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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Understanding Eigenvector

We have a matrix $A$ of size $N \times M$, where $N\le M$. Consider a vector $V$ of length $N$. Now I take product of $AV$ to get a vector $W$ of length $M$. Here I have projected the original ...
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2answers
2k views

The Schwarz Reflection Principle for a circle

I'm working on the following exercise (not homework) from Ahlfors' text: " If $f(z)$ is analytic in $|z| \leq 1$ and satisfies $|f| = 1$ on $|z| = 1$, show that $f(z)$ is rational." I already know ...
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1answer
450 views

Polynomial bounded real part of an entire function

Let $f(z)$ be an entire function whose real part is bounded by a polynomial in $|z|$. Does it follow that $f(z)$ is a polynomial? Or, without loss of generality and more suggestively ...
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2answers
1k views

What is the radius of convergence of $\displaystyle\sum z^{n!}$?

How could you find out the radius of convergence of $\displaystyle\sum z^{n!}$? I'm used to applying the ratio test to power series of the form $\displaystyle\sum a_{n}z^{n}$, but for a different ...
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1answer
255 views

Location of zeros of a sum of exponentials

Describe the approximate locations of the zeros of the function $$ f(z) = e^{iz}+e^{-iz}+e^z $$ lying outside the circle $|z|=R >>1$. Another prelim problem. For Rouche's theorem we need to ...
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2answers
763 views

All the zeroes of $p(z)$ lie inside the unit disk

Let $p(z) = c_0 + c_1z + c_2z^2 + \dots + c_nz^n$ where $0 \le c_0 \le c_1 \le \dots \le c_n$. I would like to show that all zeroes of this polynomial lie inside the unit disk by applying Rouche's ...
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2answers
1k views

Isolated zeros on closure of a domain

Let $f$ be an analytic function on the open unit disk domain $D$. Suppose also that $f$ is bounded. Since $f$ is bounded I beleive that $f$ can be continuously extended to the closed unit disk. I ...
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3answers
1k views

Using Residue theorem to evaluate $ \int_0^\pi \sin^{2n}\theta\, d\theta $

can you please guide me on evaluating this integral using residue theorem and binomial theorem $$ \int_0^\pi \sin^{2n}\theta\, d\theta $$ for $n = 1,2,3$ Honestly, I do not even know where to start, ...
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3answers
649 views

If $f^2$ and $f^3$ are analytic prove that $f$ is analytic at every point of $\mathbb{C}$. [duplicate]

Let $f : \mathbb{C} \to \mathbb{C}$ be continuous. If $f^2$ and $f^3$ are analytic prove that $f$ is analytic at every point of $\mathbb{C}$. if $f^2$ has no zero then $f=f^3/f^2$ and then it is ...
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1answer
171 views

sum of holomorphic functions

Does anyone know how prove the following? Suppose that $f,g$ are holomorphic functions on a non-empty open connected set $\Omega \subset \mathbb{C}$ and that $|f|^2+ |g|^2$ is constant on $\Omega$. ...
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227 views

Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
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2answers
186 views

Evaluate these infinite products $\prod_{n\geq 2}(1-\frac{1}{n^3})$ and $\prod_{n\geq 1}(1+\frac{1}{n^3})$

What is $\prod\limits_{n\geq 2}(1-\frac{1}{n^3})=?$ $\prod\limits_{n\geq 1}(1+\frac{1}{n^3})=?$ I am sure about their convergence. But don't know about exact values. Know some bounds as well. For ...
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2answers
633 views

Proving surjectivity of $\cos(z)$ and $\sin(z)$ and find all $z : \cos(z) \in \mathbb R$ and all $z: \sin(z) \in \mathbb R$

I am trying to solve the following two problems: 1) Prove that the functions $\cos(z)$, $\sin(z)$ are surjective over the complex numbers. 2) Find all $z \in \mathbb C$: $cos(z) \in \mathbb R$ and ...
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2answers
161 views

Show that $f(z)=0$.

Suppose that $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic on the open unit disc and continuous on the closed unit disc. Assume that $f(z)=0$ on an arc of the circle $\{z\in\mathbb{C}:|z|=1\}$. Show ...
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4answers
900 views

Improper integral of $\sin^2(x)/x^2$ evaluated via residues

I have come across another improper integral I wish to evaluate via residues. The integral is: $$\int_{-\infty}^\infty{\frac{\sin(x)^2}{x^2}}dx$$ $\sin(z)$ behaves in an uneasy way so I tried ...
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3answers
620 views

How can I show this statement.

Show that there is no holomorphic fuction $f$ in the unit disc $D$ that extends continuously to boundary of $D$ such that $f(z)=\frac{1}{z} ~for~ z\in \partial( D) $. I tried to apply maximum ...
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3answers
1k views

Proof of Cauchy-Schwarz inequality - Why select s so that so that $||x-sy||$ would be minimized?

