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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160 views

Residue at an essential singularity

Consider the function $$f(z)=\frac{e^{\frac{1}{z-1}}}{e^z -1}$$ $z_0=1$ is an essential singularity, hence $$f(z)=\displaystyle\sum_{-\infty}^{+\infty}a_n(z-1)^n$$ near to $z_0=1$ and I want to find ...
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332 views

Under which hypotheses is switching between sum and integral signs legit?

Which hypotheses are needed to change the order of sum and integral signs? Concrete example: consider the expression $$ ...
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93 views

Integration of sine^2 w.r.t. some norm

Let $||x||$ be any norm over $\mathbb R^n$. Let $B_T$ the open ball with radius $T$ w.r.t. to our norm, i.e. all $x\in\mathbb R^n$ such that $||x||<T$. Let $n\in\mathbb N$. How much ...
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52 views

Holomorphic analogue of geodesics

Let $X$ be a complex manifold with a Hermitian metric. Is there a "complex" analogue of geodesics on $X$ which is of any interest? For example, is anything known about holomorphic maps $f : \mathbb C ...
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122 views

Finding the analytic function

Find all analytic function $f: \mathbb C \rightarrow \mathbb C$ such that $|f^`(z)|$ constant on curves of the form $Ref$ constant. This is one of the past comp question. Seriously I do not know ...
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78 views

$f(r)\in\mathbb R$ $\forall$ real $r<-1$

$f$ is analytic in {$z:|z|>1$} and $f(r)\in\mathbb R$ $\forall$ real $r>1$. How can I show that the same hold $\forall$ real $r<-1$? Please don't solve it completely. I'm just looking ...
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62 views

Perturbations of algebraic varieties

Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it})$ its restriction to the torus. In the specific problem I'm considering, the set $Z=\{(s,t): ...
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320 views

Question Relating with Open Mapping Theorem for Analytic Functions

This problem is taken from Section VIII.4 of Theodore Gamelin's Complex Analysis: Let $f(z)$ be an analytic function on the open unit disk $\mathbb{D}=\{|z|<1\}$. Suppose there is an annulus ...
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180 views

Proveing that if a Holomorphic function is one to one on a circle, then it's one to one within the disk created by the circle.

I would like to prove that if a function $f(z)$ is holomorphic on $\overline{D(P,r)}$ and one to one on $\partial D(P,r)$ then $f$ is one to one on $D(P,r)$. I noticed that for $w \in f(D(P,r))$ with ...
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73 views

Complex nonlinear differential equation

I have the following nonlinear differential equation: $$\ddot z(t)-\sin(z(t))=0$$ where $z(t)$ is a complex variable. The solution of the same equation with $z(t)$ real, is a function of Jacobi ...
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774 views

Uniform distribution on the unit circle (in the complex plane)

I was trying to prove that for a standard complex Gaussian variable $Z$ it holds that $|Z|^2$ is exponentially distributed with parameter 1, $\frac{Z}{|Z|}$ is uniformly distributed on the unit circle ...
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533 views

schwarz lemma, a simple example

Let $f$ be a holomorphic function on $B(0,R)$ , R>0. Assume that there exist an $M>0$ such that $| f(z) | \le M$ $\forall z\in B(0,R)$. , and a natural number, such that : $$ 0 = f(0)=f'(0) = ... ...
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72 views

Unramified functions between Riemann surfaces

Let $F:X\rightarrow Y$ a unramified holomorphic function between two compact Riemann surfaces. I don't understand why $F$ is a covering map. By a well-known theorem $F$ is surjective; then since the ...
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221 views

Anti-holomorphic involution of $\mathbb{P}^1$

I wonder if anti-holomorphic involution of $\mathbb{P}^1$ is, up to change of coordinate, given by either $$ z\mapsto \overline{z}, \ \ \,z\mapsto -\overline{z}, \ \ \ or \ \ \ z\mapsto ...
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688 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 ...
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257 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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377 views

What are applications of Lagrange's identity?

