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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12 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 $$ ...
4
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2answers
51 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
33 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 ...
2
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1answer
36 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 $f$, ...
3
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0answers
69 views

Beautiful Closed form $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

Hi I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_2\ln^3 2 ...
3
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1answer
35 views

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

Picard's Great Theorem Every nonconstant 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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28 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 ...
2
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1answer
41 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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0answers
20 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
51 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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1answer
49 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
22 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
13 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
26 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
14 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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0answers
20 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) ...
3
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1answer
21 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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24 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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30 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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1answer
32 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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35 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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34 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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15 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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2answers
28 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
20 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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11 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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11 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
43 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
21 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
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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17 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
54 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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0answers
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 ...
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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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36 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
votes
1answer
81 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
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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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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1answer
33 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
29 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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15 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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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 ...
2
votes
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 ...
1
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2answers
29 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
37 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 = ...
0
votes
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?
3
votes
0answers
49 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 ...
1
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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 ...
1
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1answer
36 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 $\dfrac{1/3}{1-z}+\dfrac{1/3}{2+z}$]. ...