I was looking at a number of different proofs of the cauchy schwarz inequality in an inner product space ($\mathbb{R}^n$ or $\mathbb{C}^n$). All of them used the idea of $||x-sy||$ where $s$ was ...
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2answers
551 views

if a complex function $f$ is real-differentiable, then $f$ or $\overline{f}$ are complex-differentiable

This is an exercise from Remmert's Theory of Complex functions. Let $D\subset \mathbb{C}$ be a domain and $f:D\rightarrow \mathbb{C}$ a real-differentiable function. Assume that the following limit ...
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2answers
638 views

Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$ [closed]

Show that $$\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)},$$ where $0\leq r <1$. Using this, prove that $\sum_{n=0}^\infty r^n ...
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2answers
1k views

Find conformal mapping from sector to unit disc

Find a conformal mapping between the sector $\{z\in\mathbb{C} : -\pi/4<\arg(z) <\pi/4\}$ and the open unit disc $D$. I know that it should be a Möbius transformation, but other than that I am ...
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1answer
163 views

Prove the following equation of complex power series.

Show that for $|z| \lt 1$ with $z \in \Bbb C$, we have $$ \sum_0^\infty \frac{{z^2}^k}{1-{z^2}^{k+1}} = \frac{z}{1-z} $$ $$ \sum_0^\infty \frac{2^k{z^2}^k}{1+{z^2}^{k}} = \frac{z}{1-z} $$ My guess ...
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1answer
154 views

Finding the residue of function with Laurent series $\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{y^n(A+By+Cy^{-1})^k}{\beta (\beta i)^n \ k!}$

I have been trying to find the residue of $f(\omega) = \frac{e^{i \omega a} e^{\frac{-b \omega}{\omega + ib}}}{i \omega}$ at the essential singularity $\omega = -ib$ for a while, but it is giving me ...
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1answer
461 views

Integrating squared absolute value of a complex sequence

I was reading through my book in complex analysis and i encountered this problem. Given, $F=\sum_{n=0}^{\infty} a_nX^n$ is a convergent power series with radius of convergence R. We are asked to show ...
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2answers
696 views

How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?

How do I prove the error function $$ \mbox{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} dt. $$ is entire? Could you give me some scratch proof?
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1answer
255 views

Representation of Holomorphic Functions By Exponential

Let $f$ be holomorphic and nonzero on $D_{1}(0)$ the open unit disc. Can we write (for the given domain) $f(z) = e^{h(z)}$ where $h$ is holomorphic? This seems clear using a naive log argument but I'm ...
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2answers
456 views

Area of Validity of Writing an Exponential Integral as Sum of IntegralSinus and -Cosinus

I'm confused by the two online references shown below. To me, they give different areas of validity of writing an exponential integral as sum of integralsinus and -cosinus. On this Wiki page, I find ...
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1answer
100 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
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1answer
537 views

Does there exist an holomorphic function such that $|f(z)|\geq \frac{1}{\sqrt|z|}$?

I have some trouble solving this problem: Does there exist an holomorphic function $f$ on $\mathbb C\setminus \{0\}$ such that $|f(z)|\geq \frac{1}{\sqrt|z|}$ for all $z\in\mathbb C \setminus ...
6
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1answer
442 views

3 holomorphic functions, sum of absolute values does not have maximum

I have the following problem: Let $f,g,h$ be holomorphic functions (non-constant) in some domain $D$. Show that the function $F(z):=|f(z)|+|g(z)|+|h(z)|$ has no local maximum in this domain $D$. ...
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2answers
437 views

How to compute the infinite tower of the complex number $i$, that is$ ^{\infty}i$

Let $x = i^{i^{i^{i^{.^{.^{.{^ \infty}}}}}}}$. This is the solution of the equation $i^x - x = 0 $ . I used Euler's identity to find a solution. But I haven't yet found the real and imaginary parts of ...
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3answers
243 views

Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx. $$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
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1answer
631 views

Show that an entire function bounded by $|z|^{10/3}$ is cubic

Question: Let $f$ be an entire function such that $|f(z)|\leq1+2|z|^{10/3}$ for all z. Prove that $f$ is a cubic polynomial Thoughts so far: Using a corollary of Liouville's theorem, we know that we ...
5
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2answers
224 views

Domination of complex-value polynomial by highest power.