I recently proved for homework the following identity on $\mathbb{C}$: if $a_1, \ldots , a_n, b_1, \ldots, b_n\in\mathbb{C}$, then $$ \left|\sum_{i=1}^na_ib_i\right|^2 = ...
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367 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 ...
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386 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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206 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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162 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 ...
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335 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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145 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 ...
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207 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 ...
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275 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 ...
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600 views

Why is every conformal bijection between disks a linear fractional transformation?

Why is every conformal bijection between disks actually a linear fractional transformation? I thought I could justify this claim with the following idea. Suppose $f$ is a conformal bijection from a ...
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119 views

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

I am taking a Kähler geometry course this semester. The book we use is Tian's Canonical Metrics in Kähler Geometry. I got a little confused about the calculation there in. For example, $\mathbb{C}$ ...
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149 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 ...
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185 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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77 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 ...
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126 views

Local uniformity implies Riemann integrability

Theorem. Let $f_{n}:G\to \mathbf{C}, n\in \mathbf{N}$, continuous and integrable and let $f=\lim_{n\to \infty}f_{n}$ be locally uniform in $G$. Then $f$ is continuous and integrable in $G$. ...
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296 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 ...
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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. ...
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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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106 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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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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188 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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91 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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548 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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196 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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36 views

Textbook +reference book in complex analysis

Which book can be used as an introductory textbook in complex analysis? I have some choices (more suggestions are welcomed) Marsden & Hoffman J.B. Conway Ahlfors Palka Lang Stein & ...
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46 views

Series for $\sin(z) / \sin(\pi z)$

${\sin(z) \over \sin(\pi z)} = 1/\pi + {z \over \pi} \sum_{n \in {\bf Z} \setminus \{0\}} {(-1)^n \sin(n) \over n(z-n)}$ First I apply Mittag-Leffler's theorem, to see that the RHS is a mereomorphic ...
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34 views

integral of harmonic function

I'm having trouble with this one: Let $u$ be a real-valued harmonic function on $D(0,1)$, and let $\gamma$ be a closed curve in that disk. Then $\int_\gamma u=0.$ I'm supposed to prove or disprove ...
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28 views

Finding the Laurent series given the poles and residues

I am working on the following problem, suppose that $f$ has a simple pole at $-1$ with $Res(f,-1) = 1$. A double pole at $2$ with $Res(f, 2) = 2$. Also $f(0) = 7/4$ and $f(1) = 5/2$. I am supposed ...
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48 views

Computing an integral using residues

I am trying to find an integral: $$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$ I went about applying contour integral over a semicircle with diameter along $ x = ...
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34 views

Interpreting and understanding the identity $e^{iz} = \cos(z) \pm \sqrt{\cos^2(z) - 1}$

A question in my complex analysis book (Gamelin's "Complex Analysis", question I.8.7) asks me to prove that $e^{iz} = \cos(z) \pm \sqrt{\cos^2(z) - 1}$. Using the identity $\cos(z) = \frac{e^{iz} + ...
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24 views

Finding a simple residue

The residue at the $z=7$ for $$\frac{1}{(z-7)^{5}}$$ is $0$ since the residue = $$ \lim_{z \to 7} = \frac{1}{(5-1)!}\frac{d^{4}}{dz^{4}}(z-7)^{5}\frac{1}{(z-7)^{5}} $$ $$ \lim_{z \to 7} = ...
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44 views

Finding a Taylor Expansion for the following:

I have: $$\frac{1}{1-z}$$ for $z_0=i$. I have no idea how to do the Taylor Series expansion of this, around $z_0=i$, and then show it summation form. I have: $\frac{1}{1-z} = ...
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18 views

Local normal form of a (several complex variable) holomorphic map at a point?

Suppose $F\colon\Omega\subseteq\mathbb C^m\to\mathbb C^n$ is a holomorphic map. WLOG, $0\in\Omega$ and $F(0)=0$. I want to determine the local normal form of $F$, i.e. classifying $F$ up to local ...
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25 views

Mobius transform answer check for $0$ to $2$,$-2i$ to $0$, $i$ to $\frac32$

Continuation of this question Is this the correct answer for the mobius transformation corresponding to: $0$ to $2$ $-2i$ to $0$ $i$ to $\frac32$ $$\frac{az+b}{cz+d}\cong ...