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, ...

learn more… | top users | synonyms (2)

3
votes
1answer
97 views

Computing residues of $\cot(\pi z)/z(z+1)$ with symmetries

I would like to know if there is a quick way of computing the residues of $$f(z) = \frac{\cot \pi z}{z(z+1)}$$at the points $z = 0$ and $z = -1$. They are double poles. Expanding this in Laurent ...
8
votes
0answers
165 views

Help with the integral $\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$

Referring to a previous question, i want help with the integral : $$\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$$ Where $n,m$ ...
3
votes
2answers
62 views

If $f$ has an essential singularity at $0$, there is a sequence $z_n \to 0$ such that $z_n^n f(z_n) \to \infty$

Here's a problem I was just working on: Let $f$ have an essential singularity at $0$. Show that there is a sequence of points $z_n \to 0$ such that $z_n^n f(z_n)$ tends to infinity. I know ...
3
votes
2answers
37 views

Help evaluating residue with simple poles

I am having a bit of trouble evaluating $$\sum_{k=1}^3{ \rm Res}\left(\frac{\log(z)}{z^3+8};z_k\right)$$ where $z_1=2e^{i\pi}$, $z_2=2e^{i\pi/3}$ and $z_3=2e^{i5\pi/3}$. I know that each $z_k$ is a ...
1
vote
0answers
24 views

Why are rational numbers required in cusps of congruence subgroups?

While we consider the action of congruence subgroups on $\mathbb{H}$ (the upper half plane), we compactify using an additional point at infinity, that is fine. But why do we add even all rational ...
1
vote
1answer
56 views

If $|f(z)+g(z)|<|f(z)|+|g(z)|$ on $\Omega$ prove that $f$ and $g$ have the same number of zeros in $\Omega$

Let , $f$ and $g$ be analytic on $\Omega$ and continuous on $\Omega\cup \partial \Omega$. If $|f(z)+g(z)|<|f(z)|+|g(z)|$ on $\Omega$ prove that $f$ and $g$ have the same number of zeros in ...
-1
votes
0answers
33 views

Pointwise convergence to uniformly convergence on compact

Let $D \subset \mathbb{C}$ be a domain and $E$ be a subset of $D$ which has a limit point in $D$. And let $\{f_n\}$ be a sequence of holomorphic functions on D. Suppose that that $\{f_n\}$ converges ...
1
vote
0answers
20 views

A question about the differentiability conditions for a fractional power

Now consider the fractional power $$f(x)=z^{m/n}=|z|^{m/n}(\cos \frac{mArgz}{n}+i \sin \frac{mArgz}{n})$$ Since $f(x)$ satisfies the Cauchy-Riemann equations, and is therefore differentiable, we can ...
5
votes
1answer
137 views

Prove that $f$ is a polynomial

If $f(z)$ is an entire function and $|f(z)|\ge1$ for all $z$ with $|z|\ge \pi$ then show that $f$ is a polynomial. I tried to apply Lioville's theorem on $f$. For $|z|\le \pi$ , $|f(z)|\le k$ for ...
0
votes
1answer
33 views

$\frac{\partial u}{\partial x} \cos \theta + \frac{\partial u}{\partial y} \sin \theta = \frac{\partial u}{\partial r} (z_0 + r e^{i\theta})$?

Let $u$ be a function of two variable and all its partial derivative exists and fix $z_0 \in \mathbb C $ and $r>0.$ My vague question: How to show: $\frac{\partial u}{\partial x} \cos \theta + ...
3
votes
1answer
43 views

Showing pre-image under entire function is simply connected.

I am currently working on the following problem and have run into a bit of trouble: Consider an entire function $f$ s.t. $\overline{B_1(0)}\subset f(\mathbb{C}).$ Show that V, a component of ...
1
vote
1answer
29 views

Why only congruence subgroups for modular forms?

