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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18
votes
4answers
782 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, ...
8
votes
1answer
581 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
14
votes
2answers
403 views

Why can't I combine complex powers

I came across this 'paradox' - $$1=e^{2\pi i}\Rightarrow 1=(e^{2\pi i})^{2\pi i}=e^{2\pi i \cdot 2\pi i}=e^{-4\pi^2}$$ I realized the fallacy lies in the fact that in general $(x^y)^z\ne x^{yz}$. Why ...
13
votes
3answers
2k 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 ...
11
votes
1answer
3k views

A series expansion for $\cot (\pi z)$

How to show the following identity holds? $$ \displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1 $$
10
votes
2answers
384 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
2answers
4k views

Branch cut for $\sqrt{1-z^2}$ - Can I use the branch cut of $\sqrt{z}$?

I was trying to clarify some questions I had about elliptic integrals using http://websites.math.leidenuniv.nl/algebra/ellcurves.pdf There they define the map $$\phi\colon w\mapsto ...
5
votes
2answers
1k views

The existence of analytical branch of the logarithm of a holomorphic function

$\Omega$ is a convex open set in $\mathbb {C^n}$ and $f$ is an analytical function Edit: without zero point on $\Omega$, then can we define an analytical branch of $\ln {f}$ on $\Omega$ ?
12
votes
3answers
693 views

What contour should be used to evaluate $\int_0^\infty \frac{\sqrt{t}}{1+t^2} dt$

Could anyone help me decide what contour to use to evaluate this integral? $$\int_0^\infty \frac{\sqrt{t}}{1+t^2} dt$$ So we have simple poles at $i$,$-i$. Why does using a quarter of a circle in ...
11
votes
3answers
1k views

Fourier series of function $f(x)=0$ if $-\pi<x<0$ and $f(x)=\sin(x)$ if $0<x<\pi$

$$f(x) = \begin{cases}0 & \text{if }-\pi<x<0, \\ \sin(x) & \text{if }0<x<\pi. \end{cases}$$ My attempt: I went the route of expanding this function with a complex Fourier series. ...
10
votes
2answers
592 views

Entire functions such that $f(z^{2})=f(z)^{2}$

I'm having trouble solving this one. Could you help me? Characterize the entire functions such that $f(z^{2})=f(z)^{2}$ for all $z\in \mathbb{C}$. Hint: Divide in the cases $f(0)=1$ and $f(0)=0$. ...
9
votes
2answers
851 views

Show that this entire function is polynomial.

Let $f$ be an entire function such that $ |f(z)| \to \infty$ as $|z| \to \infty$. Prove that $f$ is a polynomial.
6
votes
2answers
573 views

How many ways to calculate: $\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}$ where $u \not \in \Bbb{Z}$

Today I have encounter a series: $$\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$$ where $u \not \in \Bbb{Z}$ . I have known a method to computer it (by Residue formula): ...
11
votes
1answer
262 views

Showing that $2 \Gamma(a) \zeta(a) \left(1-\frac{1}{2^{a}} \right) = \int_{0}^{\infty}\left( \frac{x^{a-1}}{\sinh x} - x^{a-2} \right) \, dx$

I want to show that $$2 \Gamma(a) \zeta(a) \left(1-\frac{1}{2^{a}} \right) = \int_{0}^{\infty}\Big( \frac{x^{a-1}}{\sinh x} - x^{a-2}\Big) \, dx \ , \ {\color{red}{-1}} <\text{Re}(a) <1. ...
9
votes
2answers
481 views

Problem on exponential of entire function

I am stuck at a problem which says : " Let $f$ and $g$ are entire functions such that $e^f, e^g$ and $1$ are linearly dependant over $\mathbb{C}$. Show that $f$, $g$ and 1 are also linearly ...
7
votes
1answer
398 views

Schwarz's Lemma, fixed points question

This is from an old qualifying examination question. If f is holomorphic in the unit disk $D$ and $|f(z)|<1$ for all $z\in D$. Suppose also that $f$ has two distinct fixed points in $D$ then ...
6
votes
1answer
5k views

How is Cauchy's estimate derived?

