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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5
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213 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
4answers
334 views

How do I calculate $\sum_{n\geq1}\frac{1}{n^4+1}$?

How do I calculate the following sum $$\sum_{n\geq1}\frac{1}{n^4+1}$$
5
votes
2answers
213 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
342 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
362 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
537 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
3answers
2k views

Some way to integrate $\sin(x^2)$?

Because the straight forward approach involves Fresnel integrals I thought about a different approach of taking the imaginary part of $\int_{-\infty}^{\infty}\exp(ix^2) $ but have no idea how to ...
3
votes
4answers
493 views

How to show that the modulus of $\frac{z-w}{1-\bar{z}w}$ is always $1$?

Let's suppose that $|z|<1$ and $|w|=1$. Show that the modulus of $\displaystyle \frac{z-w}{1-\bar{z}w}$ is always $1$. Some hint.
3
votes
1answer
117 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
106 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
142 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
193 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
335 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
54 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
344 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
303 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
244 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
65 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 ...
43
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8answers
7k 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?
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 ...
44
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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, ...
23
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6answers
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 ...
35
votes
2answers
935 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.
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2answers
2k views

on the boundary of analytic functions

Suppose I have a function $f$ that is analytic on the unit disk $D = \{ z \in \mathbb{C} : |z| < 1 \}$ that is also continuous up to $\bar{D}$. If $f$ is identically zero on some segment of of the ...
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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
475 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 ...
10
votes
2answers
267 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
405 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
950 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 ...
21
votes
1answer
302 views

Are there always singularities at the edge of a disk of convergence?

Take a function that is analytic at 0 and consider its Maclaurin Series. Here are some examples I'll refer to: $$\frac{1}{1-x} =\sum_{n=0}^\infty x^n$$ $$\frac{1}{1+x^2} ...
13
votes
1answer
2k views

Residue of $z^2 e^{1/\sin z}$ at $z=\pi$

A while back I was working through many problems in Mathews and Walker's Mathematical Methods of Physics. In the appendix is this problem: A-6. Find the residue of the function $z^2 e^{1/\sin z}$ ...
11
votes
1answer
424 views

Help with a troublesome double integral

I'm having difficulty with a double integral $$-2i\int_{0}^{\infty}\int_{0}^{\infty}\frac{dxdt}{t(e^{2\pi x}-1)(e^{2\pi t/s}-1)}\left[\cos(t\log(1-ix))-\cos(t\log(1+ix))\right]$$ where ...
11
votes
5answers
758 views

Difference between $\mathbb C$ and $\mathbb R^2$

What are the basic differences between $\mathbb C$ and $\mathbb R^2$? The points in these two sets are written as ordered pairs, I mean the structure looks similar to me. So what is the reason to ...
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 ...
23
votes
2answers
1k views

What is the image near the essential singularity of z sin(1/z)?

This was part of a homework problem from J.B. Conway's complex analysis text which I was assigned long ago but didn't get. A few years later I was a TA for a course where the problem was assigned. I ...
21
votes
2answers
756 views

Summation using residues

In reference to this question about showing that the following interesting series takes on the value $$\sum_{n=0}^\infty \frac{1}{(2n+1)\operatorname{sinh}((2n+1)\pi)}=\frac{\log(2)}{8}$$ I tried ...
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 ...
14
votes
5answers
397 views

Integral $\int_0^\infty \log^2 x\frac{1+x^2}{1+x^4}dx=\frac{3 \pi^3}{16\sqrt 2}$

This integral below $$ I:=\int_0^\infty \log^2 x\frac{1+x^2}{1+x^4}dx=\frac{3 \pi^3}{16 \sqrt 2} $$ is what I am trying to prove. Thanks. We can not expand the denominator as a series since the ...
12
votes
4answers
646 views

Why is $2\pi i \neq 0?$ [duplicate]

We know that $e^{\pi i} = -1$ because of de Moivre's formula. ($e^{\pi i} = \cos \pi + i\sin \pi = -1).$ Suppose we square both sides and get $e^{2\pi i} = 1$(which you also get from de Moivre's ...
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votes
3answers
2k views

When does a complex function have a square root?

I would like to show that there is a holomorphic $f$ on a neighborhood of zero such that $f(z)^2=1-\cos(z)$. In other words, I want to show that $1-\cos(z)$ has a complex square root. I know that ...
8
votes
4answers
488 views

Integrate: $\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)}dx$

How to evaluate the following $$\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)} dx $$ Given hints says to construct a rectangle $0\to R\to R+i\to i \to 0$ and consider $\displaystyle ...
18
votes
4answers
703 views

Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$

I know that $$\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$$ For substituting $u=2$ into $$\zeta(u)\Gamma(u)=\int^\infty_0\frac{x^{u-1}}{e^x-1}dx$$ However, I suspect that there is an easier proof, ...
15
votes
7answers
2k views

Integration by means of complex analysis

Dear all: this time I have the integral $$\int_0^\infty\frac{1-\cos x}{x^2(x^2+1)}\,dx$$and we must try to solve it using complex integration, residues, Cauchy's Theorem and the whole lot. (BTW, does ...
14
votes
2answers
600 views

A bounded holomorphic function

If $\Omega$ is a region which is dense in $\mathbb{C}$, $f\in H(\Omega)$ and is continuous on $\mathbb{C}$, moreover $f$ is bounded on $\mathbb{C}$, can we claim that $f$ is a constant?
11
votes
1answer
579 views

$\int_{0}^{\infty}\frac{dx}{1+x^n}$

My goal is to evaluate $$\int_{0}^{\infty}\frac{dx}{1+x^n}\;\;(n\in\mathbb{N},n\geq2).$$ Here is my approach: Clearly, the integral converges. Denote the value of the integral by $I_n$. Now let ...
9
votes
1answer
345 views

Harmonic functions with zeros on two lines

For which pairs of lines $L_1$, $L_2$ do there exist real functions, harmonic in the whole plane, that are $0$ at all points of $L_1 \cup L_2$ without vanishing identically? Note: This is ...
8
votes
1answer
302 views

Proving two entire functions are constant.

Let $f$ and $g$ be entire functions such that $f^n+g^n=1$, where $n\geq 3$ is an integer. Prove that $f$ and $g$ are constant. I suppose I should somehow prove that either $f$ or $g$ is bounded so ...
7
votes
3answers
559 views

Sources on Several Complex Variables

I have searched the past entries about sources on SCV but couldn't find about this topic. If I am not careful enough, sorry for this! We are using Hörmander's book which is really hard to follow. ...
6
votes
4answers
1k views

zeroes of holomorphic function

I know that zeroes of holomorphic functions are isolated,and I know that if a holomorphic function has zero set whic has a limit point then it is identically zero function,i know a holomorphic ...
5
votes
2answers
1k views

Cauchy's residue theorem with an infinite number of poles

Is it possible to apply Cauchy's residue theorem to a function which has an infinite number of isolated singularities within the contour of integration (say a semicircle whose radius goes to ...