When we define modular forms, why do we restrict ourselves to congrumence subgroups? Why not any subgroup of finite index? Or, even more generally why not any subgroup? Is it just a matter of ...
1
vote
2answers
35 views

Schwarz reflection principle, understanding the conjugated function:

Given a symmetric region $\Omega$, say, symmetric w.r.t. the real line, and f(z) defined and analytic only on $\Omega^{+}$, we can analytically continue the function to $\Omega^{-}$ with the analytic ...
0
votes
1answer
27 views

Why does a conformal mapping create a full tiling of semi-infinite strips in the w-plane?

I know that, specifically for linear fractional transformations, symmetric points get mapped to symmetric points. So, if the real line gets mapped to a circle, then under a LFT, points symmetric ...
4
votes
0answers
36 views

Laplace transform of the logarithmic integral function

What is the Laplace transform of the logarithmic integral function $\text{li}(t)$. Meaning, how to compute the integral : $$\int_{0}^{\infty}\text{li}(t)e^{-st}dt$$
0
votes
0answers
23 views

One side half twist mobius band are one to one transformation on extended complex plane

I try to understand how one side half twist mobius are one to one transformation on extended complex plane $C^{*} = \mathbb{C} \cup \infty$ I don't ask for full proof, but I do want "something" ...
5
votes
0answers
40 views

Asymptotic behavior of the generalized polygamma function

The generalized polygamma function$^{[1]}$$\!^{[2]}$ is defined as $$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1$$ where ...
2
votes
0answers
98 views

A difficult integral $\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$

Here is an integral that I want to see a different approach: $$\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$$ Well, for someone who is deeply aware of the exponential integral function and the ...
3
votes
3answers
90 views

What is the value of $\sum e^{-n} \sin^2 n$?

Clearly the series $\sum_1^\infty e^{-n} \sin^2 n$ converges. If I put it into Maple, I get an exact value: $$ -\frac {{\rm e}^1 ( {\rm e}^1 (\cos(1))^2 + (\cos(1))^2 - {\rm e}^1 -1 ) }{-4{\rm e}^2 ( ...
1
vote
1answer
30 views

how to prove complex rational function has this property

Let's say we have a rational function $f$ (i.e polynom divided by polynom) , and assume that $f$ has no poles in the upper plane $\{z;Imz \geq 0\}$. we have to prove that: $$sup\{|f(z)|; Imz \geq ...
3
votes
3answers
59 views

Continuity of distance function

I wonder if this is obvious because it does not appear to me obvious at all: Reference: [Hormander: An introduction to Complex Analysis in Several Variables], page 37: Here is the quote Now, let ...
2
votes
1answer
32 views

Existence of an entire function with certain property

Let $\{a_n\}$ and $\{b_n\}$ be two sequence of complex numbers such that $|a_n|\to\infty$ as $n\to\infty$. Prove that there exists an entire function $f:\Bbb C\to\Bbb C$ (i.e. $f$ is complex ...
1
vote
2answers
97 views

map the UHP to an equilateral triangle

Explain how the upper half-plane can be mapped one-to-one and conformally onto an equilateral triangle. Thanks,
1
vote
2answers
47 views

Integration in a semicircle

Evaluate : $$\int_\Gamma(z^2+3\bar z)\,dz$$ where $\Gamma$ is the upper half of the unit circle from $-1$ to $+1$. First put , $z=e^{i\theta}, 0\le\theta\le\pi$ and take a negative sign as the ...
0
votes
1answer
38 views

Composition of analytic functions is analytic

I want to find a proof that shows the composition of two analytic functions is analytic. I know I should prove this using Cauchy-Riemann equations, but I wasn't able to use them in the proper way in ...
0
votes
1answer
18 views

Describe the Riemann surface for $w^2=z^2-1$.