Cauchy's integral formula says $$ f^{(n)}(z)=\frac{n!}{2\pi i}\int_C\frac{f(\zeta)d\zeta}{(\zeta-z)^{n+1}}. $$ If we let $C$ be the circle of radius $r$, such that $|f(\zeta)|\leq M$ on $C$, then ...
5
votes
1answer
1k views

$\lambda-z-e^{-z}=0$ has one solution in the right half plane

Let $\lambda > 1$ , want to show that the equation $$\lambda-z-e^{-z}=0$$ has exactly one solution in the right half plane $\{z:Re(z)>0\}$. Moreover, the solution must be real.I tried to use ...
5
votes
5answers
20k views

Proof of Cauchy Riemann Equations in Polar Coordinates

How would one go about showing the polar version of the Cauchy Riemann Equations are sufficient to get differentiability of a complex valued function which has continuous partial derivatives? I ...
4
votes
2answers
255 views

which of the following is/are true for the entire function $f$?

Let , $f$ be an entire function. Let, $g(z)=\overline{f(\bar z)}$. Let, $D=\{z:Im(z)=0\}\cup\{z:Im(z)=a\}$ for some $a>0$. Then which are correct ? (A) If $f(z)\in \mathbb R$ for all $z\in \mathbb ...
3
votes
1answer
400 views

Modification of Schwarz-Christoffel integral

I found two different formulations of the Schwarz-Christoffel formula (e.g. Link1, p.20 and Link2, p. 9). The first is \begin{align*} ...
2
votes
1answer
103 views

A Tough Problem about Residue

I tried my best to solve this problem from what I learned in residues, but the solution seems very far from what I was doing!! Is there any way other than using Laurent series expansion? Here is the ...
2
votes
3answers
9k views

How to find a Laurent Series for this function

How do I give a Laurent Series on various ranges of $|z|$? I need to find the Laurent series expansion for $$f(z)=\frac{1}{z(z-1)(z-2)}$$ for the following ranges of $|z|$: $0<|z|<1$ ...
1
vote
3answers
335 views

Integral of $\log(\sin(x))$ using contour integrals

I know the integral is possible with a simple fourier series expansion of $-\log(\sin(x))$ But I am interested in complex analysis, so I want to try this. $$I = \int_{0}^{\pi} \log(\sin(x)) dx$$ ...
18
votes
2answers
2k views

What is the intuition behind the Wirtinger derivatives?

The Wirtinger differential operators are introduced in complex analysis to simplify differentiation in complex variables. Most textbooks introduce them as if it were a natural thing to do. However, I ...
11
votes
3answers
2k views

An entire function is identically zero?

I'm preparing for a PhD prelim in Complex Analysis, and I encountered this question from an old PhD prelim: Suppose $f(z)$ is an entire function such that $|f(z)| \leq \log(1+|z|) \forall z$. Show ...
8
votes
2answers
938 views

Characterize entire functions $f$ such that $|f(z)| \leq |\sin(z)|$ [duplicate]

I want to determine all entire functions $f$ such that $|f(z)| \leq |\sin(z)|$. I searched around on MathSEx and I found the following question from which I tried to get inspired but I think it ...
6
votes
2answers
991 views

Is a meromorphic function always a ratio of two holomorphic functions?

Suppose $D$ is a region (connected open set) in complex plane, and $f$ is a meromorphic function on $D$. Question: Does there always exist two holomorphic function $g$ and $h$ such that ...
5
votes
1answer
260 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 ...
5
votes
1answer
205 views

When is the composition of a function and a harmonic function harmonic?

I was looking at a comprehensive exam, and I found the this question. Can anyone help me out? If $u$ is a harmonic function, which type of function $f$ is needed so that $f(u)$ is harmonic?
4
votes
2answers
390 views

Branch point-what makes a closed loop around it special?

