give me some scratch proof?
1
vote
1answer
244 views

Representation of Holomorphic Functions By Exponential

Let $f$ be holomorphic and nonzero on $D_{1}(0)$ the open unit disc. Can we write (for the given domain) $f(z) = e^{h(z)}$ where $h$ is holomorphic? This seems clear using a naive log argument but I'm ...
1
vote
2answers
444 views

Area of Validity of Writing an Exponential Integral as Sum of IntegralSinus and -Cosinus

I'm confused by the two online references shown below. To me, they give different areas of validity of writing an exponential integral as sum of integralsinus and -cosinus. On this Wiki page, I find ...
0
votes
1answer
94 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
13
votes
1answer
3k views

Why is every meromorphic function on $\hat{\mathbb{C}}$ a rational function?

I know that an analytic function on $\mathbb{C}$ with a nonessential singularity at $\infty$ is necessarily a polynomial. Now consider a meromorphic function $f$ on the extended complex plane ...
7
votes
1answer
502 views

Does there exist an holomorphic function such that $|f(z)|\geq \frac{1}{\sqrt|z|}$?

I have some trouble solving this problem: Does there exist an holomorphic function $f$ on $\mathbb C\setminus \{0\}$ such that $|f(z)|\geq \frac{1}{\sqrt|z|}$ for all $z\in\mathbb C \setminus ...
6
votes
1answer
415 views

3 holomorphic functions, sum of absolute values does not have maximum

I have the following problem: Let $f,g,h$ be holomorphic functions (non-constant) in some domain $D$. Show that the function $F(z):=|f(z)|+|g(z)|+|h(z)|$ has no local maximum in this domain $D$. ...
6
votes
2answers
434 views

How to compute the infinite tower of the complex number $i$, that is$ ^{\infty}i$

Let $x = i^{i^{i^{i^{.^{.^{.{^ \infty}}}}}}}$. This is the solution of the equation $i^x - x = 0 $ . I used Euler's identity to find a solution. But I haven't yet found the real and imaginary parts of ...
5
votes
3answers
235 views

Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx. $$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
5
votes
2answers
218 views

Domination of complex-value polynomial by highest power.

The problem: Let $P(n)$ be a polynomial of degree $n$. Let $$M(r):= \underset{|z|\le r}{\mbox{sup}} \hspace{2mm} \left|P(z)\right|.$$ I desire to establish that $$r\mapsto \frac{M(r)}{r^n}$$ for ...
4
votes
5answers
355 views

Fourier transform of $f(x)=\frac{1}{x^2+6x+13}$

How to find the Fourier transform of the following function: $$f(x)=\frac{1}{x^2+6x+13}$$
4
votes
1answer
393 views

An entire function with two periods

Can anybody help me with this question: If $f(z)$ is an entire periodic function and it has to periods $2$ and $2i$, how can I find all other periods?
4
votes
1answer
607 views

Show that an entire function bounded by $|z|^{10/3}$ is cubic

Question: Let $f$ be an entire function such that $|f(z)|\leq1+2|z|^{10/3}$ for all z. Prove that $f$ is a cubic polynomial Thoughts so far: Using a corollary of Liouville's theorem, we know that we ...
3
votes
1answer
125 views

convergence of a particular series

Let be $ \Lambda\subseteq \mathbb C$ a lattice, I don't understand why the series $$\sum_{\lambda\in\Lambda\setminus\{0\}} \frac{1}{|\lambda|^s}$$ converges for $s>2$. Can someone help me?
2
votes
1answer
79 views

Infinite sum involving ascending powers

I was working on a problem involving finite differences and came across the following sum as a general solution to a problem I was working with $$ 1 + x + \frac{1}{1 + a}x^2 + \frac{1}{1 + a} ...
2
votes
1answer
115 views

a generalization of normal distribution to the complex case: complex integral over the real line

How to prove $\int_{\mathbb{R}} e^{-\frac{(x+it)^2}{2}}dx=\sqrt{2\pi}$ for any $t\in \mathbb{R}$? I only obtained the case that $t=0$, $\int_{\mathbb{R}} e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}$. Thanks.
2
votes
1answer
151 views

If $f(z)g(z) = 0$ for every $z$, then $f(z) = 0$ or $g(z) = 0$ for every $z$.

This is for homework, and I would really appreciate a hint. The question states "If $f$ and $g$ are holomorphic on some domain $\Omega$ and $f(z)g(z) = 0$ for every $z \in \Omega$, then $f(z) = 0$ ...
2
votes
2answers
211 views

criterions for holomorphic functions

What are the criterions for holomorphic functions except $\frac{\partial f}{\partial \overline z}=0$ and $f$ has a power series extension? I was considering the problem, which is the extension of a ...
2
votes
2answers
810 views

Why there is no continuous argument function on $\mathbb{C}\setminus\{0\}$?

