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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Question about branches of functions (complex power)

I have the following question, I really appreciate if someone can help me to clarify ideas and I apologize if is a stupid question: This is from Conway's complex analysis book: Let $f: G \to ...
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63 views

holomorphic complex function such that $f(\frac{1}{n})=n\space$ but $f$ is not identically $1/z$

Question: Find a function $f(z)$ holomorphic on $\{0<|z|<1\}$ such that $f(\frac{1}{n})=n\space$ for each integer $n >1 $, but so that $f$ is not identically $1/z$. I attempted to solve ...
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123 views

Differentiability implies continuity in $R^2$

Let F be a function from $R^2$ to $R^2$. F is differentiable at a point (a,b) in $R^2$, prove that F is continuous at this point. Can i write F(x,y)= F(a,b)+ c(x-a)+ d(x-b)+e where c,d,e are real ...
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77 views

Strange Holomorphic Function

Let $f$ be holomorphic function on unit disk and it is continuous on boundary of the disk. It is well-known that $f$ is constant and equal to zero if $f$ is vanishing on sub-arc of boundary (Maximum ...
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49 views

Prove that $\frac{1}{z+i} +\sin(z)=0$ has infinite solutions over $\mathbb{C}$

Prove that $\frac{1}{z+i} +\sin(z)=0$ has infinite solutions over $\mathbb{C}$ Can someone give me a clue?
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1answer
28 views

On Compactness in Runges theorem

Let $f$ be holomorphic function in an open set $\Omega$ in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions, converging uniformly to $f$ on $\Omega$. For each $f_n$, let ...
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70 views

Can $\int_{-a}^{a}\frac{\sqrt{a^2-x^2}}{\log(\frac{4}{b}\sqrt{a^2-x^2})}e^{ikx}dx$ be found in closed form?

I am trying to see if it is worth pursuing to try to calculate the following integral analytically: \begin{align} \int_{-a}^{a}\frac{\sqrt{a^2-x^2}}{\log(\frac{4}{b}\sqrt{a^2 ...
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49 views

Analytical evaluation of complex root at real extrema

The following is based on my belief/sense: If a Circle radius R is shifted up along y-axis to $ (0,h) \; [ h>R $] , then two complex roots exist due to imaginary intersection of Circle with ...
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1answer
50 views

Show that $\sum_{n=1}^{\infty}n^{i(z^2+a)}$ represents an analytic function.

Let, $a\in \mathbb R$ be fixed. Find the set of $z\in \mathbb C$ for which $$\sum_{n=1}^{\infty}n^{i(z^2+a)}$$ represents an analytic function. I know that if the radius of convergence of a power ...
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234 views

Does there exist an analytic function $f$ such satisfying the following three conditions?

Does there exist an analytic function $f:\{z\in \mathbb C:|z|<1\}\to \{z\in \mathbb C:|z|<1\} $ such that, $f(0)=1/2$ , $f(1/2)=1/3$ , $f(1/3)=1/4$ ? I tried through the Schwarz-Pick lemma ...
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1answer
74 views

evaluating $\int_0^{k!}e^{i\frac{t^k}{k!}} dt$

How to evaluate the following integral?$$ \int_0^{k!}e^{i\frac{x^k}{k!}} dx $$ Here $k$ can be any positive integer. When $k=2$, I can square it and use polar coordinates. But I've no idea about the ...
2
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3answers
60 views

Evaluate $(-1)^x$ by hand.