Describe the Riemann surface for $w^2=z^2-1$. [Editorial note: I'm posting this to save an answer on another question, which actually answers this question instead.]
3
votes
2answers
460 views

Compute this integral, using a method other than the Residue Theorem,

$\int_0^\infty$ $\frac{1}{1+x}$$\frac{dx}{\sqrt{x}}$ Part (a) asks to compute the integral by means of the residue at x = -1. I have done this just now, and the answer is $\pi$. Part (b) asks, "can ...
1
vote
1answer
18 views

Bound on imaginary parts of family of analytic functions

Question: Let $ F$ be the set of holomorphic maps $f$ from the unit disc into the upper half plane, such that $f(0)=i$. Show that the supremum of the imaginary parts, $\sup_{f\in F}$ Im[$f(\frac i ...
2
votes
1answer
29 views

What does complexification mean in complex analysis, .e.g., in residue calculus,

I've learned complexification formally in a graduate linear algebra class. But what does the word mean in the setting of complex analysis? If I consider a real integral on the positive half line, ...
6
votes
1answer
114 views

A particular integral: $\int_{-\infty}^{+\infty}\frac{\sin(\pi x)}{\prod_{k=-n}^{n}(x-k)}\,dx$

I have to show summability, then compute the following integral: $$\int\limits_{-\infty}^{+\infty} \frac{\sin(\pi\,x)}{\prod_{k = - n}^n (x - k)}\,dx = \frac{(-4)^n}{(2\,n)!}\,\pi $$ for every $n\in ...
1
vote
1answer
26 views

Simple bijective map from upper half plane to whole plane

What is a simple bijective map from the upper half plane to the whole plane? (Either the complex plane or $\mathbb{R}^2$ can do.) I think it can be done via a map from upper half plane to unit disc ...
1
vote
0answers
32 views

Show that Riemann Theorem does not hold when set is not simply connected

Riemann Theorem states that for any simply-connected domain in $\mathbb{C}$ (which is not whole $\mathbb{C}$) there exists biholomorphic map onto the open unit disk. I find it hard to show that we ...
0
votes
2answers
45 views

when does the complex square root exists?

Let $\Omega\subset \mathbb{C}$ be an open connected subset not containing the origin. Define $f(z):\mathbb{C}\to \mathbb{C}$ by $f(z)=z^2$. What conditions on $\Omega$ do we need so there exists a ...
0
votes
2answers
34 views

Radius of convergence of complex series

I need help for this exercise: We consider the following sequence of function $(f_n)_{n\ge0}$: $$f_n:\mathbb{C} \rightarrow \mathbb{C}$$ $$z \mapsto \frac{1}{p_n}[z(1-z)]^{4^n}$$ where $p_n$ is the ...
2
votes
3answers
63 views

Evaluate the given limit in $C_r=\{re^{i\theta}:0\le \theta \le \pi\}$

Let , $C_r=\{re^{i\theta}:0\le \theta \le \pi\}$ denotes the semicircle traversed clockwise. Show that $$\lim_{r\to 0}\int_{C_r}\frac{e^{iz}}{z(z^2+1)}\,dz=-\pi i$$ I can not use the Jordan's ...
5
votes
0answers
85 views

An open and connected subset $U\subseteq \mathbb C$ is still connected if you remove a curve that lies entirely in $U$

Let $U\subseteq \mathbb C$ be open and connected. If $f:[0,1]\rightarrow U$ is continuous with $f(0)\neq f(1)$ and $f(s)\neq f(t)$ for $s\neq t$, then $U\setminus f([0,1])$ is connected. This seems ...
4
votes
3answers
51 views

Rouché's theorem and maximum modulus principle

I have a problem: Suppose $f$ is analytic on closed disk $(\bar{\mathbb{D}} = \{z \in \mathbb{C} : |z| \leq 1 \})$. Assume that $|f(z)| = 1$ for $|z| = 1$, and that $f$ is not constant. Show that ...
0
votes
1answer
20 views

Complex Polynomial with roots in uppar half plane.

