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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7
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2answers
79 views

Question regarding usage of residue theorem in a specific case

I'm looking over the solution of an exercise in a course I'm taking and there's something I simply don't understand. Let $f(z)=\pi\cot(\pi z)$ and $\varphi(z) = \frac{1}{z^2}$. $f$ has poles of order ...
0
votes
1answer
57 views

Possible application of maximum modulus princple

Suppose that $f,g$ are analytic on $\{ |z| \leq 1\}$ with $g \neq 0$ on $\{ |z| < 1\}$. Prove that $|f(z)| \leq |g(z)|$, $\forall z \in \{z = 1 \}$, implies $|f(0)| \leq |g(0)|$. Here is my ...
0
votes
2answers
297 views

simple tools to extract Re,Im,Abs… of any complex function

I've developped my own set of simple yet powerful tools to work on complex functions. I would like to know if these simple tools are currently used in complex analysis. Let's $z = x + i y = |z| ...
5
votes
2answers
136 views

How many values does $1^{\alpha}$ have for $\alpha$ irrational?

One such value is $\displaystyle\cos\left(2\pi\alpha\right)+i\sin\left(2\pi\alpha\right)$, by Euler's theorem. On the other hand, we can choose an arbitrary sequence $S=(a_n)_n$ of rational numbers ...
1
vote
1answer
103 views

Prove $\sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} $is divergent

How do I show that the following power series is divergent? $$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$ where $t$ is complex 1-dimensional, $x$ ...
0
votes
1answer
88 views

Ahlfors' method of calculating the fundamental group of a punctured disk

In Ahlfors' complex analysis text, page 297 he determines the fundamental group of the punctured disk $\{0<|z|<\rho \}$. His approach is to divide any loop $\gamma$ into short enough arcs, by ...
0
votes
1answer
61 views

Uniform and absolute convergence of complex series to $\log(1+z)$

Show that the series $\displaystyle\sum_{n=1}^{\infty} (-1)^{n-1}\frac{z^n}{n}$ converges uniformly and absolutely to $\log(1+z)$ on the open disk where $\log(\rho e^{i\theta})=\log(\rho)+i\theta$ ...
0
votes
1answer
37 views

Computing integral

Can anyone help me with working out the following integrals: $\int_{\alpha} f(z)dz$ and $\int_{\alpha}|f(z)||dz|$ for $f(z)=(Rez)^2$ and $f(z)={\overline{z}}^n$. I tried to work out it with ...
1
vote
2answers
103 views

Infinite product convergence

Prove that $$\prod_n\left(1+\frac{i}{n}\right)$$ diverges. But $$\prod_n\left\vert 1+\frac{i}{n}\right\vert$$ converges. I know the theorem $\prod (1+z_k) $ converges $\iff$ $\sum\log (1+z_k)$ ...
2
votes
1answer
162 views

Weierstrass $\wp$-function: $(\partial_z \wp(z,\omega))^2$

Let $\vartheta(z,\omega)$ be the Riemann theta function. For $j \in \mathbb{Z}$ let $c_j$ be the coefficient of $z^{j}$ in the Laurent expansion of $\partial_z \log \vartheta \left(z + \frac{1 + ...
0
votes
1answer
193 views

Dirichlet Problem on the unit disk

Find a C-harmonic function in the unit disk with boundary values $x^3-xy$. I know the answer is $u(x,y)=\frac{(x^3-3xy^2)}{4} + \frac{3x}{4} - xy$ but don't know how to solve it Any hint or help is ...
1
vote
1answer
105 views

Convergence radius: $R = \lim_{n \rightarrow \infty} \frac {\mid a_n \mid} {\mid a_{n+1} \mid}$ (incl. $\infty$) when $R = 0$ and Ratio test

I have read the following proof of a theorem in a textbook of mine, and I've been wondering why the proof holds when $$R = \lim_{n \rightarrow \infty} \frac {\mid a_n \mid} {\mid a_{n+1} \mid} = 0$$ ...
3
votes
2answers
94 views

entire function with $f(n) = f'(n)$ for every integer $n$

Show that there exists an entire function $f$ such that $f(n) = f'(n)$ for every integer $n$, and such that the range of $f$ includes both $0$ and $1$. I have tried quite a few different things but ...
1
vote
1answer
31 views

