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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13 views

If $I=\int_{0}^R \frac {1}{1+x^{2n}}dx$, why$ \int_{0}^R \frac {1}{1+(x e^{\frac{i\pi}{n}})^{2n}}d(xe^{\frac {i\pi}{n}})=e^{\frac {i\pi}{n} }I$?

Just read a proof in the textbook Basic Complex Analysis, and there is one point that I do not understand: Let $I=\int_{0}^R \frac {1}{1+x^{2n}}dx$, where $x$ is real. We have the following: $$ ...
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1answer
474 views

Complex Contour Integration Contour Deformation

Let $C$ be the unit circle $|z| = 1$ traversed once counterclockwise and then once clockwise, starting from $z = 1$. Construct a function $z(s, t)$ which deforms $C$ to the single point $z = 1$ in any ...
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13 views

Definition of an inverse-powerseries

Let $t(q)=\sum_{n=0}^{\infty}t_n q^n$ be a complex powerseries convergent for all $|q|<1$. Assume $t_0=0$ and $t_1\neq0$. Not it says Let $q(t)$ be the local inverse of $t(q)$ with $q(0)=0$. ...
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1answer
87 views

Prove an inequality using complex analysis

If $f:\mathbb{D}\rightarrow\mathbb{D}$ is holomorphic then prove that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq|f(z)| \leq\frac{|f(0)| + |z|}{1 - |f(0)||z|} $$ I have been wracking my brain for ...
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1answer
27 views

Extending function to holomorphic

Good day, I am having some problems with this question. Our supervisor's notes gave an answer, but she doesn't know how to get to it (it was set by an old lecturer). Given a continuous function ...
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2answers
29 views

Show : A holomorphic function is harmonic if $\frac{\partial f}{\partial \overline{z}}=0$

Let's consider a "new" basis of the partial differential operators (of order 1) on $\mathbb{R^2}\approx\mathbb{C}$ defined by : $\frac{\partial}{\partial z}:= \frac{1}{2}(\frac{\partial}{\partial ...
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1answer
17 views

Laplace transformation inequality

$f$ is a piecewise continuous function that is absolutely integrable on $[0,\infty)$. I want to show that $$\max_{\operatorname{Re} p\geqslant 0} |\mathcal{L}[f](p)|=\max_{\operatorname{Re} ...
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2answers
38 views

Holomorphic function multiplied by real number

I would like to ask if a holomorphic function (say $z$) multiplied by some real constant is still a holomorphic function. It seems a bit obvious but I'm searching for a good argument. Thanks
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2answers
32 views

Proving, that $\text{Arg}(-i\sin(x))=\pi/2\text{sgn}(x)$ on $(-\pi,\pi)$

Alright. I thought, that $\text{Arg}(-i\sin(x))=3\pi/2$, however, the Wolfram Alpha tells a different story. I am sure that it must be kind of true, because $\text{Arg}(\sin(x))$ is the result of sum ...
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25 views

Nice exercise with Rouche theorem

let $P_n(z) = 1 + z/1! + z^2/2! + ... + z^n/n!$ prove that for all $R>0$ sufficiently large, there exists $N\in \mathbb N$ such that for all $n \ge N$, all the roots of $P_n$ lie in $D(0,R)$ I ...
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15 views

Values of modular functions on the cusps

I have the meromorphic modular function (weight $0$) on $\Gamma_1(6)$ $$y(z)=\frac{\eta(6z)^8\eta(z)^4}{\eta(2z)^8\eta(3z)^4},$$whereby $\eta$ is the Dedekind eta function. The set of cusps of ...
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1answer
21 views

Solution of Schrodinger equation for free particle - How to eliminate mass variable

Probably missing something very obvious, sorry if this is a stupid question. I have to show the function $\psi(x, y) = \frac{1}{\sqrt{2\pi i t}} \int_{-\infty}^{\infty} e^{i(x-y)^2 / 4t} \psi_0 (y) ...
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1answer
23 views

Suspected Rouche's Theorem Question?

