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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14
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408 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 ...
12
votes
3answers
3k 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
5k 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 \int_0^w\frac{\...
9
votes
2answers
974 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.
3
votes
1answer
448 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*} z=w(\zeta)=&A+C\int\limits^{z}\prod\limits_{k=1}^n\left(\zeta-...
12
votes
3answers
721 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 ...
7
votes
1answer
2k 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 ...
6
votes
1answer
6k 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
2answers
3k 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
3answers
2k views

definition of winding number, have doubt in definition.

could any one tell me why in the definition of index number or winding number of a curve $\gamma(t)$ around some point $a$ we take this integral : $$\frac{1}{2\pi i}\int_{\gamma}\frac{1}{z-a} $$ why ...
2
votes
1answer
105 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
10k 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<|z|...
13
votes
1answer
282 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. \tag{1}...
11
votes
1answer
800 views

Radius of convergence of power series

Given a meromorphic function on $\mathbb{C}$, is the radius of convergence in a regular point exactly the distance to the closest pole? As Robert Israel points out in his answer, that this is of ...
9
votes
1answer
756 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 \{0\}...
8
votes
2answers
484 views

Exist domains in complex plane with only trivial automorphisms?

Does exist open domain in $\mathbb C$ who has only identity for holomorphic automorphism? Related question: does exist open domain in $\mathbb C$ so that every holomorphic automorphism has fixed ...
8
votes
2answers
982 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 ...
8
votes
2answers
6k views

An entire function whose real part is bounded must be constant.

Greets This is exercise 15.d chapter 3 of Stein & Shakarchi's "Complex Analysis", they hint: "Use the maximum modulus principle", but I didn't see how to do the exercise with this hint rightaway, ...
7
votes
2answers
1k 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 $f=\frac{g}{...
7
votes
2answers
4k views

Prove that the zeros of an analytic function are finite and isolated

Let us assume that the zeros of $f = \{Z_1,\ldots,Z_n,a\}$ are infinite and converge towards $a$. The book which I am reading says that any neighborhood of $a$ will contain infinite zeros. Since $f$ ...
6
votes
1answer
2k views

Analytic functions of absolute value 1 on the boundary of the unit disc

Is there a characterization of analytic functions $f$ on the unit disc such that $|f(z)|=1$ for $|z|=1$? If $f$ only has a zero $a\in D(0,1)$ of order $n$, then $f(z)=\phi_a(cz^n)$ for some constant $|...
5
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 ...
5
votes
1answer
264 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
3answers
653 views

A difficult integral evaluation problem

How do I compute the integration for $a>0$, $$ \int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx? $$ I want to find a complex function and integrate by the residue theorem.
5
votes
1answer
1k views

Equality of triangle inequality in complex numbers

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda ...
5
votes
5answers
172 views

Intuitive Explanation why the Fundamental Theorem of Algebra fails for infinite sums

We know that for every polynomial of n.order Fundamental Theorem of Algebra guarantees n complex roots. Lets consider the complex exponential function $f(z)=\exp(z)$. As $f(z)$ is holomorphic, we are ...
5
votes
1answer
173 views

Finding a trigonometric polynomial

I'm trying to solve exercise 5 in chapter 14 of Rudin's Real & Complex Analysis: Suppose $f$ is a trigonometric polynomial, $$f(\theta) = \sum_{k=-n}^n a_k e^{ik\theta}$$ and $f(\theta) &...
4
votes
2answers
404 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 ...
4
votes
2answers
218 views

proving $\csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8$

How can I prove the following identity using complex variables $$ \begin{align*} 1) & \csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8 \\...
3
votes
3answers
1k views

Proving theorem connecting the inverse of a holomorphic function to a contour integral of the function.

