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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321 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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92 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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49 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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121 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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301 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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178 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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68 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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720 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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66 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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215 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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741 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 ...
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593 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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237 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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351 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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363 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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199 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 ...
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305 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 ...
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199 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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242 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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107 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, ...
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147 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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180 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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73 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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282 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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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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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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181 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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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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531 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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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 ...
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21 views

Claim about the z-transform of a discrete function

Claim: $\lim_{k\to\infty} x[k]$ exist and if finite is $X(z)$ the Z-transform of $x[k]$ has no pole in $|z|>1$ and at most 1 pole at $z = 1$ Attempt: \begin{align*} X(z) &= ...
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27 views

Prove there is no branch of arg $z$ on $0 < z < 1$.

If $G$ is an open connected subset of $\mathbb{C}$ that does not contain the origin, we call a continuous function $\alpha$ satisfying $\alpha(z) = \text{arg} z$ for all $z \in G$ a branch of arg $z$. ...
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27 views

What are conditions for an infinite sum with a complex parameter not to be analyitically extendable?

I'm looking for a sequence $f(n)$, so that $g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$ with $z$ so that this converges classically, defines a function which can not be ...
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15 views

How to determine the order of the poles of this particular function?

Consider the function $$f(z) := \frac{1}{z^{2n}+1} \quad .$$ The denominator is equal to zero when $z^{2n} = -1$, so its zeros $z_{k}$ are located at $z_{k} = e^{\frac{\pi i}{2n}+\frac{k \pi i}{n}} $. ...
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20 views

How to visualize bilinear transform of the form $S(z) = \frac {T}{2} \frac {z+1}{z-1}$

Note that this is the bilinear transform from a z-domain as appears in Z-transform to s-domain in Laplace transform Recall that bilinear transform has form $M(z) = \frac{az+b}{cz+d}$ with and has to ...
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23 views

Proper holomorphic map of unit disc

My problem is to prove that every non-constant proper holomorphic map of unit disc into itself is product of finitely many disc automorphisms. As a hint I have: If $f$ is proper and $f=Mg$ where $M$ ...
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60 views

Is this a legit way to visualize complex functions?

I am doing laplace transform in a class and I hate how there seems to be no graphical support when things are transformed to laplace domain i.e. nobody cares what they look like in laplace domain But ...
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27 views

Are there any applications for complex analysis in population dynamics?

Strange question: could complex analysis be used to understand population dynamics? I'm interested in modelling dominance hierarchies, mating relationships, and illness behaviour in ancient ...
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34 views

Prove that $\mathrm{Re}(z_1\overline{z_2})=|z_1||z_2|\;\mathrm{iff}\;\arg(z_1)-\arg(z_2)=2n\pi$, where $n\in\mathbb{Z}$.

I've been sitting here for an hour thinking about how to approach this problem. I can't seem to connect $z_1$ $z_2$ with its arguments, I mean can't $z_1$ and $z_2$ be anything? Can someone please ...
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42 views

When does $\wp$ take real values?

Whittaker and Watson mention that when the invariants of Weierstrass $\wp$ function are such that $g_2^2 - 27g_3^2 > 0$, and if $2\omega_1$ and $2\omega_2$ are its periods then the function takes ...
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48 views

How it comes that integral of odd function is not even?

I know a rule that integral of an odd function on the complex plane should be an even function. For instance, $$\int \sin z\, dz=-\cos z +C$$ $$\int z\, dz=\frac {z^2}2 +C$$ etc. Yet, integral of ...
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80 views

What is the value of the integral $ \int_{-\infty}^{\infty}\frac{\sin(2x)}{x^3}dx$?

I tried to evaluate the integral $$ \int\limits_{-\infty}^{\infty}\frac{\sin(2x)}{x^3}dx$$ using residues but the answer comes out to be a negative value, $-2 \pi$, which seems strange. Any help on ...
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74 views

Evaluating a sum $-\zeta'(2)$

Is it possible to obtain any closed-form expression for the infinite sum $$\sum_{n=1}^{\infty}\frac{\log(n)}{n^{2}}$$ by Residue calculus? My thought was to try to integrate $$f(z) ...
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25 views

Difficulties with the “correspondence” between essential singularities and formal Laurent series

To fix terminology that I'm not 100% sure is universal, let the ring of Laurant series about $0$ be $\mathbb{C}[[z]][z^{-1}]$, and the ring of formal Laurant series about $0$ be ...