An argument function $\phi$ on $\mathbb{C}\setminus\{0\} = \mathbb{R}^2\setminus\{0\}$ is a function such that for every $z\neq 0$ it holds that $$z = |z|\exp(i\phi(z)).$$ Is there an elementary and ...
2
votes
2answers
347 views

How to derive the following identity?

The book Irresistible Integrals by George Boros and Victor Moll on page 204 has the following identity $\displaystyle \frac{1}{1+x}=\prod_{k=1}^{\infty}\left(\frac{k+x+1}{k+x} \times ...
1
vote
1answer
57 views

complex integration, how to evaluate it?

I have this exercise: Show that if $|a| < r <|b|$, then $\int_\gamma \! \frac{1}{(z-a)(z-b)} \, \mathrm{d}z=\frac{2\pi i}{a-b}$, where $\gamma$ denotes the circle centered at the origin, ...
1
vote
1answer
382 views

On a conformal mapping

I was asked to find a one-to-one analytic map $f$ of unit disc $\mathbb{D}\subset \mathbb{C}$ so that $\mathbb{D}$ is mapped to $\{(x,y):y<x^2\}$. I thought the core procedure could be done by ...
1
vote
2answers
364 views

How to construct this Laurent series?

How do I construct the following Laurent series (clipped off Wolfram Alpha)? I know that the numerator can be written as $-1+\frac{\pi}2 z-...$ Alternatively (without the Laurent series), how can I ...
1
vote
1answer
273 views

The meaning of a notation from complex analysis

I have read the book Function Theory of Several Complex Variables of Krantz.But there is a notation I don't know what's it meaning. Can somebody give me a definition.Thank you. The notation is ...
0
votes
2answers
66 views

Understanding the Definition of $\int_\gamma f\ \overline{dz}$

Definitionally, we have that $$ \int_\gamma f\ \overline{dz} = \overline{\int_\gamma \overline{f}\ dz} $$ Now let $\int_\gamma f\ dz = w = x +yi$. Question 1: Is it not the case that $\int_\gamma ...
46
votes
7answers
8k views

Can someone please explain the Riemann Hypothesis to me… in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
27
votes
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 ...
28
votes
8answers
3k views

Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
46
votes
1answer
1k views

Fractal behavior along the boundary of convergence?

The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, ...
35
votes
2answers
989 views

Topology and analytic functions

Is there a topology T on the set of complex numbers such that the class of T-continuous functions and the class of analytic functions coincide.
14
votes
2answers
4k views

How does a branch cut define a branch?

I am studying complex analysis and I have problem understanding the concept of branch cut. The lecture draw this as some curve that starts from a point and goes on and on in some direction (for ...
12
votes
2answers
497 views

Integral $\int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma x)\,dx$

I am having trouble showing this equality is true$$ \int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma ...
11
votes
2answers
291 views

From the series $\sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)$ to $\zeta(\frac{1}{2}+it)$

Here is a pretty series $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \quad (*) $$ where $H_{n}:=\sum_{1}^{n} ...
17
votes
3answers
414 views

What would be the value of $\sum\limits_{n=0}^\infty \frac{1}{an^2+bn+c}$

I would like to evaluate the sum $$\sum_{n=0}^\infty \frac{1}{an^2+bn+c}$$ Here is my attempt: Letting $$f(z)=\frac{1}{az^2+bz+c}$$ The poles of $f(z)$ are located at $$z_0 = ...
10
votes
4answers
984 views

Zeros of a holomorphic function

Suppose $\Omega$ is a bounded domain in the plane whose boundary consist of $m+1$ disjoint analytic simple closed curves. Let $f$ be holomorphic and nonconstant on a neighborhood of the closure of ...
18
votes
2answers
1k views

Does constant modulus on boundary of annulus imply constant function?

Suppose I have a function $f:\mathbb{C}\rightarrow \mathbb{C}$, holomorphic on some neighborhood of an annulus $r\le|z|\le R$, $r<R$. If, for $z\in\{|z|=r\text{ or }|z|=R\}$, $|f(z)|=C$ for some ...
11
votes
9answers
2k views

half iterate of $x^2+c$

I'm looking for literature on fractional iterates of $x^2+c$, where c>0. For c=0, generating the half iterate is trivial. $$h(h(x))=x^2$$ $$h(x)=x^{\sqrt{2}}$$ The question is, for $c>0,$ and ...