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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44
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1answer
991 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, ...
26
votes
8answers
2k 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 ...
23
votes
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
885 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.
18
votes
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 ...
12
votes
2answers
470 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
253 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
946 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
301 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
416 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
726 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
715 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 ...
12
votes
4answers
626 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 ...
10
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
475 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
693 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
589 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?
12
votes
5answers
366 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 ...
11
votes
1answer
573 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 ...
10
votes
2answers
315 views

The Laurent series of the digamma function at the negative integers

To find the Laurent series of $\psi(z)$ at $z= 0$, I would first find the Taylor series of $\psi(z+1)$ at $z=0$ and then use the functional equation of the digamma function. Specifically, ...
9
votes
1answer
335 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
292 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
545 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
3answers
321 views

Improper integration involving complex analytic arguments

I am trying to evaluate the following: $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$ Any help will be much appreciated.
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 ...
5
votes
3answers
2k views

Evaluating the contour integral $\int_{0}^{\infty}\frac{\sin^{3}(x)}{x^{3}}\mathrm dx$

I am trying to show $$\int_{0}^{\infty}\frac{\sin^{3}(x)}{x^{3}}\mathrm dx = \frac{3\pi}{8}.$$ I believe the contour I should use is a semicircle in the upper half plane with a slight bump at the ...
12
votes
3answers
788 views

Prove: The positive integers cannot be partitioned into arithmetic sequences (using Complex Analysis)

An arithmetic sequence of step $d$ is a set of the form: {$a, a+d, a+2d, a+3d, ...$} where $a, d$ are positive integers. Show that the positive integers cannot be partitioned into a finite number ...
11
votes
3answers
186 views

For fixed $z_i$s inside the unit disc, can we always choose $a_i$s such that $\left|\sum_{i=1}^n a_iz_i\right|<\sqrt3$?

Let $z_1,z_2,\ldots,z_n$ be complex number such that $|z_i|<1$ for all $i=1,2,\ldots,n$. Show that we can choose $a_i \in\{-1,1\}$, $i=1,2,\ldots,n$ such that $$\left|\sum_{i=1}^n ...
11
votes
2answers
1k views

Showing that $\int_0^1 \log(\sin \pi x)dx=-\log2$

I need help with a textbook exercise (Stein's Complex Analysis, Chapter 3, Exercises 9). This exercise requires me to show that $$\int_0^1 \log(\sin \pi x)dx=-\log2$$ A hint is given as "Use the ...
11
votes
2answers
707 views

entire bijection of $\mathbb{C}$ with 2 fixed points

Besides the identity map, is there an entire function $f$ that is a bijection from $\mathbb{C}$ to $\mathbb{C}$ and has 2 fixed points? Thank you for the help.
10
votes
3answers
902 views

Complex Analysis Question from Stein

The question is #$14$ from Chapter $2$ in Stein and Shakarchi's text Complex Analysis: Suppose that $f$ is holomorphic in an open set containing the closed unit disc, except for a pole at $z_0$ on ...
9
votes
2answers
214 views

Is $f(z)$ a constant in $\Bbb{C}$?

Suppose $f:\Bbb{C}\rightarrow \Bbb{C}$ is holomorphic, $f(z+1)=f(z)$ for all $z \in \Bbb{C}$ and $$|f(z)|\leq \exp\left(\frac{2}{\pi}|z|\right).$$ Is $f(z)$ a constant in $\Bbb{C}$?
9
votes
5answers
4k views

How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?

I'm doing a bit of self study, but I'm uncomfortable with a certain idea. I want to show that $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is analytic, and by analytic I mean ...
9
votes
5answers
489 views

Visualising functions from complex numbers to complex numbers

I think that complex analysis is hard because graphs of even basic functions are 4 dimensional. Does anyone have any good visual representations of basic complex functions or know of any tools for ...
7
votes
1answer
167 views

Closed form for $\sum_{n=-\infty}^\infty \frac{1}{(z+n)^2+a^2}$

I want to express $$\sum_{n=-\infty}^\infty \dfrac{1}{(z+n)^2+a^2}$$ in closed form. What comes to mind is the formula $$\pi\cot\pi z = \dfrac{1}{z}+\sum_{n\ne ...
7
votes
1answer
386 views

Suppose that $ f $ is entire and that for each $ z $, either $ |f(z)| \leq 1 $ or $ |f^\prime (z) |\leq 1 $. Prove that $ f $ is a linear polynomial.

My question is in the title. I'm a little lost in how to solve this problem. There is a hint associated with the problem that states the following: Use a line integral to show that $ |f(z)| \leq A + ...
5
votes
2answers
310 views

Proving that if $f: \mathbb{C} \to \mathbb{C} $ is a continuous function with $f^2, f^3$ analytic, then $f$ is also analytic

Let $f: \mathbb{C} \to \mathbb{C}$ be a continuous function such that $f^2$ and $f^3$ are both analytic. Prove that $f$ is also analytic. Some ideas: At $z_0$ where $f^2$ is not $0$ , then $f^3$ ...
0
votes
2answers
291 views

simple tools to extract Re,Im,Abs… of any complex function

I've developped my own set of simple yet powerful tools to work on complex functions. I would like to know if these simple tools are currently used in complex analysis. Let's $z = x + i y = |z| ...
15
votes
2answers
836 views

Finding $f$ such that $ \int f = \sum f$

Please see the problem 5 of the given link: http://www.artofproblemsolving.com/Forum/resources.php?c=2&cid=59&year=2005&sid=722231ab4ec5ce280584eb8f24f07656 It asks us to prove that ...
10
votes
4answers
319 views

contour integration of logarithm

I must compute the following integral $$\displaystyle\int_{0}^{+\infty}\frac{\log x}{1+x^3}dx$$ Can someone suggest me the right circuit in the complex plane over which to do the integration? I ...
10
votes
1answer
575 views

Pointwise convergence of sequences of holomorphic functions to holomorphic functions

Let $(f_{n})_{n \in IN}$ be a sequence of holomorphic functions on the open unit disc $D$ in $\mathbb{C}$, and suppose that this sequence converges pointwise to a function $f$. By Osgood's theorem one ...
8
votes
1answer
256 views

Integral $\int_0^{\infty} \frac{\ln \cos^2 x}{x^2}dx=-\pi$

$$ I:=\int_0^{\infty} \frac{\ln \cos^2 x}{x^2}dx=-\pi. $$ Using $2\cos^2 x=1+\cos 2x$ failed me because I ran into two divergent integrals after using $\ln(ab)=\ln a + \ln b$ since I obtained ...
5
votes
1answer
224 views

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

Let $$f(x)=\large \frac{1}{e^x+e^{-x}+2}$$ Compute the Fourier transform of $f$. We can factor the denominator to get $$f(x)=\frac1{(\exp(x/2)+\exp(-x/2))^2}=\frac1{(2\cosh(x/2))^2}$$ I'm thinking ...