Part 1: Let $g$ be analytic in $|z|<1$ and $|g(z)|\leq 1$ for $|z|<1$. If $g(0)=g'(0)=0$, prove that $|g(z)|\leq|z|^2$ for $|z|<1$. Part 2: Assume further that $|g(i/2)|=1/4$, anything we ...
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11 views

Every cycle in a domain $D$ is null homolog

Let $D$ be a domain, where every cycle is null homolog and $f$ be a biholomorphism. Proof that every cycle $c$ in $f(D)$ is nullhomolog. Let $c$ be a cylce in D, it is null homolog, if ...
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1answer
26 views

Complex variable, multiplication of numbers

Question: Let a and b be complex numbers with $a \neq 0.$ Describe the set of points $az + b $ as $z$ varies over the first quadrant, $\{z = x+iy: x>0 \,and \,y>0\}$ Solution: Let $a = ...
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1answer
14 views

Arg function algebra

I have a question regarding the Arg function. Now I know, that $\text{Log}(z)=\log(|z|)+i\text{Arg}(z)$ for $z\in\mathbb{C}$. So it holds, that $\Im(\text{Log(z)})=\text{Arg}(z)$. My problem arises ...
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2answers
44 views

Calculate $\int_{-\pi}^{\pi} \frac{xe^{ix}} {1+\cos^2 {x}} dx$

So I'm trying to calculate $$ \int_{-\pi}^{\pi} \frac{xe^{ix}} {1+\cos^2 {x}} dx $$ knowing that if $f(a+b-x)=f(x)$ then $$ \int_{a}^{b} xf(x)dx=\frac{a+b}{2} \int_{a}^{b} f(x)dx, $$ but it doesn't ...
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20 views

Power series expansion of $z\mapsto \frac{\mathrm e^{z}}{1-tz}$ and $z\mapsto \tan z$

Determine the power series expansion and radius of convergence of $z\mapsto \frac{\mathrm e^{z}}{1-tz}$ around $0$ with $t\in\mathbb C$. Determine the radius of convergence and the first three ...
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34 views

Power series expansion of $z\mapsto \frac{1}{1+z^2}$ around arbitrary point $x\in \mathbb R$

Determine the power series expansion of $z\mapsto \frac{1}{1+z^2}$ around $x\in\mathbb R$ with the respective radius of convergence. At first I tried working with Cauchy's integral formula to ...
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14 views

Closed curve is homologous to zero in simply connected domains

If I am in a simply connected domain, are all the closed curves homologous to zero? i feel that this is true because a simply connected domain has no holes so there cant be a point outside the ...
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1answer
25 views

Is the set $\mathbb C - \{z:: \Re z \le 0$ and $\Im z = -1\}$ simply connected?

Is the set: $\mathbb C - \{z:: \Re z \le 0$ and $\Im z = -1\}$ a simply connected domain? it looks like it is because any closed curve i can imagine can be shrunk to a point. is this right? ...
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1answer
29 views

Help to understand the generalization of the Argument Principle

I'm reading Conway's complex analysis book and I'm trying to prove this theorem left to the reader on page 124: I tried to use integration by parts without success. I need some hint how prove this ...
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51 views

Is $z^z$ continuous everywhere in $\mathbb C$ except at 0?

Is $z^z$ continuous everywhere in $\mathbb C$ except at 0? further more is the discontinuity at 0 removable? update : the problem in this question is about multivaluedness of $z^z$, can we separate ...
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1answer
55 views

find set of complex numbers where $Arg(\frac{1}{z}) \neq -Arg(z)$

find set of complex numbers where $Arg(\frac{1}{z}) \neq -Arg(z)$ Def $Arg(z)=\theta :z=re^{i \theta } \wedge -\pi < \theta \leq \pi$ for z in first, second quadrant it holds except for ...
3
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3answers
74 views

How to show that $f$ is constant by using Liouville's theorem?

