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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557 views

Solve $\int \cos^{2n}\theta d\theta$

I am trying to solve the integral $\int_0^{2\pi} \cos^{2n}\theta d\theta$ using residues. I get the wrong answer so could you please say what I am doing wrong? We start with the substitution $z = ...
3
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0answers
719 views

If $|f(z)|=1$ , $f$ is holomorphic then $f$ is constant

Let $G\subset \mathbb{C}$ be a region, $f$ a holomorphic function. Then it does hold that: If $f(G) \subset \mathbb{R} \Rightarrow f$ is constant If $|f(z)|=1$ for all $z \in G$, then $f$ is ...
3
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0answers
579 views

Integration of nontrivial trigonometric functions

First an example which I know how to solve. If we have the following integral $$\int_{-\pi}^{\pi}\frac{1}{1+3~\cos^2(t)}dt$$ there is a very practical way to evaluate it by interpreting it as some ...
3
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0answers
228 views

Properties of the lemniscate functions as meromorphic functions on $\mathbb{C}$

We consider the following function. $$u(x) = \int_{0}^{x} \frac{dt}{\sqrt{1 - t^4}}$$ $u(x)$ is defined on $[-1, 1]$. Since $u'(x) = \frac{1}{\sqrt{1 - x^4}} > 0$ on $(-1, 1)$, $u(x)$ is strctly ...
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343 views

Family of One-to-One Analytic Functions is Normal

I've been working on the following problem: Let $D$ and $D'$ be simply connected plane domains, each different from the whole plane. Suppose that $z_1 \in D$ and $z_2 \in D'$. Let $\mathcal{F}$ be ...
3
votes
0answers
358 views

Uniform convergence of analytic functions

Let $f_{n}(z), g(z)$ be entire functions, for all $n\geq 1$. Suppose that $g(x)$ doesn't vanish on $\mathbb H\cup\mathbb R$ (so we have $\frac{f_{n}(z)}{g(z)}$ analytic on $\mathbb H\cup\mathbb R$). ...
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198 views

complex analysis question

Prove that if $f$ is an analytic function in an open set containing the closed unit disk and if $k$ is a positive integer, then there exists a $z_0$ with $|z_0|=1$ and ...
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158 views

Equivalent Definitions of the Weierstass $\wp$-Function

I've come across two equivalent definitions of the Weierstrass $\wp$-function, but don't know how to prove that they are equivalent. Definition 1 $\wp(z)=cf(z)+d$ where $f$ is the elliptic function ...
3
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0answers
299 views

Jensen's Inequality for complex functions

Jensen's inequality states that if $\mu$ is a probability measure on $X$, $\phi$ is convex, and $f$ is a real-valued function, then $$ \int \phi(f) \, d\mu \geq \phi\left(\int f \, d\mu\right).$$ Is ...
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144 views

Does $\displaystyle \lim_{m \to +\infty}f_{2,m}(x)$ converge?

This is related to a previous question where, as stated there, $f_{2}(n)$ gives the greatest power of $2$ that divides $n$. Specifically the sequence $\lbrace ...
3
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194 views

Function in the Hardy space $\mathbb H^{2}$

Given a function $f(z)$, $z=x+iy, x,y\in \mathbb R$, which belongs to $\mathbb H^{2}(\mathbb C^{+})$, where $\mathbb C^{+}$ is the upper half plane Im$(z)>0$ and $f(a_{n})=0$, for all $n\in \mathbb ...
3
votes
0answers
235 views

Etymology of the word “pole”?

In his book Control System Design, Bernard Friedland writes (section 4.2, page 115): The roots of the denominator [of a rational function] are called the poles of the transfer function because ...
3
votes
0answers
105 views

Length of $\frac{\partial }{\partial z}$ in Kaehler geometry.

I am taking a Kaehler geometry course this semester. The book we use is Tian's Canonical Metrics in Kaehler Geometry. I got a little confused about the calculation there in. For example, ...
3
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0answers
146 views

Analytic function that provide $f^2(z)=z$

I am trying to solve this problem: Does there exist a function $f(z)$, that is analytic at $E=\{x+iy :x>y\}$ and provides $f^2(z)=z$ for every $z \in \mathbb C$. I have seen a solution that ...
3
votes
0answers
179 views

Constructing normalizations of algebraic curves vs constructing Riemann surfaces of functions

This question is sort of a further extension to this question I have been asking, Relation between n-tuple points on an algebaric curve and its pre-image in the normalizing Riemann surface It seems ...
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72 views

Relation between n-tuple points on an algebaric curve and its pre-image in the normalizing Riemann surface

This question is sort of an extension to this previous question of mine, Hyperellipticity (or not!) of a Riemann surface and the singularities of the curve If one knows the multiplicity of a ...
3
votes
0answers
279 views

Limit at infinity of a complex function

If $f(z)$ is an entire function such that $$ \lim_{x \rightarrow -\infty}\frac{f(x)}{|f(x)|}=1$$ where $|f(x)|$ is the modulus of $f$, and $f(x)$ is just evaluating $f$ at real $x$. What can we say ...
3
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0answers
82 views