How to calculate the value for $(-1)^x$ for any $x$ by hand. Using Mathematica I kind of figured $(-1)^x=\cos(\pi x)+i\sin (\pi x)$. but how can I prove this. This is my first question here. Sorry ...
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4answers
119 views

How to show $\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$

I need to show the following: $$\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$$ For all complex numbers z. I don't know how to start this. Should I use l'Hopitals rule somehow?
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1answer
33 views

Is this right? Difference between evaluating and expressing to cartesian form $z= 1 + \sqrt{3} i $

Am i going right with this? $$z= 1 + \sqrt{3} i $$ i need to i) evaluate $z^9$ and ii) express in cartesian for $z^5$... Which i'm a bit confused with. First what i did was find the polar form... ...
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100 views

Recursion Formula Euler Numbers

I am trying to derive the formula $$\displaystyle\sum_{k=0}^{n}{2n\choose 2k}E_k = \displaystyle\sum_{k=0}^{n}{n\choose k}^2E_k=0$$ Where $E_k$ are the Euler Numbers. The approach that I have taken ...
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1answer
45 views

help proving function is constant

Let $f\in O(C)$ be entire function. Assume there is a disk $D_r(z_0)$ such that $f(C)\cap D_r(z_0)=\emptyset$ prove that f is constant. I need help starting the proof. I tried to using Liouville ...
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1answer
88 views

Question about Eremenko's paper on iteration of entire functions

I have two questions about Eremenko's paper On the iteration of entire functions On the second page, it says "The family $\{f^n\}=\{\underbrace{f\circ f\circ\ldots\circ f}_{n}:n\in \mathbb{N}\}$ is ...
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2answers
102 views

Consider $\left(\frac{z-2}{z}\right)^4 = 1$ then solve $z^3-3z^2+4z-2=0$

Consider $\left(\frac{z-2}{z}\right)^4=1$ then solve for the roots of $z^3-3z^2+4z-2=0$ First consider $$\left(\frac{z-2}{z}\right)^4=1$$ Let $$w^4=\left(\frac{z-2}{z}\right)^4=1$$ $$w^4=1$$ Now ...
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1answer
76 views

Prove that $f=u+iv$ is differentiable if and only if $\lim_{r→0} \frac{1}{πr^2 } \int_{C(z_0,r)}f(z)dz=0$

Suppose that $u,v$ are real-valued function that having continuous partial derivative of first order in the neighborhood of $z_0=x_0+iy_0$ . Prove that $f=u+iv$ is differentiable if and only if ...
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62 views

laurent series convergence

Find the Laurent Series representations in powers of $z$ for i) $\frac{ \cos{z}}{z}$ ii) $z^4 \cosh{\frac{1}{z^2}}$ Where do they converge? I found the Laurent Series for each of the functions, ...
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72 views

Suppose $\lim\limits_{z\to z_0} g(z) = B \neq0$. Prove that There exists $\delta > 0$ such that $|g(z)|> 1/2|B|$ for $0 < |z-z_0|< \delta$

Suppose $\lim\limits_{z\to z_0} g(z) = B \neq0$. Prove that There exists $\delta > 0$ such that $|g(z)|> 1/2|B|$ for $0 < |z-z_0|< \delta$ I wanted to know if my proof is correct. ...
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1answer
45 views

Evaluate $∫_γ \frac{z^2+1}{z(z^2+4)} dz$ Where $γ(t)=re^{it}$ with $0≤t≤2π$ for all possible value of $r$, $0<r<2$ and $2<r<∞$

Evaluate $∫_γ \frac{z^2+1}{z(z^2+4)} dz$ Where $γ(t)=re^{it}$ with $0≤t≤2π$ for all possible value of $r$, $0<r<2$ and $2<r<∞$ Theorem: Let $f: G \to \mathbb C$ be analytic, suppose ...
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1answer
140 views

Does there exist an analytic function s.t. $f\left(\frac{1}{n}\right)=2^{-n}.$

Prove that there does not exist an analytic function $f$ in an unit disc containing $0$ such that $$f\left(\frac{1}{n}\right)=2^{-n}.$$ I tried by using Identity theorem. Suppose that $f$ is ...
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1answer
82 views

Complex Taylor Series Circles of Convergence

I am trying to find the Taylor Series and circles of convergence for three different functions. i) $\frac{\sin{z}}{z}$ which I determined the Taylor series to be $\sum_{n=0}^\infty ...
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75 views

A necessary condition for a multi-complex-variable holomorphic function. [closed]