Prove that $S_R^+(b)$ and $S_r^+(a)$ are homotopic in a domain $D$.

currently I'm working on the following exercise: Let $D \subset \mathbb C$ be a domain. Let $a,b \in \mathbb C$ and $r,R > 0$ such that $B_r(a) \subset B_R(b)$ and \begin{align*} A := \{z ...
0
votes
1answer
58 views

Relationship between two power series.

Problem: If $f(z) = \sum_{n =1}^\infty a_n z^n$, what is $\sum_{n = 1}^\infty n^3 a_n z^n$ ? I can prove the radius of convergence of the power series $\sum_{n = 1}^\infty n^3 a_n z^n$ will be same ...
2
votes
3answers
131 views

Prove f has at least one fixed point on the boundary

Let $$f(z) = \frac{a-z^2}{1- \bar a \cdot z^2}$$ where $a \in D=\{|z| <1\} $. Denote the boundary as $S =\{|z| =1\} $. Show that $f$ has at least one fixed point $w \in S$. Obviously, $f(S) ...
0
votes
1answer
29 views

Angle between of a sum of unit vectors on a plane

Assume we have N unit vectors $a_1, \dots, a_N$ on a plane. Let $\arg a_i$ be an angle between x axis and vector $a_i$ (vectors $a_i$ can be treated as complex numbers). Is it true that $$\arg ...
3
votes
1answer
784 views

Number of roots of trigonometric polynomial

Exercise 1.8.5 of Berenstein-Gay "Complex variables" asks to count the number of zeroes in $(0,2\pi)$ of certain trigonometric polynomial. Towards the exercise, the book suggests to first show the ...
0
votes
1answer
27 views

Expectation of argument of a complex random variable.

Let $\xi$ be a complex random variable, assume that argument of $\xi$ is always between $-\pi$ and $\pi$. Is next true? $$ Earg(\xi) = arg(E\xi)$$
2
votes
1answer
66 views

Can we integrate $\int_a^b t^i f(t) \, dt$

If we are given reals $a$ and $b$, and we have a function of $t$, $f(t)$, we can analyze the integral: $$\int_a^b t^i f(t) \, dt$$ ...where $i$ is the imaginary number. I'm wondering if we can ...
1
vote
1answer
222 views

Contour Integral of Exponential

I want to show the following for $a > 0$: $$e^{-a} = \int_{0}^{\infty}{\frac{e^{-x}}{\sqrt{x}}e^{-a^{2}/(4x)}dx}.$$
2
votes
2answers
191 views

Behavior of holomorphic functions on the boundary of the unit disk

$\textbf{Problem.}$ Suppose $f$ is holomorphic on the unit disk $\mathbb{D}$. Show there are points $a_n\in \mathbb{D}$, $a\in \partial \mathbb{D}$, and $b\in \mathbb{C}$ such that $a_n\to a$ and ...
1
vote
1answer
101 views

Formula for $\Gamma (\frac{1}{2} + i t)$

I have been working on the following problem for my complex analysis class involving Euler's Gamma function: For $$\Gamma (s) := \int_0 ^{\infty} t^{s-1} e^{-t} \,dt \ , \ Re(s)>0$$ Show that ...
1
vote
2answers
91 views

Find a formula for $\Gamma(\frac{n}{2})$ for positive integer n.

