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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483 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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157 views

Hölder Condition Implying Uniform Convergence

Define $g(z)=\frac{1}{2\pi i}\int_{-k}^k \frac{h(\zeta)}{\zeta-z}d\zeta$, where $h$ is continuous and defined on $[-k,k]$. Let $|h(x)-h(y)|\leq |x-y|^\alpha$ for all $x,y\in[-k,k]$ and for some ...
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260 views

circle of convergence

How do you analyse for convergence/divergence on the circle of convergence of a particular power series?
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186 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 ...
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216 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 ...
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46 views

Any other operators that may convert algebraic function into transcendental ones

As we know, the integral may convert or map a rational function or algebraic function into a transcendental one. Are there any other operators that may convert a rational function or algebraic ...
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35 views

Is this growth condition satisfied by Dirichlet series?

Suppose that we have $a_n=\mathcal{O}(n^k)$ for some $k \in \mathbb{R}$. Thus, the following Dirichlet serie : $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ is absolutly convergent in the ...
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68 views

What is the limit of this sequence of complex numbers?

Let $z_1$ and $z_2$ be two complex numbers in the upper half-plane. Does the sequence $c_n = \exp^n\left(\sqrt{\log^n(z_1)*\log^n(z_2)}\right)$ converge to a fixed point as $n\to\infty$? If so, what ...
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67 views

another question about singularity and converge uniformly.

a)Prove if $n\in \mathbb{Z}$ then, $\frac{1}{\sin^2{z}}-\frac{1}{(z-\pi n)^2}$ has a removable singularity at $z=\pi n$ when $n=0$. Actually this is true for all $n$. b) Prove that ...
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56 views

Evaluate $\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx$

Is there a closed form for the integral $$\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx?$$ where $\lambda>0$, $a>0$, $d>0$ and where $b$, ...
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39 views

Why does the Riemann Xi function $(\xi(s))$ have order of growth 1

Why does $s(s-1)\xi(s)$, have order of growth 1? In other words, why is it that $\forall \epsilon > 0 $ $\exists A_{\epsilon},B_{\epsilon} \in \mathbb R_+$ so that $\forall s \in \mathbb C$, ...
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22 views

A question on normal family

Let $f$ be an entire function. Suppose that the sequence $F_0=\{f(0),f'(0),f''(0),\cdots,f^{(k)}(0),\cdots \}$ is bounded. Show that $F_z=\{f(z),f'(z),f''(z),\cdots,f^{(k)}(z),\cdots \}$ is a normal ...
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53 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
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44 views

$\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.

Suppose that $A$ is a bounded linear operator on a complex Banach space $X$ with resolvent set $\rho(A)$. If $C$ is a simple closed smooth curve in $\rho(A)$ such that $$ ...
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69 views

Finding the limit of sum $\sum \frac{1}{n^4}$

I'm trying to use the reside theorem to find the limit of $\sum \frac{1}{n^4}$. So I am considering the function $f(z) = \frac{\pi \cos(\pi z)}{\sin (\pi z)z^4}$ on a square contour. Now I am ...
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47 views

Complex Contour Integration - Complex Analysis

I'm just practising for my upcoming exam, and I've come across a question I'm having a bit of difficulty with. I've been asked to show the following; $$\int_{0}^{\infty} \frac{dz}{\cosh(z)} = ...
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114 views

What is ${\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$?

This is a new integral that I propose to evaluate in closed form: $$ {\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$$ where $\log (z)$ denotes the principal value of ...
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29 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ ...
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36 views

Is the image of the Riemann sphere a Riemann surface?

I am looking for conditions on $f$ such that $f:S^2 \to \mathbb C$ is such that the image, $f(S^2)$ is a Riemann surface. Must $f$ be analytic, or something stronger? Or are there no simple conditions ...
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101 views

Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form ...
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35 views

an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$ I simply wish ...
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0answers
26 views

Show that $P_a(z)=0$ iff $z=N(a)$ for polynomial $P_a$

Let for $a_0=(a_0,a_1,...,a_n)\in\mathbb C^{n+1}$ the polynomial $P_{a_0}=\sum_{k=0}^na_kz^k$ and $z_0\in\mathbb C$ with $P_{a_0}(z_o)=0$ and $DP_{a_o}$ (the differential matrix) invertible. Show ...
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40 views

Integration of irrational Functions

Integrate the irrational Functions $$f(z)=\int_{0}^{z}\{(\xi-1)(\xi+1)(\xi-i)\}^{-\frac{2}{3}}d\xi=?$$ PS:This question comes from Schwarz–Christoffel mapping, ...
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22 views

Does this series converge to a rational function?

Consider the series $$\sum_{k=0}^{\infty}\frac{1}{k!(z-k)^k}$$ Is there any domain of the complex plain where this series converges to a rational function? Any hints will be appreciated
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24 views

How to choose branches of function for computer code

I need to code up the correct way my computer (in python) views the function $\sqrt(z^2+1)$. I want the branch cuts to extend from $i \to i\infty$ and from $-i \to -i\infty$. I was hoping someone ...
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0answers
26 views

Does sum over primes of $p^{-z}$ diverge for all Re(z) = 1?

Let the function q(z) of one complex variable z be the sum over all primes p of (1/p^z). I was wondering about the complex zeros of q(z) [hoping that this problem might be much easier than the same ...
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33 views

Fractional linear transformations and the extended complex plane in a more abstract context?

