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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What are examples of functions with these property?

Complex Analysis - Silverman p. 121 Exercise 4 (a) Let $n\in\mathbb{Z}^+$ and $r\in\mathbb{R}\setminus\{0\}$. Construct a complex valued function such that $\lim_{z\to 0} f(z)=0$ along ...
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1answer
65 views

Finding an upperbound for $|f^{(n)}(z)|$

Ahlfors: If $f(z)$ is analytic and $|f(z)| \le M$ for $|z| \le R$, find an upper bound for $|f^{(n)}(z)|$ in $|z| \le \rho < R$. Attempt: Let $z$ satisfy $|z| \le \rho < R$ so that ...
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1answer
56 views

Extending bounded functions on $\mathbb{R}$ to $\mathbb{H}$ with the Poisson kernel.

Let $h(\phi):\mathbb{R}\to\mathbb{R}$ be a bounded piecewise continuous function on the real line. Define a function on the upper half-plane by the formula $\tilde{h}(s+it):=\int_{-\infty}^{\infty} ...
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1answer
64 views

What does this exercise want me to solve?

This is an exercise in "Complex analysis -silverman" Let $y=y_0$ be a line parallel to the $x$-axis in the complex plane. What is the image of it under Log ($z^2$)? (Principal value) Simple ...
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32 views

Laplace transform, when $s \rightarrow \infty$

I'm reviewing lecture notes on Laplace Transform and there's one step that I don't understand: Find the solution to: $$x y'' + y' + xy = 0, y(0) = 1, y'(0) \mbox{ finite}$$ Taking the Laplace ...
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1answer
65 views

Computing $\int_{|z|=2} z^n(1 - z)^m\ dz$

My two questions are bolded below. Hypothesis: Let $\gamma$ denote the circle about the origin of radius $2$. Goal: Compute $$ \int_{\gamma} z^n(1 - z)^m\ dz $$ Attempt: We have that $$ ...
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2answers
567 views

Intuition Behind Maximum Principle (Complex Analysis)

Let $D$ be an open set in the complex plane and $f(z)$ be a non-constant holomorphic function on D. Then $|f(z)|$ has no local maximum on D. I can follow the proof fine - usually if I don't ...
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2answers
57 views

How to show that the $\phi $ and $\varphi$ satisfy the Cauchy Riemann equation [closed]

When $u=(u,v)=(\frac{\partial\varphi}{\partial y},-\frac{\partial\varphi}{\partial x})$ and $u(x)=\text{grad}\;\phi(x)=\nabla\phi(x)$, how can you use the equations above to prove that the $\phi$ ...
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2answers
57 views

Does $f^{(n)} = 0$ imply that complex $f$ is a polynomial?

Let $f$ be a complex function with the property that $f^{(n)} = 0$. Does this imply that $f$ is a polynomial? If so, why? Upon thinking about this problem myself, I can easily observe that every ...
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1answer
70 views

The $\frac{1}{x+i\varepsilon}$ distribution.

I read that the distribution defined as: $$ \lim_{\varepsilon \rightarrow 0}\frac{1}{x+i\varepsilon}$$ is equal to $$p.v. \frac{1}{x} -i\pi \delta(x)$$ So that for any test function $f$, ...
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304 views

Can one give me some concrete examples explaining Picard's Great Theorem

Picard's Great Theorem Every non-constant entire function attains every complex value with at most one exception. Furthermore, every analytic function assumes every complex value, with possibly one ...
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37 views

Showing that $\int_C 1\ dz = 2 \pi r$

Let $\gamma$ be a circle centered at $a$ of radius $r$. Parameterize $\gamma$ via $\gamma(t) = a + re^{it}$ on $0 \le t \le 2 \pi$. This yields us that $\gamma'(t) = ire^{it}$. I suspect the line ...
3
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1answer
96 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...
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1answer
168 views

Geometric Derivation of the D-Bar Operator $\frac{\partial}{\partial z} = \frac{1}{2}(\frac{\partial }{\partial x} - i\frac{\partial }{\partial y})$

This picture from Visual Complex Analysis is all you need to derive the Cauchy-Riemann equations, i.e. from the picture we see $i \frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}$ ...
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1answer
81 views

Evaluate $\int_0^{\infty}\frac{x^4e^x}{(e^x-1)^2} \, dx$

I am trying to find the value of the integral below. Can anyone let me know how to evaluate this integral? $$\int_0^{\infty}\frac{x^4e^x}{(e^x-1)^2} dx$$
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2answers
141 views

Compute $\int_0^\infty \frac{\ln x}{(1+x)^3}\,\mathrm{d}x$

Compute $$\int_0^\infty \frac{\ln x}{(1+x)^3}\,\mathrm{d}x$$ Well by comparison test the integral is convergent. I tried to use residue theorem, with the positive real axis being the branch ...
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1answer
28 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
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1answer
46 views

How do i show that $w=z^2$ maps $|z-1|\leq 1$ onto $R=2(1+\cos \theta)$?