If $f$ is entire and $\mbox{Arg}(f(z))=-\frac{\pi}{2}$,when $|z|=1$ then show that $f$ is constant. All I need to prove is that f is bounded but I can't figure out how.I'd like someone's help.
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1answer
31 views

Simply Connected sets

In my textbook it states, that the Union of two open docs is simply connected but not connected Why is this. I know simply connected means any closed path or loop can be shrunk to a point ...
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1answer
500 views

Prove the complex function is entire

The function $$ f(z) = e^{x^2 - y^2} (\cos 2xy + i \sin 2xy )=e^{z^2} $$ Then, how to prove it's analytic everywhere of complex plane of the exp function...?
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1answer
595 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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2answers
42 views

Using the Weierstrass M-test, show that the series converges uniformly on the given domain

$\sum_{k \geq 0} \frac{z^k}{z^k+1}$ on the domain $\overline{D}[0, r]$, where $0 \leq r < 1$ I'm honestly not sure how to do this. My text mentions the Weierstrass M-test but the example they ...
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24 views

Proof that in any set A such that A contains a circumference centered at zero you can't find a continuous square root function.

The following is an exercise in my textbook on complex analysis: Proof that in any set A such that A contains a circumference centered at zero you can't find a continuous square root function. ...
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2answers
59 views

why can't we define a branch of $\log f(z)$ in the whole complex plane?

My question is really simple. The only problem to define a branch of $\log f(z)$ in the whole complex plane is because we can have $f(z)=0$ for some $z\in \mathbb C$? In fact I think I don't ...
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43 views

Does this function belong to $H^1(\mathbb D)$?

$\mathbb D$ is the unitary disk centered at $0$. Does the following function belong to $H^1(\mathbb D)$? .$$f_\epsilon(z) = \frac{1}{(1-z)\left(\frac{1}{z}\log\frac{1}{1-z}\right)^{1+\epsilon}}, ...
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17 views

Exercise for proving maximum modulus principle

I have this exercise: let $D$ be a domain and $a \in D$ such that $D'(a,r)$ is a subset of $D$ (here $'$ for closure). suppose that $f$ is holomorphic on $D$ and let $A = \max|f(z)|$ with $|z - a| = ...
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1answer
27 views

Möbius transformation mapping problem [on hold]

Let $a,b,c,d\in \mathbb{R}$ be such that $ad-bc>0$. Consider the Möbius transformation $$T_{a,b,c,d}(z)=\frac{az+b}{cz+d}$$ and define ...
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45 views

A curious trigonometric equality

Let's consider the following expression: $(1)\cos(15\sqrt{2}^\circ) = \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}+\frac{i}{2}} +  \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}-\frac{i}{2}}$ The left ...
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48 views
+50

radius of convergence of Taylor series, function with branch cuts

Let $f(z) $ being the analytic continuation of some holomorphic function, having many branch points and isolated singularities at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$ is the radius of convergence ...
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1answer
48 views

Help in this proof of the argument principle

I'm reading Conway's complex analysis book and on page 123 he made the following comment: Suppose that $f$ is analytic and has a zero of order $m$ at $z=a$. So $f(z)=(z-a)^mg(z)$ where $g(a)\neq ...
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1answer
18 views

Find the Laurent Series for $\frac{z-2}{z+1}$ around $z = -1$

Find the Laurent Series for $\frac{z-2}{z+1}$ around $z = -1$ I'm not sure how to do this because it is not something with a simple numerator. If it was something like $\frac{1}{(z-2)(z+1)}$ I ...
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0answers
26 views

$\frac{1}{2\pi}\int_0^{2\pi}\log|re^{it}-\zeta|dt$?

