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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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 ...
3
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1answer
13 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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2answers
157 views

Double Euler sum $ \sum_{k\geq 1} \frac{H_k^{(2)} H_k}{k^3} $

I proved the following result $$\displaystyle \sum_{k\geq 1} \frac{H_k^{(2)} H_k}{k^3} =- \frac{97}{12} \zeta(6)+\frac{7}{4}\zeta(4)\zeta(2) + \frac{5}{2}\zeta(3)^2+\frac{2}{3}\zeta(2)^3$$ After ...
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11 views

is it all right my pf?

PB: Give a proof that the image of a circle under a linear transformation is a circle. (Let $z$ be a $z=z_{0}+Re^{it}$, $t$ is a angle.) I tried it. Can you check my pf? (is it all right?) My Pf) ...
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26 views

$\sum_{j,k=1}^\infty \frac{H_j(H_{k+1}-1)}{jk(k+1)(j+k)}=-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5)$

Hi I am trying to calculate the infinite double sum $$ S:=\sum_{j,k=1}^\infty \frac{H_j(H_{k+1}-1)}{jk(k+1)(j+k)}=-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5),\quad H_n:=\sum_{k=1}^n\frac{1}{k}\ \ ...
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11 views

How to apply Little Picard Theorem to prove that $a$ is unique

Little Picard Theorem: If a function $f : ℂ→ℂ$ is entire and non-constant, then the set of values that $f(z)$ assumes is either the whole complex plane or the plane minus a single point. Assume that ...
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12 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 singularity ...
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25 views

Common misconceptions in Complex analysis. [on hold]

Well, i recently started studying complex analysis and found that my concepts were not very strong. Once my teacher told me about some relation between analyticity and differentiability and thanks to ...
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33 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 ...
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29 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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14 views

Prove that equation satisfies Laplace's Eq.

I'm trying prove that the equation $T(x,y)$ satisfies Laplace's Equation where $T(x,y)$ is given as $T(x,y) = -Im\Omega(z)$ where $\Omega = 1/\omega$ and we are told to use the substitution $z = ...
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1answer
17 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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2answers
39 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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26 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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23 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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10 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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0answers
10 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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3answers
34 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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20 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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2answers
451 views

Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

To everyone: don't bother writing up another answer, i'm giving this bounty Antonio's answer. It just doesn't let me yet (24 hours delay). If you map the nth roots of unity $z$ with the function ...
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0answers
16 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
27 views

Understanding why $\int_\gamma {dz \over z - a} = k 2\pi i$ for $\gamma$ a closed curve not passing through $a$

The following is a paraphrased proof from Ahlfors. I bolded the part that is confusing me and asked a question about it at the bottom of this post. Hypothesis: Let $\gamma$ be a closed curve that ...
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1answer
21 views

Definition of an integrand

General Question: Say we have integral $$ \int f(z)\ dz $$ Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its ...
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1answer
50 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 ...
6
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1answer
343 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
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10 views

Laplace transform of a majorated function

I have the following problem. I have an analytic function and I want to show that it is majorated by a convenient function. To do that, it is very helpful to solve the transformed equation. I have a ...
3
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1answer
78 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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2answers
39 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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66 views

Independently analytic and continuous, but not jointly continuous?

In Bak/Newman's "Complex Analysis", they write: 17.9 Theorem Suppose $\phi(z,t)$ is a continuous function of $t$, with $b \ge t \ge a$, for fixed $z$ and an analytic function of $z \in D$ for ...
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3answers
365 views

Where is $\log(z+z^{-1} -2)$ analytic?

I need some help in determining where $\log(z+z^{-1} -2)$ is analytic, where $z$ is a complex number and $\log(z)=\ln|z|+\arg(z+2k\pi),k\in\mathbb{Z}$. Thank you in advanced.
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3answers
40 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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1answer
31 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 ...
1
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1answer
27 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
42 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
16 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 ...
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0answers
14 views

Taylor expansion and expansion in powers of z-1

I am trying to expand $z^2/(z+1)^2$ as a Taylor Series. I have acquired its partial fraction decomposition of $z^2/(z+1)^2$ = $(1/6)*(1/(z+1)) + (5/6)(1/(z-5))$. The first term is in the form ...
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35 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 = ...
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25 views

Level set of a real valued harmonic fucntion

Let $f$ be a real valued harmonic function defined on a neighborhood $U$ of origin in $\mathbb{R}^2$. And $f$ is such that its gradient vanishes at origin. Then how do i show that the set given by ...
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2answers
28 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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0answers
25 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 ...
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1answer
20 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
48 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
18 views

Univalent function and one-to-one function

What is the difference between univalent function and one to one function? I do not know how to be rigorous for this problem. I would appreciate if someone can prove this rigorously?
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1answer
458 views

Fourier transform of a function of compact support

My professor occasionally assigns optional difficult problems which we do not turn in from Stein and Shakarchi's Complex Analysis. I am currently studying for a test in that class and try to get all ...
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2answers
30 views

Laplace equation on semidisk

I am interested in the solution of the following boundary value problem on the semidisk $D=\{(r,\theta): 0<r<1, 0<\theta<\pi\}$: $$u_{xx}+u_{yy}=0 \mbox{ in } D, $$ $$u(1,\theta)=0 \mbox{ ...
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1answer
35 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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0answers
11 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 ...
2
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1answer
26 views

Inverse of the function $\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$

It can be proved that the function $f:[-1,1]\to \mathbb{C}$ defined by $$f(x)=\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$$ maps the interval $[-1,1]$ one to one onto the lower part of the unit circle. ...
0
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1answer
66 views
+50

About the branch-cut in the complex logarithm

Say I have the function $log (f(z))$, then does the imaginary part of the value go down by $-i 2 \pi$ on crossing the branch-cut of the log function even if $f(z)$ is not crossing the branch-cut? ...
5
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65 views
+250

Formula for decomposing a form into $(p,q)$ forms

Let $L: \mathbb{C}^n \to \mathbb{C}$ be a real linear map. In other words, $L(a\vec{v}_1+b\vec{v_2}) = aL(\vec{v}_1)+bL(\vec{v}_2)$ for all $a,b \in \mathbb{R}$. Then $L$ decomposes uniquely into a ...