Let $p$ be a complex one variable polynomial. Suppose all zeros of $p$ are in upper half plane. Then which of the following is/are true 1.$ Im\frac{p^{\prime}(z)}{p(z)} >0 ~for~ z\in \mathbb{R} ...
0
votes
1answer
28 views

How do I recognize branch points?

For instance, $z^2$ - 1 has branch points at i and -i, but that doesn't seem obvious at all - and writing this function using the exponential and complex logarithm functions doesn't seem to help ...
1
vote
3answers
60 views

Evaluatig: $\int_{0}^{\infty}{e^{ax^2}\cos(bx)dx}$

Evaluatig: $$\int_{0}^{\infty}{e^{ax^2}\cos(bx)dx}$$ Where $a, b\in \mathbb R^+$ What i have done: Because $\cos(bx)=\Re(e^{ibx})$, we can note that: ...
1
vote
2answers
85 views

Show that $\left|\dfrac{z-a}{1-\bar a z}\right|=r$ represents a circle

Suppose $|a|<1$ and $r\in (0,1)$. Show that the set of complex number $z$ satisfying $\left|\dfrac{z-a}{1-\bar a z}\right|=r$ is a circle in complex plane. Find the centre and radius of this ...
0
votes
2answers
41 views

$f$ is an entire function satisfying the given condition . Show that the function is constant

If an entire function $f(z)$ satisfies $$|f(z)|\le \frac{1+|z|}{1+\sqrt {|z|}}$$ for all $z\in \mathbb C$ then show that $f=c$ with $|c|\le 2(\sqrt 2-1)$. First we consider a function ...
1
vote
0answers
54 views

Integrals of the type $f'(z)/f(z)$

I am having trouble understanding integrals of the form: $$\int_\gamma\frac{f'(z)}{f(z)}\,{\rm d}z$$I am aware that there are problems with the complex logarithm, and we have the formula: ...
2
votes
2answers
65 views

Evaluating: $\int_{-\infty}^{\infty}{\frac{1}{\cosh(kx)}dx}$

How can you integrate: $$\int_{-\infty}^{\infty}{\dfrac{1}{\cosh(kx)}dx}$$ I know that: ...
3
votes
1answer
55 views

finding such function

Is there a function $f(z)$ satisfying: (1) $f$ is analytic in some region containing $|z|\leq 1$ (2) The only zero of $f$ in $|z|\leq 1$ occurs at $1/2+i/2$ and it has order 3. (3) $|f(z)|=1$ in ...
1
vote
1answer
67 views

Localizing the zeros of $z^4+z+1$

Consider the complex polynomial $p(z) = z^4+z+1$. I want to check that this polynomial has exactly one zero in each quadrant of $\Bbb C$. I do not want to solve the equation $z^4+z+1 = 0$ (as I ...
0
votes
1answer
59 views

Out of all the proofs of the PNT, which one is the most accessible?

I have been studying the continuation of the Riemann zeta function $\zeta(s)$ for the past while. I can prove that all the zeroes must lie in the critical strip.I am currently in the process of using ...
1
vote
1answer
33 views

Identity theorem - holomorphic functions

Let $f$ and $g$ be holomorphic functions on some connected open set $D\subset\mathbb{C}$. Then if set $A=\{z\in\mathbb{C} : f(z)=g(z) \}$ has limit point in $D$, then $f\equiv g$ on $D$. But what if ...
5
votes
4answers
133 views

calculate $\int_{0}^{\pi} \frac{dx}{a+\sin^2(x)} $using complex analysis

where $a>1$ calculate $$\int_{0}^{\pi} \dfrac{dx}{a+\sin^2(x)}$$ I tried to use the regular $z=e^{ix}$ in $|z|=1$ contour. ($2\sin(x) = z-\frac1z)$, but it turned out not to work well because ...
1
vote
0answers
25 views

Questions on Levi pseudoconvex domain

Here are some of the exercise questions which I am stuck: Question 1: Give an example of a real hypersurface $M\subset\mathbb{C}^n$ such that $0\in M$, such that $M$ has a polynomial defining ...