Prove that if $\zeta \in \mathbb{C}$ and $r>0$ then $\frac{1}{2\pi}\int_0^{2\pi}\log|re^{it}-\zeta|dt=\log |\zeta|$ if $r\leq |\zeta|$, and it is $\log r$ if $r> |\zeta|$. My Try: First I ...
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0answers
19 views

Complex analysis: given the radius of convergence of one series, how can I find radii of convergence for other series?

Suppose I know that: $\sum_{k = 0}^{\infty} c_k z^k$ is $R$. How can I use this to find the radius of convergence for something like: $\sum_{k = 0}^{\infty} 3^k c_k z^k$ $\sum_{k = 0}^{\infty} ...
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3answers
83 views

Evaluating an integral using the gamma function

My question regards an integral $$\int_0^\infty \frac{\sin(x^p)}{x^p}\mathrm{d}x$$ The answer should be $$\frac{1}{p-1}\cos(\frac{\pi}{2p})\Gamma(\frac{1}{p})$$ and I roughly know that I should apply ...
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0answers
11 views

Differentiation between the unit spheres and the hypersurfaces in $\mathbb C^n$

Let $\Sigma ^{n-1}$ be the complex unit sphere in $\mathbb C^n$, $$\Sigma^{n-1}=\{(z_1,...,z_n)\in \mathbb C^n; z_1\bar {z_1}+...+z_n\bar {z_n}=1\}$$ and let $S^{n-1}_{\mathbb C}$ be the hypersurface, ...
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2answers
86 views

closed path, winding number, Jordan contour

If $ D$ is a domain in $\Bbb C$, $z_0\in \Bbb C\setminus D$, and $\gamma$ is a closed, piecewise smooth path in $ D$ for which the winding number $n(\gamma, z_0)\ne0$, show that there is a Jordan ...
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1answer
66 views

What happens when $\beta_1 + \beta_2=1$ and when $0<\beta_1 + \beta_2<1$?

I have the following example of the Scwarz-Christoffel integral formula: $$S(z)=\int_0^z w^{-\beta_1}(1-w)^{-\beta_2}dw$$ with $0<β_1 <1, 0<β_2 <1$, and $1<β_1 +β_2 <2$ and I know ...
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16 views

Finding where a complex series converges absolutely, uniformly.

I need to figure out where the series converges absolutely and uniformly. I know that once I have absolute convergence on a region, then I know I also have uniform convergence on that region, so I ...
2
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4answers
61 views

Evaluation of the principal value of $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3} \, dx$

I'm trying to evaluate an integral $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3}\,dx$ using Cauchy's theorem. Considering an integral from $-R$ to $-\epsilon$, then a semicircular indentation ...
2
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2answers
64 views

Is $\sin z/z$ analytic at the origin?

For $z\in\Bbb C$ let $$ f(z) = \frac{\sin z}{z} $$ Along the real line this is well behaved, and approaches $1$ as $z\to 0$. But is $f(z)$ analytic at the origin ($z=0$)? I tried explicitly checking ...
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1answer
34 views

Evaluating $\int R(X)sin(x) dx$ with residue theorem.

The integral I am trying to evaluate is: $$I = \int_{-\infty}^\infty \frac{x}{1+x^2}\sin x\ dx = \int_{-\infty}^\infty f(x)\ dx$$ The standard approach to this is to realise $\sin x$ as the complex ...
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0answers
25 views

Integrate by parts $\int_0^\infty w' \bar w$; any nice expression for $w$ complex-valued?

Let $w$ be a complex-valued function of $t \in [0,\infty)$. At $t \to \infty$, it decays to zero. And $w_t(0)$ is prescribed. Is there any nice expression for the integral $$\int_0^\infty w' \bar w$$ ...
2
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2answers
61 views

Proving that two branch cuts can cancel out

Define the following functions $\mathbb{C}\to\mathbb{C}:$ $$u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z}\quad \left[-\pi\leqslant\arg \left(z+\tfrac12\right)<\pi\right];\quad v(z)=\frac{\log ...