How do i show that $w=z^2$ maps $|z-1|\leq 1$ onto the cardioid $R=2(1+\cos \theta)$?
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1answer
103 views

Mean value theorem does not hold for the complex function $f(z)=z^3$

Consider $f(z)=z^{3}$, two point $z_{1}=1$ and $z_{2}=i$. show that Do Not exist a point $c$ on the $y=1-x$ between $1$ and $i$ such that Do Not satisfying ${f(z_{2})-f(z_{1})\over ...
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1answer
29 views

How do i prove this property of bilinear mapping?

Let $T$ be a Mobius transformation which takes real line onto the unit circle. Assume $T(z_0)=w_0$. Then how do i prove that $T(\overline{z_0})=\frac{1}{\overline{w_0}}$? (Silverman complex ...
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127 views

Prove that this holomorphic function is constant

Suppose $f$ is a non-vanishing continuous function on $\bar{\mathbb{D}}$ that is holomorphic in $\mathbb{D}$. Prove that if $$|f(z)|=1~~~\text{whenever}~~~|z|=1$$ then $f$ is constant. I have proved ...
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66 views

How to prove that $a$ is unique

Assume that $f : ℂ→ℂ$ is a non-constant non polynomial and entire function and there exist $a∈ℂ$ such that the fiber $f⁻¹(a)$ is finite. My question is: How to prove that $a$ is unique.
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1answer
102 views

Inverse Laplace transform of $s^{k}$

How can I find the inverse Laplace transform of $s^{k}$ where $k$ is non-integer and negative? I know that $$\mathcal{L}^{-1}[s^k] = \frac{1}{2\pi i}\int e^{st} s^k ds$$ and since we have ...
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1answer
498 views

$-4\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5)=S$

EDIT: Due to the solution below, I edited the answer of the post. Thanks!!!! Hi I am trying to calculate the infinite double sum $$ S:=\sum_{j,k=1}^\infty ...
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1answer
284 views

Why a holomorphic function satisfying these conditions has to be linear?

Let $\Omega$ be a bounded open subset of $\mathbb{C}$ and $f:\Omega\rightarrow\Omega$ be holomorphic in $\Omega$. Prove that if there exists a point $z_0$ in $\Omega$ such that ...
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2answers
42 views

Why is $\int_C {dz \over z - a} = 2 \pi i$ not a counter-example to Cauchy's theorem in a disk?

Cauchy's theorem in a disk states that if $\Delta$ is an open disk and $f$ is analytic on $\Delta$, then if $\gamma$ is a closed curve inside $\Delta$ we have that $$ \int_\gamma f(z)\ dz = 0 $$ ...
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1answer
93 views

Cauchy's theorem in a disk (Proof Verification)

Consider the following proof of Cauchy's theorem in a disk. My question is pasted at the bottom of the picture. (Note that in the proof below, a reference is made to "Theorem 2". In my textbook ...
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23 views

Finding residue from a Laurent Series

I know that you can calculate the residue of a Laurent Series by looking at the coefficient of the z^-1 term ie eg for this series ...
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25 views

Question about Classifying singular points and finding corresponding residues from Laurent Series

I wanted to check if I had the right idea : Singularities have 3 classification 'essential'.'removabe' and 'pole order x' a singularity is essential if when you expand it,it is a never ending series ...
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2answers
75 views

Computing $\int_\gamma { |dz| \over |z-a|^2}$

Goal: Compute $$ \int_{|z|= \rho} {|\mathrm{d}z| \over |z-a|^2} $$ under the condition $|a| \ne \rho$. Ahlfors' Hint: make use of the equations $z \bar{z} = \rho^2$ and $$ |\mathrm{d}z| = -i ...
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0answers
94 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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3answers
53 views

Complex analysis. Manipulation of conjugates, fractions and modulus.

Let $a,b,c \in \mathbb C $ with $|b|<1$ and $z\neq \bar a$ and $$\left|\frac {z-a}{z-\bar a}\right| \le |b| $$ Show that, $$|z| \le |a| \frac{1+|b|}{1-|b|}$$ This a revision question I'm ...
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63 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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1answer
87 views

Prove that $\frac{1}{\sqrt{1-z}}=\sum_{n=0}^{\infty}\frac{1}{4^{n}}\binom{2n}{n}z^{n}$ using Cauchy product

need to prove using Cauchy product for series for all $\left|z\right|<1$ that $$\frac{1}{\sqrt{1-z}}=\sum_{n=0}^{\infty}\frac{1}{4^{n}}\binom{2n}{n}z^{n}$$ (with appropriate branch of the root ...
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2answers
67 views

How do i prove that $| \arg z|<\pi/2$?