Do the solutions to the unit equation lie dense in the complex numbers

Let $S\subset \overline{\mathbf{Q}}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers. ...
3
votes
0answers
102 views

Translating coordinates on a Riemann surface

Let $U\subset X$ be an open subset of a connected Riemann surface $X$. Let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. Let $P\in U$ be the ...
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102 views

Fermat-like equation

I'm looking for information on this kind of reverse-Fermat problem: Given $a,b,c \in \mathbb{C}$, find a $z \in \mathbb{C}$ such that $a^z + b^z = c^z$? When does such a $z$ exist? Is anything known? ...
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0answers
148 views

Describing a complex function

If $p,q,r\in \mathbb{C}$, how would one describe the curve $$\mathrm{Re}\left(pz^2+qz+r\right)=0.$$ If I write $p=p_1+ip_2$, $q=q_1+iq_2$, $r=r_1+ir_2$, and $z=x+iy$, then ...
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180 views

Change of variables in line integral with abs. value

Let $\gamma : I \rightarrow \mathbb C$ be a path. Let $g: \mathbb C \rightarrow \mathbb C$ be a biholomorphic map. Let $f$ be a holomorphic function. Consider the integral $$ \int_{g\circ \gamma} ...
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432 views

An obscure explanation in Conway's Complex analysis

I don't understand a paragraph in Conway's complex analysis at the beginning of Chapter VI page 128 (Maximum Modulus Theorem). He says: "Note that in Theorem 1.2 we did not assume that $G$ is ...
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votes
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83 views

The canonical (1,1)-form on a compact Riemann surface gives locally a subharmonic function

Let $X$ be a compact connected Riemann surface of genus $g>0$. We have the so called canonical (1,1)-form $\mu$ on $X$ defined as follows. Choose an orthonormal basis $(\omega_1,\ldots, \omega_g)$ ...
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votes
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526 views

Laplace inverse transform formula and Cauchy's integral formula

The question is about Laplace Transform and the inverse transform formula. Can the inverse transform formula be proved using Cauchy's integral formula?
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0answers
194 views

proving the saddle point method in a specific case

$1:$ the problem Let $f : U \to \Bbb{C}$ be analytic on some open set $U$ that includes the closed unit ball. Define a path $\gamma$ by: $\gamma(t) = e^{i t}$, -$\pi < t \leq \pi$ I want to ...
3
votes
0answers
227 views

Please recommend good text on complex Fourier series/analysis

I am looking for some good text/reference on complex Fourier series resp. Fourier analysis for complex (in particular holomoprhic) functions (of one variable). The more it contains on this particular ...
2
votes
0answers
13 views

Integral Asymptotics for inhomogenous phase

I'm looking for asymptotics for an integral of the form: $$F(n):=\int_{1/2-i\infty}^{1/2+i\infty} e^{\phi(n,z)}dz$$ where $\phi(n,z)=(n-n^3)\log(1-z)+n^2\log(1+z)-n\log(z)$. One can solve for the ...
2
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0answers
36 views

Need help with holomorphic functions on a domain interval removed.

I want to prove that for a region $\Omega$ with interval $I=[a,b]\subset\Omega$, if $f$ is continuous in $\Omega$ and $f\in H(\Omega-I)$, then actually $f\in H(\Omega)$. Is this problem related to ...
2
votes
0answers
31 views

Using residue theorem along a branch cut to evaluate the inverse Laplace transform

I am trying to find the inverse Laplace transform of $f(z)$ using the residue theorem. Can you please check to see if what am doing below is correct? I am not really sure about what I am doing. ...
2
votes
0answers
27 views

Calculate a complex integral using residues

Let $f(z)= \frac{2(e^\frac{1}{z}-1)(\sin^2z)}{z^3}$. Calculate $\int\limits_{\partial B_+(O,1)} f(z)\operatorname{d}z$ Could someone confirm my solution? Solution? I try to calculate the ...
2
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0answers
32 views

Riemann Zeta Function and Laurent Expansion

In the wikipedia page "1+2+3+4+..." http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF (and specifically in the section "Zeta Function Regularization")it is stated without reference ...
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0answers
20 views

Riemann's proof of his mapping theorem

I'm reviewing the proof of the Riemann mapping theorem, for which many books mention that Riemann's original proof is flawed because he didn't show the Dirichlet principle necessary for the existence ...
2
votes
0answers
21 views

An $f\in H^{1/2}$ with self-convolution, showing it is an $C^1$ function.

If $f\in H^{\frac{1}{2}}(\mathbb{R})$ is a Sobelev 1/2 function that $f=f*f$, then how do you show that $f\in C^1$ with a bounded derivative.
2
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0answers
54 views

What is the angle at critical point $z=1$ of $\left|z-\frac{i-1}{2}\right|=\frac{\sqrt{5}}{\sqrt{2}}$ under the Joukowski transform?

