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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2
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2answers
468 views

What are the reasons for using a semi-circle in upper half plane of $\mathbb{C}$ for contour integration?

Why is it that when one in considering contour integration of a real function, such as $$ \int_{-\infty}^{\infty} \frac{dx}{1+x^2}$$ the contour in the complex plane used is the following: ...
2
votes
2answers
813 views

Must a holomorphic function from $D(0,1)$ to $D(0,1)$ have a fixed point?

Must every holomorphic function $f:D(0,1)\longrightarrow D(0,1)$ have a fixed point? I know that any holomorphic function with two fixed points is the identity: $f=Id$, but I can't find out an ...
1
vote
1answer
58 views

I want to find the image under M of the region R which is the intersection of disks $|z+1|\le 2$ and $|z-1|\le 2$

Let $M$ be möbius transformation. $$M(3^{1/2}i)=\infty$$$$ M(-3^{1/2}i)=0$$$$M(0)=-1$$ I want to find the image under M of the region R which is the intersection of disks $|z+1|\le 2$ and $|z-1|\le ...
-1
votes
1answer
117 views

Complex Variables Quesion: determine how many zeros of the function $z^3-3z+1$ lie in the given annulus $1<|z|<2$

Can anyone help me with the problem? I am not sure how to approach it. Determine how many zeros of the given function lie in the given annulus. $$z^3-3z+1$$ lie in $$1<|z|<2$$
2
votes
1answer
63 views

What does the author (Ahlfors) mean here (modular $\lambda$ function)

In Ahlfors' complex analysis text, page 281 it says By reflection the region $\Omega'$ that is symmetric to $\Omega$ with respect to the imaginary axis is mapped onto the lower half plane, and ...
3
votes
2answers
96 views

prove that if $f$ is not surjective, then there exist $g\in M$ s.t. $|g'(z_0)|\gt |f'(z_0)|$

$N$ is aconnected, simply connected, nonempty, proper open subset of $\Bbb C$ Let $z_0\in N$ is assumed to be fixed. Let M = the set of all analytic injective functions $f: N \to D$ for unit disk ...
1
vote
0answers
30 views

If $f$ is an entire function with a non dense image then it is constant [duplicate]

Is it true that if $f$ is entire with a non dense image then it is constant? Can anybody help with the proof? Or give a counter example. Thank you
1
vote
1answer
56 views

Type of zeros of $\cos(z)$

Are the zeros of $cos(z)$ simple or multiple? How to find out the integer k so that $\cos(z)=(z-a)^kf(z)$ where a is a zero of the $\cos(z)$ ? Thank you.
2
votes
1answer
98 views

new ArcTan serie working for any x?

Using Log series, we can write: $$ \frac{1}{2 i} \left( \text{Log} \left( 1 + e^{\text{i2t}} \right) - \text{Log} \left( 1 + e^{\text{-i2t}} \right) \right) = \frac {1} {2 i}\left ( \sum _ {k = ...
1
vote
2answers
124 views

Singularity of the power function

I am reading some complex analysis and I am confused with the power function. So I understand that $z^a$ has a branching point at $0$ if $a$ is not integer and that the number of branches can be ...
2
votes
1answer
133 views

Calculate the Fourier transform on $\mathbb{R}$ of $\frac1{(1+x^2)^2}$ and $\frac{x}{(1+x^2)^2}$.

Calculate the Fourier transform on $\mathbb{R}$ of $\frac1{(1+x^2)^2}$ and $\frac{x}{(1+x^2)^2}$. Also calculate the Fourier transforms of $\frac1{(1+ix)^2}$ and $\frac{\cos(\pi x/2)}{1-x^2}$. ...
1
vote
0answers
224 views

Euler reflection formula via Wielandt theorem

I would like to prove Euler's reflection formula $$\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$$ using Wielandt's theorem: Let $f$ be a function that is bounded on the strip $1 \le ...
3
votes
1answer
74 views

Residues computation when we need power series

I'm trying to compute residues in situation where we need to manipulate power series to get it, but I can't find a good way. Indeed for the sake of example, consider the residue of the following ...
4
votes
1answer
65 views