Let $\Omega\subset \mathbb{C}^n$ be an open unit ball, $f:\Omega \to\mathbb{C}$ is a bounded function. For $a \in \mathbb{C}^n$, define $$ ...
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38 views

Differential equation of the form to find

Lets $f(z)$ is some analytic function on complex plane and $y(z)$ is known analytic function on complex plane. Problem statement: find all $f(z)$ that: $$f(z) = f(z\frac{\partial}{\partial z})y(z)$$ ...
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1answer
99 views

Prove f,g are holomorphic constant functions

Let $\Omega$ be an open (and convex) subset of $\mathbb{C}$ and $f,g$ $\in$ $\mathcal{H}$$(\Omega)$. If $ $ $| f(z_0)|$+ $|g(z_0)|$ $\geq$ $|f(z)|$+$|g(z)|$ $ \forall z$ $\in \Omega$ $ $ with ...
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44 views

An idenity related to Gamma function

I know that for gamma function we have $$\int_0^{\infty}v^{k}e^{-av}dv=a^{-k-1}\Gamma(1+k),$$ given that $\Re(k)>-1$ and $\Re(a)>0$. Question: Now considering ...
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1answer
67 views

Radius of convergence $\sum\limits_{n\ge0}q^{n^2}z^n$

Radius of convergence of the power series $\sum\limits_{n\ge0}q^{n^2}z^n$ By a theorem the radius is $\limsup \lvert q^{n^2}\rvert^{1/n}=\limsup\lvert q^n\rvert=\begin{cases}0,&\text{if}\ ...
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115 views

If $f:D\to D-\lbrace 0 \rbrace$ is Holomorphic with $f(0)=1/2,$ then $\vert f(1/2)\vert\ge 1/8$

Let $D$ denote the open unit disc centered at $0\in \mathbb{C}$ and suppose $f:D\to D-\lbrace 0 \rbrace$ is analytic and $f(0)=1/2$. Show $\vert f(1/2)\vert \ge 1/8$. The only techniques that ...
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499 views

Evaluating $\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx$ using complex analysis

I'm taking a complex analysis course and doing some practice computing residues & evaluating integrals. I pulled out an old book called "The Cauchy Method of Residues: Theory and Applications, ...
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18 views

For all $M \gt 0$ there exists $z \in G$ such that $|z|^3+z+\overline{z} = M$

I am trying to show that if $G$ is an open, unbounded, connected set and if $0 \in G$ then for every $M \gt 0$ there exists $z \in G$ such that $|z|^3+z+\overline{z} = M$. Here's what I did and I ...
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28 views

Represent the following curves in the z-plane in the form z(t)

Represent the following curve in the z-plane in the form z=z(t). $$y = {x^2}\quad 1 \le x \le 3 $$ I know the answer is $$z(t) = t + i{t^2}\quad 1 \le t \le 3 $$ however I don't have any idea how to ...
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23 views

Usage: Holomorphic Functions

This isn't a math question, but rather a question is word usage in mathematics. Why do people say " $f$ is an isomorphism," or "$f$ is a diffeomorphism," but in complex analysis, we say that "$f$ is ...
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1k views

About the integral $\int_{0}^{+\infty}\sin(x\,\log x)\,dx$

It is an interesting exercise to show that the function $f(x)=\sin(x\log x)$ is Riemann-integrable over $\mathbb{R}^+$ (as shown by robjohn in this related question, for instance). Even more ...
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60 views

Show that $\lim_{r\to 0} \frac{1}{r^2}\int_{C_{r}}f(z)dz = 2 \pi i\frac{\partial f}{\partial \bar{z}}(z_0)$

Well, after spending hours on this problem, I'm still stuck, so I thought I'd turn to you guys. The problem statement is as follows. Let $f$ be a complex-valued function that is $C^1$ in the disk $|z ...
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1answer
61 views

Simple Complex analysis integration

If we let $\gamma$ be the circline path from $ 0$ to $1$, how do we list all possible values of $$\int_{\gamma} z^3dz$$ One of which, I think, could be over the real axis, s.t. $$\int_{\gamma} ...
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1answer
55 views

Using Gaussian function in the proof of Weierstrass approximation theorem.