Question: What is the angle at the critical point $z=1$ of the image of the circle $|z-\frac{i-1}{2}|=\frac{\sqrt{5}}{\sqrt{2}}$ under the Joukowski transform? The Joukowski transform is defined ...
2
votes
0answers
40 views

Asymptotic expansion of integral (Laguerre)

Consider $$L_n = \frac{1}{2\pi i } \oint_{C'} \frac{1}{(1-t)^{\alpha+1} t^{n+1}} e^{-\frac{xt}{1-t}} dt\,\,\,\,(1)$$ where $C'$ is an anticlockwise contour around zero. Now set $\alpha = n$ and I want ...
2
votes
0answers
21 views

Invert a somewhat tricky characteristic function to find density function

I am interested in find the probability density function corresponding to the characteristic function $\phi(t) = \left(\frac{1 - i b t}{1 - i t}\right)^c$ where $c > 1$ and and $0< b < 1$. ...
2
votes
0answers
51 views

Roots of a polynomial equation where coefficients follow a geometric progression

Given a positive constant $a\in\mathbb{R}$, , and a positive integer $n$, I am interested in the roots of $x^n + \sum_{i=0}^{n-1} a^i x^{n-i-1} = x^n + x^{n-1} + a x^{n-2} + a^2 x^{n-3} +\cdots + ...
2
votes
0answers
43 views

Continuity in the complex plane

I was reading a book where it is claimed that a sufficient condition for \begin{equation} f(x)=\frac{1}{2\pi}\left|\sum_{j=0}^{\infty}\theta_je^{ix j}\right|^2 \end{equation} to be continuous and is ...
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0answers
25 views

Determine the set of points $M$ of affix of $z\in\mathbb{C}$

Determine the set of points $M$ of affix of $z\in\mathbb{C}$ such that there exists at least one real $t$ satisfying $z^2=t(t-i)$ My attempt: We look for the form $z=x+yi$ and we want there is a ...
2
votes
0answers
36 views

Meromorphic function with a simple pole and a simple zero, and satisfies an inequality. What can it be?

Describe all meromorphic functions f(z) in the complex plane with a simple pole at z=1, a simple zero at z=-1, and for which $$|f(z)|\le M|z|,$$ for $|z|\ge 2$ for some $M>0$. I know that, ...
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votes
0answers
47 views

Laplace transform via complex analysis

Let $Y(s) = \frac{2e^{-s}}{s(s^2 + 3s + 2)}$. Then the inverse Laplace transform is \begin{align} y(t) &= \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{2e^{s(t - 1)}}{s(s^2 + 3s + ...
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0answers
50 views

Is there a closed-form expression for this trigonometric Cauchy Principal Value-type integral?

Consider the following definite integral, $I(n; \theta)$. $$ I(n; \theta) = \int_{0}^{\pi} \frac{\cos(n\phi)}{\cos\phi-\cos\theta} d\phi \quad \text{where } n \in N $$ When $0 < \theta < \pi ...
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votes
0answers
20 views

Find the Schwarz-Christoffel transformation of the upper half-plane $U$ onto the region

Find the Schwarz-Christoffel transformation of the upper half-plane $U$ onto the region what I have done is considering the other region: let $x_1=-1,\ x_2=0,\ w_1=ia,\ w_2=-R$, then we have ...
2
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0answers
32 views

Misplaced complex analysis intuition on Riemann Surfaces

Next week I will be giving a lecture, based on Chapter 2.6 from Jost's book Compact Riemann Surfaces. He states the following theorem: Theorem 1 (Jost Theorem 2.6.2) Let $S$ and $\Sigma$ be Riemann ...
2
votes
0answers
26 views

Contour Integral of sin(z)/(z^2-z)

Find the integral $\int_{\lambda}\frac{\sin(z)}{z(z-1)}$ where $\lambda(t) = 10e^{it},t\in[0,2\pi]$ We notice that there are poles at $z = 0$ and $z=1$. So we can use residue theorem but I am ...
2
votes
0answers
26 views

How to integrate $\int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}$?

In Lancaster & Blundell's QFT book they show that \begin{equation}A:= \int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}\end{equation} returns a nonzero value for $x$, $t$ and $m$ ...
2
votes
0answers
55 views

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent. I use a root test: $\lim_{n\rightarrow\infty} |\frac{e^{nz^2}}{n}|^{1/n}=\lim_{n\rightarrow\infty} ...
2
votes
0answers
25 views

Solution of gaussian integral with hyperbolic cotangent

I was wondering if the integral $$I=\int_{-\infty}^{\infty}d\omega \omega e^{-(\omega/a)^2}\coth(\frac{b\omega}{2})\cos(\omega c)$$ where $a,b,c>0$ can be solved using complex countour ...
2
votes
0answers
15 views

Proof check: Continuity of an integral

$\gamma$ is a rectifiable curve in $\mathbb{C}$; For some open $G$, we have a continuous function $\phi:\{\gamma\}\times G\rightarrow \mathbb{C}$. $g(z):=\int_\gamma \phi(w,z) dw.$ We wish to show ...