I am having difficulty understanding the concept of a branch point of a multifunction. It is typically explained as follows:branch point is a point such that the function is discontinuous when going ...
3
votes
1answer
686 views

Characterization of Harmonic Functions on the Punctured Disk

The following is an old qual problem I came across. If $h$ is harmonic on $D-\{0\}$, where $D$ is the unit disk, show that $h(z) = \Re(f(z)) + c \log|z|$ for where $f$ is analytic on $D- \{0\}$. ...
3
votes
2answers
750 views

Prove that $\frac{1}{\sin^2 z } = \sum\limits_{n= -\infty} ^ {+\infty} \frac{1}{(z-\pi n)^2} $

I have the following problem: Find the constants $c_n$ so that $$ \frac{1}{\sin^2 z } = \sum_{n= -\infty} ^ {+\infty} \frac{c_n}{(z-\pi n)^2} $$ and the series converges uniformly on every ...
1
vote
1answer
189 views

An issue with approximations of a recurrence sequence

By trying to give an approximation to a given recurrence sequence I encountered a problem. To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
13
votes
6answers
1k views

Natural derivation of the complex exponential function?

Bourbaki shows in a very natural way that every continuous group isomorphism of the additive reals to the positive multiplicative reals is determined by its value at $1$, and in fact, that every such ...
9
votes
1answer
544 views

Polynomial bounded real part of an entire function

Let $f(z)$ be an entire function whose real part is bounded by a polynomial in $|z|$. Does it follow that $f(z)$ is a polynomial? Or, without loss of generality and more suggestively ...
7
votes
2answers
1k views

What is the radius of convergence of $\displaystyle\sum z^{n!}$?

How could you find out the radius of convergence of $\displaystyle\sum z^{n!}$? I'm used to applying the ratio test to power series of the form $\displaystyle\sum a_{n}z^{n}$, but for a different ...
6
votes
3answers
1k views

Using Residue theorem to evaluate $ \int_0^\pi \sin^{2n}\theta\, d\theta $

can you please guide me on evaluating this integral using residue theorem and binomial theorem $$ \int_0^\pi \sin^{2n}\theta\, d\theta $$ for $n = 1,2,3$ Honestly, I do not even know where to start, ...
6
votes
2answers
521 views

Entire, $|f(z)|\le1+\sqrt{|z|}$ implies $f$ is constant

I am stuck on the following question. Given that $f$ is an entire function with $|f(z)|\le1+\sqrt{|z|}$ for all $z\in \mathbb{C}$, show that $f$ is constant. Can anyone give me a hint to get me ...
5
votes
1answer
374 views

Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$

I need some help with this problem: Let $f\colon D \to D$ analytic and $f(z_1)=0, f(z_2)=0, \ldots, f(z_n)=0$ where $z_1, z_2, \ldots, z_n \in D= \{z:|z|<1\}$. I want to show that $$|f(z)| ...
4
votes
2answers
923 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
3answers
2k views

Complex Integration poles real axis

In class my professor said that $$ \int_{-\infty}^{\infty}\frac{e^{iax}}{x^2 - b^2}dx = -\frac{2\pi}{b}\sin(ab) $$ where $a,b > 0$. However, since the poles are on the real axis, isn't the integral ...
3
votes
2answers
2k views

Laurent-series expansion of $1/(e^z-1)$

Find the Laurent series for the given function about the indicated point. Also, give the residue of the function at the point. $$ \frac{1}{e^z - 1} $$ at $z_0=0$(four terms of laurent series). I ...
3
votes
2answers
144 views

What is a simple form of this integral?

This integral reminds me of something familiar but I cannot remember the rule to make it simple. $$\int_{-\infty}^{+\infty} \frac{\exp(i a \cdot v)}v \mathrm d v$$ where $a$ is a scalar for ...
3
votes
2answers
179 views

The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive ...
3
votes
2answers
601 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
100 views

Prove the function is continuous, exercise from Conway's “Functions of One Complex Variable I”

For the first proof of Cauchy's integral formula, Conway in his book "Functions of One Complex Variable" (Chapter IV, section 5.4) uses the following claim: Let $G$ be an open subset of $\mathbb ...
2
votes
1answer
458 views

Integration of $\ln $ around a keyhole contour

I want to evaluate the following integral: $$\int_{0}^{\infty}\frac{\ln^2 x}{x^2-x+1}{\rm d}x$$ I use the following contour in order to integrate. I considered the function $\displaystyle ...
2
votes
1answer
407 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
1
vote
2answers
782 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?