I am asked to prove this theorem: If $f:U \rightarrow C$ is holomorphic in $U$ and invertible, $P\in U$ and if $D(P,r)$ is a sufficently small disc about P, then $$f^{-1}(w) = \frac{1}{2\pi i} \...
1
vote
3answers
440 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. $$I = \int_{0}^{\pi} \log(\sin(x)) dx$$ ...
1
vote
1answer
130 views

Suppose $f$ is entire and $|f(z)| \leq 1/|Re z|^2$ for all $z$. Show that $f $ is identically $0$.

This is a problem from my complex analysis textbook. The hint is to consider $g(z)=(z-iR)^2(z+iR)^2 f(z)$ and to show that $|g(z)| \leq 8R^2$. This is what i have tried: Consider $Re z \geq 0$, then ...
10
votes
2answers
6k views

Value of Summation of $\log(n)$

Context: I am learning Dijstra's Algorithm to find shortest path to any node, given the start node. Here, we can use Fibonnacci Heap as Priority Queue. Following is few lines of algorithm: ...
8
votes
1answer
1k views

On the growth of the Jacobi theta function

So, I ran into this exercise from Stein & Shakarchi. CA, Chapter 5: Show that if $\tau$ is fixed with positive imaginary part, then the Jacobi theta function $$\theta(z | r) = \sum_{n=-\infty}^{\...
8
votes
3answers
2k views

Prove the open mapping theorem by using maximum modulus principle

The open mapping theorem says a non constant analytic function maps open sets to open sets. The maximum modulus principle says if $f$ a non constant analytic function on an open connected set $D\...
6
votes
2answers
589 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 ...
6
votes
3answers
2k 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, ...
5
votes
3answers
1k views

erf(a+ib) error function separate into real and imaginary part

Is there an easy way to separate erf(a+ib) into real and imaginary part?
5
votes
1answer
492 views

Mean value property implies harmonicity

It is fairly easy to show that harmonic functions satisfy the mean value property, but it seems harder to show the converse. I've seen the following theorem without proof: If $u \in C(\Omega)$ ...
5
votes
1answer
220 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?
5
votes
2answers
2k views

The family of analytic functions with positive real part is normal

I'm having difficulty with the following exercise in Ahlfors' text, on page 227. Prove that in any region $\Omega$ the family of analytic functions with positive real part is normal. Under what ...
5
votes
1answer
659 views

Proving that a Function is Analytic Given that it is Equal to the Complex Conjugate of an analytic Function

So I'm working a problem that states: A function $f$ is analytic in an open set $U$. Define $g$ by $g(z)=\overline{f(\overline{z})}$ (just because the notation can be hard to read, this is the the ...
4
votes
1answer
802 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 $D- \{0\}$. ...
4
votes
2answers
760 views

Convergence power series in boundary

Say I have a power series $\sum_{k=0}^\infty a_k z^k$ with radius of convergence $0<R<\infty$. What can be said topologically about the set $\{z\in\Bbb C\mid |z|=R\,\mbox{ and }\sum_{k=0}^\infty ...
1
vote
1answer
161 views

$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$

Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. I want to demonstrate that if $$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$$ then $f$ is linear. I know this is a well-known ...
1
vote
1answer
64 views

The line integral $\int_{\gamma}\frac 1z$ and branchs of logarithm

Fix $w=re^{i\theta}\neq 0$ and let $\gamma$ be a rectifiable path in $\mathbb{C}\setminus\{0\}$ from $1$ to $w$. Show that there is a $k\in\mathbb{Z}$ such that $\displaystyle\int_{\gamma}\dfrac 1z=\...
1
vote
1answer
191 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 ...
18
votes
1answer
513 views

Where is the topology hiding in this theorem on entire functions?

There is a standard theorem that says if $f$ is analytic in the whole complex plane and never zero, then it has the form $e^g$ where $g$ is analytic in the whole plane; i.e. we can define a function $\...
9
votes
1answer
599 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 $$(f(z)=\sum_{k=...
7
votes
2answers
1k views

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

How to find the radius of convergence of $\sum z^{n!}$? I'm used to applying the ratio test to power series of the form $\sum a_{n}z^{n}$, but for a different power of $z$, I am a bit stumped. What ...