Does anyone know of an "abstract algebra-esque" treatment of the extended complex plane and the Mobius transformations? I am studying complex analysis now, and I am a little frustrated that my ...
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19 views

Extensions of the Hermite Bielher and Hermite-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...
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52 views

Convergence of $\sum_{n=1}^{\infty}(n+1)z^n$

Consider the series $\sum_{n=1}^{\infty}(n+1)z^n$ A) For which complex numbers $z$ does this series converge? B) For those $z$, let $f(z)$ be the sum of the series and find $f(z)$. C) Evaluate ...
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99 views

An Integral possibly related to Legendre polynomials

Consider the integral $$\int_0^1\frac{(t^2-1)^a}{(t-u)^{b+1}}dz$$ where $b\gg a$, with $a,b$ integers and $u>1$. I know you can write this integral as the sum of two hypergeometric functions but ...
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0answers
33 views

Why is it that linear fractional transformations cannot take the upper half-plane to the first quadrant?

Problem: Is there a linear fractional transformation that takes the upper half-plane to the first quadrant? If yes, write it down. If no, give proof. Attempt: We have that a linear ...
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0answers
38 views

Complex integral and parametrization of a circle

I am trying to compute the following integral of $$\int \frac{1}{z^3+3} dz$$ over a circle of radius $2$, centerd at $(2,0)$. Thus I am trying to compute the residue and have found that the function ...
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44 views

When to Interchange Limit & Integral

I got stuck in the proof of Cauchy's Integral Formula for higher derivatives: Under what Conditions over a function $f$, we can infer that : $\displaystyle\lim_{h\rightarrow ...
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35 views

Order of entire function.

Show that $$1)\ \ \ \ \ \ \ f(z)=\frac{\Gamma^2(1+d)}{\Gamma(1+d+z)\Gamma(1+d-z)}, \ \ d\in\mathbb R$$ $$2)\ \ \ \ \ \ \ f(z)=\frac{\Gamma^2(1+\bar d)}{\Gamma(1+\bar d+z)\Gamma(1+\bar d-z)}, \ \ \bar ...
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38 views

Moving the branch cut of the complex logarithm

The complex logarithm is defined as $\log z:=\operatorname{Log} |z|+i\arg z$ , with the branch cut on the non-negative real axis. Determine a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$ ...
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62 views

Changing research area in grad school

I'm a PhD student about to close out my third year. My current research area is operator algebras. At the beginning of this semester I completed my qualifying exams (this was accomplished a semester ...
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28 views

Image of $\{z:|\mathfrak{Re}(z)|<1\}$ under $f$

If I have a set of complex numbers, $S=\{z:|\mathfrak{Re}(z)|<1\}$ and I apply the mapping $$f:\mathbb{C}\rightarrow\mathbb{C}$$ $$f(z)=(1+i)z+1$$ How can I write the image set? I know that the ...
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0answers
25 views

Clarification on the absolute convergence of Mellin transform

I have a question I haven't been able to find a direct answer to that I presume is true but I am unable to show. We know these two following results on the mellin transform. If $$\int_0^\infty ...
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0answers
53 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
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74 views

the partial fraction of ${\pi}\cot{\pi}z$ from the partial fraction of $\frac{\pi^{2}}{\sin^{2}{\pi}z}$

I want to deduce the equation $${\pi}\cot{\pi}z=\frac{1}{z}+ \sum_{n=1}^{\infty} \frac{2z}{z^{2}-n^{2}}$$ where the convergence is uniform on compact subsets of $\mathbb{C}-\mathbb{Z}$ from the ...
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100 views

Good book on analytic continuation?

For my Bachelor's thesis, I am investigating divergent series summation methods. One of those is analytic continuation. There are quite a few books on complex analysis that include a chapter or two on ...
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0answers
32 views

This $\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{in x}dx}{1+\tan^m(x)}$ integral: does a closed form exist?

$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{in x}dx}{1+\tan^m(x)}$$ Does a closed form for the above exist, ideally for $n,m\in\mathbb{C}$ (most bounds probably removed at one point using ...
2
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0answers
73 views

Coefficients of the Weierstrass $\wp$'s Laurent expansion

I am trying to study the Weierstrass $\wp$ function using a combination of texts from Alfors; Cartan; Freitag and Busam; and Siegel. But I am having some trouble because I would like to try to avoid ...
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0answers
48 views

Why does the space of germs construction correspond to the gluing construction of Riemann surfaces?

I know this might be too broad / vague a question, but still looking for somebody to write something meaningful about this. When constructing the Riemann surfaces, why does the space of germs ...
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245 views

How do I study Stein & Shakarchi's Complex Analysis

I'm currently self-studying some complex analysis. My background is limited: single- and multivariable calculus, linear algebra, introductory Fourier analysis and matrix theory. Each course, with ...
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44 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
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72 views

Question on the Fourier Transform, specifically concerning polynomials

Suppose $P$ is a polynomial of degree $\ge 2$ with distinct roots, none lying on the real axis. Calculate: $$\int_{- \infty}^{\infty}\frac{e^{-2 \pi i x \xi}}{P(x)}dx,\space \space \space\xi \in ...
2
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0answers
43 views

Integral on the circle

It's a standard fact that to calculate integrals of the form $$\int_{0}^{2\pi}\mathcal{R}(\cos(\theta),\sin(\theta)) \ d\theta$$ with $\mathcal{R(x,y)}$ a rational function in two variables without ...
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0answers
87 views

On Goldbach conjecture

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq ...
2
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0answers
92 views

Perhaps obvious integration that I do not see

Context: This is the Laurent Series section of my text, which can be found here on pages 129-131. It is right after he derived the expression for Laurent coefficients for a function that is analytic ...