I am trying to understand the details of one version of the proof of the Weierstrass approximation theorem. In order to prove it, we use a lemma that states that if our function $f : ...
2
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1answer
65 views

If h is a holomorphic non-vanishing function in the complex plane then h is the exponential of another function.

Show if $h(z)$ is a holomorphic function such $\forall$$z\in$$\mathbb{C}$ $h(z)\neq0$ then $\exists$ $H(z)$ such $h(z)=e^{iH(z)}$. I think I should define the Taylor series of $h(z)$ and then ...
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1answer
76 views

Detail in proof of Hartog Theorem

I am stuck in the middle of what you can see below: when the book says "we can repeat the same construction [...] horizontal strip arbitrary close to $z_2=0$": We define different sets $E_l$ on any ...
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160 views

Generalization of the argument principle

This exercise is from big Rudin: Let $f \in H(U)$ and $D(a,r)\subset U$ be a disk s.t. $f$ has no zero on the boundary of the disk. Let $\gamma$ be a curve parametrizing the boundary of ...
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112 views

Characterization of simply connected domain

Let $D$ be a domain (open and connected) in $\mathbb{C}$. Then show that the following are equivalent: (1) $D$ is simply connected (in homotopy sense); (2) $\mathbb{C} \cup \{\infty\}$ is connected; ...
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25 views

Boundary for coefficients of a complex polynomial

I'm trying to solve the following problem: prove that if $ f(z) = \sum\limits_{k=0}^m a_k z^k, ~~ \sup\limits_{|z| = 1} |f(z)| = M $, then $ \sum\limits_{k=0}^m |a_k|^2 \leq 2 \pi M^2$ I don't ...
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120 views

Winding number of composition of curve

Let $f$ be analytic in a "simply connected domain" D $\subseteq \mathbb{C}$ and let $\gamma$ be a closed piecewise differentiable path in $D$. Set $\beta = f \circ \gamma.$ Show that $\frac{1}{2\pi ...
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1answer
56 views

Power Series with singularities in {z: |z|=1}

Prove that all the points in $D=\left\{ z \in \mathbb{C} : \mid z\mid=1 \right\}$ are singularities of the function $$ f(z)=\sum_{n=0}^{\infty} \frac{z^{n!}}{n!} $$ This was easy for the ...
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2answers
65 views

How to calculate the argument and its limit for the sequence $z_n=-2+i\frac{(-1)^n}{n^2}$

I am trying to show that the limit of the sequence $$z_n=-2+i\frac{(-1)^n}{n^2}$$ exists, using the polar representation. Note that $\lim_{n\rightarrow \infty }z_n=-2$. $$$$I am finding difficulty in ...
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1answer
47 views

How do we know that $\tan z$ or $\cos z$ don't have singularities off of the real axis?

Since $\sin z$ is bounded, $\tan z = \frac{\sin z}{ \cos z}$ has singularities when $\cos z = 0$. We know $\cos z = 0$ for $z = \frac{\pi}{2} + n\pi$ for $n \in \mathbb{Z}$, but could it not also be ...
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1answer
29 views

Equation of a Circle Mobius Transformation

I am able to prove that w= $e^{i\theta}$ $\frac{z+\alpha}{1+\bar{\alpha} z}$ maps circles to circles for a given problem, however I am not to familiar with this particular equation, I would appreciate ...
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3answers
32 views

Cauchy Riemann to find differentiable function

Use Cauchy-Reimann equation to determine all differentiable function that satisfy $$\text{Im} f(z)=x^2-y^2$$ what I did: $$v(x,y)=x^2 - y^2$$ $$u_x= v_y =-2y$$ $$u(x,y) =-x^2 + K(y)$$ ...
2
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1answer
60 views

radius of convergence of a complex power series

Can you tell me what you think about my solution to this problem? In case it's wrong, or needs changes, just something like "try looking at this", "consider that"... a hint is enough, please no ...