Find a formula for $\Gamma(\frac{n}{2})$ for positive integer n. I know the following relations; $\Gamma (z+1)=z\Gamma (z)$ and $\Gamma(n+1)=n!$ Please give me a way how to show this. Thank ...
2
votes
1answer
66 views

Cauchy's Theorem for some region A\{z,w…}

Does Cauchy's Theorem hold when we integrate $f$ over a region $A$ such that $f$ is continuous on all the points of $A$ but not analytic (holomorphic) on a finite number of points $z,w...$ contained ...
0
votes
0answers
43 views

Cauchy's Theorem for complex integrals with a condition

Does Cauchy's Theorem for complex integrals hold if I have the following scenario: I integrate a function $f$ which is analytic (holomorphic) over a region $A$ except for a point $z_0$. If that ...
1
vote
1answer
212 views

Contour Integration (Choice of Contour)

Let $ \alpha \le 0 $ and $\sigma > 0$ . I want to choose a contour, including $ [\sigma - iR, \sigma+iR] $ , such that i can apply Cauchy's Residue theorem and evaluate: $$ \lim_{R \rightarrow ...
2
votes
2answers
156 views

How to plot complex functions on the paper by your hand?

I want to know the exact method of plotting complex function used by human, computer, and whatever who can do mathematics. For example how should I plot this : $w = u+iv$ , $z = x+iy$ , $w= f(z)= z^2$ ...
1
vote
2answers
90 views

Integrating $f(x)=e^{2x}+\cos(4x)$ from $0$ to $ \pi$ with complex variables?

Is it possible to integrate $f(x)=e^{2x}+\cos(4x)$ from $0$ to $ \pi$ with complex variables? I read in the book "Calculus with Complex numbers - John B.Reade", that you can replace $\cos(4x)$ with ...
1
vote
2answers
78 views

Identity Principle for square roots

Suppose that on a Domain, $D,$ there exists analytic functions $f(z), g(z)$ so that $\Re \sqrt{f(z)} = \Re \sqrt{g(z)}$ and $f(0)=g(0)=0$ on $D.$ Here the branch is taken to be the principal branch ...
4
votes
2answers
393 views

Conformal map from disk with a slit to the upper half plane

Find a conformal map from the set $\{|z|<1, \Re{z} > 0\}\backslash [0,1/2]$ to the upper half plane. The main problem I am encountering is that the boundary of the given domain is comprised of ...
-1
votes
1answer
53 views

question on holomorphic functions [closed]

Let f be a nonconstant holomorphic function in the unit disc {|z|<1} such that f(0)=1. Then it is necessary that there are infinitely many points z in the unit disc such that |f(z)|=1 f is ...
2
votes
2answers
167 views

Evaluating the sum $\sum_{n=1}^{\infty}\dfrac{(-1)^{n}}{n^{2}}$

I am tasked to evaluate the sum $$\sum_{n=1}^{\infty}\dfrac{(-1)^{n}}{n^{2}}$$ Using contour integration. This is what I've done so far. Let $C_{N}$ be the square defined by the lines ...
4
votes
1answer
134 views

Prove that $\Gamma'(1)=-\gamma$

Use the product formula for $1/\Gamma(z)$ to prove that $$\Gamma'(1)=-\gamma$$ I know that for Euler constant $\gamma$, $$\frac{1}{\Gamma(z)} =ze^{\gamma z}\prod _{k=1}^{\infty} ...
1
vote
1answer
139 views

The range of the complex function $f(z)=\frac{z}{(1-z)^2}$ on the unit disk

how can I find the range of the function $$f(z)=\frac{z}{(1-z)^2}$$ over the unit disk $\{z\in C;|z|=1\}$. I could not get anything by writting $z$ as $x+iy$ or $re^{i\theta}$.
1
vote
1answer
87 views

estimate of a holomorphic function in a unit disc

Prove that if $a\not \in \bar{\mathbb{D}}$, then $$\inf_{c\in\mathbb{C}}\left(\sup_{z\in\mathbb{D}}\left|\frac{z-c}{z-a}\right|\right)=\frac{1}{|a|}$$ My Observation: If $a\not \in ...
1
vote
2answers
175 views

Gamma function in complex analysis.

Prove that $$ \Gamma\left(z\right) = \lim_{n\to \infty}\int_{0}^{n}t^{z - 1}\left(1 - {t \over n}\right)^{n}\,{\rm d}t \quad\mbox{for}\quad \Re z \gt 0 $$ I know that $$ {\rm e}^{-t/n} = 1 - {t ...
0
votes
2answers
99 views

How to find the complex function $u + iv$ if $u - v$ is given?