Let $|1-z|<1$. Then how do i prove that $| \arg z| < \pi/2$? This is geometrically trivial, but i dunno how to prove this precisely
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2answers
46 views

Showing distance from $z_0$ to the line parametrized by $z(t)=w_0+te^{i \theta}$ is

Suppose $w_0$ and $z_0$ are in $\mathbb{C}$ and $\theta$ is a fixed angle with $0 \le \theta \le 2 \pi$. Show that the distance from the point $z_0$ to the line parametrized by $z(t)=w_0+te^{i ...
3
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1answer
125 views

Contour method to solve $\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx$

Prove the following using complex analysis $$\tag{1}\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx=\frac{\pi}{2}\ln(2)$$ I found this problem in Schaum's outlines of complex variables. I thought that we ...
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3answers
58 views

Computing $\int_{|z|=2} {dz \over z^2 + 1}$

Goal: To compute $$ \int_{|z|=2} {dz \over z^2 + 1} $$ by decomposition of the integrand in partial fractions. Attempt: Let $\gamma$ be the circle around the origin of radius $2$. Let us ...
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79 views

Showing iterates of a complex function on the upper half plane converges uniformly on compact sets

The following is an irksome problem that my complex analysis class is having trouble solving: Let $*$ be an operator that takes a function $F:\mathcal{H}\to\mathcal{H}$ to a function ...
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1answer
34 views

Computing ${\mathrm{d} \over \mathrm{d}t}\left(e^{it}\right)$

Let $t \in \mathbb{R}$. Is the following elementary calculation correct? $$ {\mathrm{d} \over \mathrm{d}t}\left(e^{it}\right) = \underbrace{{\mathrm{d} \over \mathrm{d}t}\left(it\right) \cdot ...
2
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1answer
82 views

Showing the winding number of the unit circle is $1$

Let $\gamma$ denote the unit circle parameterized on the domain $[0,2\pi]$. I'm trying to compute $n(\gamma, 0)$ as follows: $$ n(\gamma,0) = {1 \over 2\pi i}\int_\gamma {dz \over z} = {1 \over 2 ...
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1answer
60 views

Computing $\int_{|z|=1} {e^z \over z}\ dz$

Goal: Let $\gamma$ be the unit circle. Then I aim to compute $$ \int_{|z|=1} {e^z \over z}\ dz = \int_{\gamma} {e^z \over z}\ dz $$ Attempt: Consider that $\gamma$ is a closed curve. Let $a = 0$. ...
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1answer
76 views

Question regarding pluriharmonic function

A real valued function $f$ defined on an open subset $U$ of $\mathbb{C}^n$ is said to be Pluriharmonic if $$\frac{\partial^2}{\partial z_i\partial\bar{z_j}}f\equiv0,$$ for $1\leq i,j \leq n.$ I was ...
2
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2answers
42 views

Can anyone explain a residue in fairly simple terms?

I'm studying Complex Analysis and everything up to this point has been pretty straightforward to visualise, but I can't get my head around residues, especially as they seem to have two very different ...
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2answers
51 views

Consistent branch choice

I found in my class notes the following comment regarding branch choice: It is important to choose a branch consistently, otherwise one can get absurd results, for example: $-1 = i^2 = ...
3
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0answers
89 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
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0answers
82 views

Linear Fractional Transforms maps the upper half unit disc onto the first quadrant

Since the LFT(Linear Fractional Transform)preserves the angles, and since $\{|z|=1,\operatorname{Im} z>0\}$ intersects $[-1,1]$ at $-1$ and $1$. So we must map one of the two right angles to the ...
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1answer
114 views

Complex Taylor and Laurent expansions

Let $f(z):=\dfrac{1}{2-z-z^2}, z\in\mathbb{C}\setminus\left\{ {1, -2}\right\}$. i) Express $f$ in the form $\dfrac{A}{1-z}+\dfrac{B}{2+z}$. [Answer to this is $\dfrac{1/3}{1-z}+\dfrac{1/3}{2+z}$]. ...
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1answer
82 views

Example for an entire function of finite order but of infinite type

I'm currently racking my brains for an example as described in the question. I have an example $$e^{e^z}$$ which is of infinite order and infinite type. Question is, does there exist an (entire) ...
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0answers
41 views

Bounded on an union of squares

I would like to do this exercise : Let $\displaystyle h(z) = \pi \mathrm{cotan}(\pi z) = \pi \frac{\cos(\pi z)}{\sin(\pi z)}$. And for $q \in \mathbb{N}^{*}$, let $C_{q}$ be the square in the ...