The problem: Let $P(n)$ be a polynomial of degree $n$. Let $$M(r):= \underset{|z|\le r}{\mbox{sup}} \hspace{2mm} \left|P(z)\right|.$$ I desire to establish that $$r\mapsto \frac{M(r)}{r^n}$$ for ...
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5answers
358 views

Fourier transform of $f(x)=\frac{1}{x^2+6x+13}$

How to find the Fourier transform of the following function: $$f(x)=\frac{1}{x^2+6x+13}$$
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1answer
399 views

An entire function with two periods

Can anybody help me with this question: If $f(z)$ is an entire periodic function and it has to periods $2$ and $2i$, how can I find all other periods?
3
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1answer
126 views

convergence of a particular series

Let be $ \Lambda\subseteq \mathbb C$ a lattice, I don't understand why the series $$\sum_{\lambda\in\Lambda\setminus\{0\}} \frac{1}{|\lambda|^s}$$ converges for $s>2$. Can someone help me?
2
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1answer
86 views

Infinite sum involving ascending powers

I was working on a problem involving finite differences and came across the following sum as a general solution to a problem I was working with $$ 1 + x + \frac{1}{1 + a}x^2 + \frac{1}{1 + a} ...
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1answer
117 views

a generalization of normal distribution to the complex case: complex integral over the real line

How to prove $\int_{\mathbb{R}} e^{-\frac{(x+it)^2}{2}}dx=\sqrt{2\pi}$ for any $t\in \mathbb{R}$? I only obtained the case that $t=0$, $\int_{\mathbb{R}} e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}$. Thanks.
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1answer
681 views

Is my proof correct? (Conformal equivalence of two circular annuli)

I want to show that the two annuli $$A=\{r<|z-z_0|<R\} $$ $$A'=\{r'<|z-z_0'|<R'\} $$ are conformally equivalent (i.e. there exists a biholomorphic map between the two) iff ...
2
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1answer
153 views

If $f(z)g(z) = 0$ for every $z$, then $f(z) = 0$ or $g(z) = 0$ for every $z$.

This is for homework, and I would really appreciate a hint. The question states "If $f$ and $g$ are holomorphic on some domain $\Omega$ and $f(z)g(z) = 0$ for every $z \in \Omega$, then $f(z) = 0$ ...
2
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2answers
215 views

criterions for holomorphic functions

What are the criterions for holomorphic functions except $\frac{\partial f}{\partial \overline z}=0$ and $f$ has a power series extension? I was considering the problem, which is the extension of a ...
2
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4answers
576 views

Introductory books on complex analysis? [duplicate]

I'm a senior in my undergrad. years of college, and I haven't taken Complex Analysis yet. I have taken Real Analysis I (covered properties of $\mathbb{R}$, set theory, limits of sequences and ...
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2answers
829 views

Why there is no continuous argument function on $\mathbb{C}\setminus\{0\}$?

An argument function $\phi$ on $\mathbb{C}\setminus\{0\} = \mathbb{R}^2\setminus\{0\}$ is a function such that for every $z\neq 0$ it holds that $$z = |z|\exp(i\phi(z)).$$ Is there an elementary and ...
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2answers
351 views

How to derive the following identity?

The book Irresistible Integrals by George Boros and Victor Moll on page 204 has the following identity $\displaystyle \frac{1}{1+x}=\prod_{k=1}^{\infty}\left(\frac{k+x+1}{k+x} \times ...
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1answer
59 views

complex integration, how to evaluate it?

I have this exercise: Show that if $|a| < r <|b|$, then $\int_\gamma \! \frac{1}{(z-a)(z-b)} \, \mathrm{d}z=\frac{2\pi i}{a-b}$, where $\gamma$ denotes the circle centered at the origin, ...
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1answer
396 views

On a conformal mapping

I was asked to find a one-to-one analytic map $f$ of unit disc $\mathbb{D}\subset \mathbb{C}$ so that $\mathbb{D}$ is mapped to $\{(x,y):y<x^2\}$. I thought the core procedure could be done by ...
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2answers
375 views

How to construct this Laurent series?

How do I construct the following Laurent series (clipped off Wolfram Alpha)? I know that the numerator can be written as $-1+\frac{\pi}2 z-...$ Alternatively (without the Laurent series), how can I ...
1
vote
1answer
279 views

The meaning of a notation from complex analysis

I have read the book Function Theory of Several Complex Variables of Krantz.But there is a notation I don't know what's it meaning. Can somebody give me a definition.Thank you. The notation is ...
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2answers
67 views

Understanding the Definition of $\int_\gamma f\ \overline{dz}$

Definitionally, we have that $$ \int_\gamma f\ \overline{dz} = \overline{\int_\gamma \overline{f}\ dz} $$ Now let $\int_\gamma f\ dz = w = x +yi$. Question 1: Is it not the case that $\int_\gamma ...
28
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7answers
3k views

Picard's Little Theorem Proofs

Picard's little theorem says that If there exist two complex numbers $a,b$ such that $f: \Bbb{C} \to \Bbb{C}\setminus \{a,b\}$ is holomorphic then $f$ is constant. I am interested in proofs for ...