Find the complex function $u + iv$ if $u - v$ is given, where $u$ and $v$ are the component functions of the complex function $$f(z)=u(x,y) + iv(x,y).$$ Here, $z = x + iy$.
1
vote
4answers
380 views

The complex gamma function

Show that $$\Gamma (z+1)=z\Gamma (z)$$ $\forall z\in \Bbb C$ except for $z=-n$ where $n\in \Bbb N$. I know that the gamma function is defined as $\Gamma (z)=\int_{0}^{\infty}e^{-t}t^{z-1}dt$ And ...
2
votes
1answer
110 views

Upper bound for coefficients of a power series

I am doing the following problem. Suppose $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is an analytic function on the unit disc $|z|<1$. Let $0<r<1$. Prove that $$|a_n|r^n\leq \max\{4A(r),0\}-2Ref(0),$$ ...
0
votes
1answer
49 views

The univalent domain of $\cos z$

I have seen in an exercie of Chapter 1 of "Concise Complex Analsysis" by Sheng Gong that the domain $$D=\{z\in C; a<\Re z<a+\pi\}$$ for any fixed $a\in R$ is an univalent domain of $\cos z$. But ...
1
vote
2answers
122 views

The range of complex -valued functions

How can I find the range of $\phi(z)=\frac{1}{2}(z+1/z)$ for $z$ in the upper plane $Im z>0$, or for $z$ outside the unit sphere. It is really difficult to do it...as I have seen Finding the Range ...
1
vote
1answer
139 views

$\arccos$ of an imaginary number

How can I solve a $\arccos$ of an imaginary number? like: $$\cos x = 0.9i$$ Because I can't make the $\arccos$ of a imaginary number
8
votes
1answer
266 views

Entire function $f(z)$ bounded for $\mathrm{Re}(z)^2 > 1$?

Let $z$ be a complex number and $\mathrm{Re}$ denote the real part. Does there exist a nonconstant entire function $f(z)$ such that $f(z)$ is bounded for $\mathrm{Re}(z)^2 > 1$ ?
3
votes
1answer
87 views

What can you conclude about the holomorphic function f?

The problems is as follows: Suppose $0<r<1, A=\{z\in\mathbb{C}: r<|z|<1\}, f:\bar{A}\to \mathbb{C}$ is continuous, and $f$ is holomorphic on $A$, and vanishes on the unit circle. What can ...
0
votes
0answers
43 views

Meaning of degenerate complex plane

Let $f : N \to C^3 \times C^3$ be a holomorhic map where $N$ is an open subset of $C^2$ and $\phi : U \to N$ such that $q= \phi(z)$ be a solution to the equation $\langle{f(q),z}\rangle = 1 $ where ...
1
vote
0answers
64 views

Are this maps a conformal?

Are this maps (As a maps of the extended complex plane) a conformal? 1) $w = \operatorname{Ln} z,\ \lim_{t \to0}{w(1 + it)} = 2\pi i,\ D = \{0 < y < 1, x > 0\}$ 2) $w = z^3,\ D = ...
1
vote
2answers
293 views

Find the Laurent series at the given point and its residue

a) $\displaystyle \frac{\sin(z)}{(z-\pi)^2}$ at $z_0=\pi$ b) $\displaystyle \frac{1}{1-\cos(z)}$ at $z_0=0$ I'm having trouble understanding Laurent series, please help!
1
vote
2answers
257 views

Determining Laurent Series

This question is from Complex Variables and Applications by Brown & Churchill, 8ed. Section 62, #2. Determine the Laurent Expansion of: $$\frac{\exp(z)}{(z+1)^2}$$ for the interval, $ 0 < ...
3
votes
1answer
232 views

Univalent functions normal family.

Prove that: The family of $S$ of univalent functions on the unit disc with f(0)=0, f'(0)=1 is a normal family. I'm pretty sure i have to do it with Zalcmans Lemma: a family of analytic functions on ...