Show $\int_{\gamma}e^{iz}e^{-z^2}dz$ same value on every line parallel to $\mathbb{R}$

From an old qualifier: Show that $$\large\int_{\gamma}e^{iz}e^{-z^2}\mathrm dz$$ has the same value on every straight line path $\gamma$ parallel to the real axis. Justify the estimates involved. My ...
1
vote
0answers
58 views

Integration of rational function on Banach algebra

I do not follow the proof of this Theorem Theorem Suppose$R(\lambda) = P(\lambda) + \sum_{m,k}c_{m,k}(\lambda - \alpha_m)^{-k}$ is a rational function with poles at the points $\alpha_m$. ($P$ ...
1
vote
1answer
33 views

Determine if a given function is a Polynomial [duplicate]

How i can show that if $\lim_{z\rightarrow\infty}\frac{|f(z)|}{|z|^{m}}=0$ then $f(z)$ is a polynomial of degree $< m$? Note: $f(z)$ is a polynomial in $\mathbb{C}[x].$ Thanks,
1
vote
1answer
93 views

Zeros of a Polynomial and maximum principle

Let $P: \mathbb C \to \mathbb C$ be a non-constant polynomial and $c>0$. Let $\Omega =\{z\in\mathbb C : |P(z)|<c\}$. I can't understand how does the maximum principle implies that every ...
1
vote
0answers
50 views

Where on the border of convergence circle series converges and where diverges?

I have power series of $ \sum\limits_{k=2}^{\infty} (\ln k)^{\alpha} z^k$. Alpha is a parameter. I've found the radius of convergence. R = 1. If $alpha \geq 0$ then series diverges for z from boundary ...
0
votes
1answer
45 views

automorfism of a simply connected region with two conditions

Be $U\subseteq \mathbb{C}$ simply connected region with $U\neq\mathbb{C}$ and $a,b\in U$, $a\neq b$. Is there an biholomorphism $f:U\longrightarrow U$ with $f(a)=b$ and $f(b)=a$? I know that, by the ...
1
vote
0answers
23 views

Calculate $ \int_{\partial D}\left ( 1+z+z^{2} \right ) (e^{\dfrac{1}{z}} +e^{\dfrac{1}{z-1}}+e^{\dfrac{1}{z-2}} ) dz$

Calculate the following integration ($ z\in\mathbb{C}$ ) $$ \int_{\partial D}\left ( 1+z+z^{2} \right )\left (e^{\frac{1}{z}} +e^{\frac{1}{z-1}}+e^{\frac{1}{z-2}}\right) dz$$ Where $D: |z|<3$
4
votes
2answers
248 views

Problem of Harmonic function.

If H is a harmonic function on an unit disk; And $H=0$ on $R_1\cup R_2$, here $R_1, R_2$ are radius of $D(0,1)$. The angle between $R_1$ and $ R_2$ is $r\pi$; here $r\in (0,1]$. If $r$ is an ...
2
votes
3answers
114 views

Is $xy = 1$ connected ? [duplicate]

The graph of $xy = 1$ is connected in $\mathbb{C}^2$. The above statement is true. Why? Please show reason. In $\mathbb{R}^2$ $xy = 1$ is not connected as it has two disjoint components in $1$-st ...
21
votes
1answer
470 views

Derivative of the Meijer G-function with respect to one of its parameters

Are there any approaches that allow to find a derivative of the Meijer G-function with respect to one of its parameters in a closed form (or at least numerically with a high precision and in ...
13
votes
5answers
2k views

Solving $(z+1)^5 = z^5$

The question says to solve this equation: $(z+1)^5 = z^5$ I did. Just want to find out if I did it properly and if my run-around logic makes sense. First I begin my writing the equations as: $$ ...
1
vote
3answers
148 views

$\int_{-\infty}^{\infty} \frac{\cos(αx)}{(x^2+1)(x^2+4)} \mathrm dx$ using Complex methods

$$\int_{-\infty}^{\infty} \frac{\cos(αx)}{(x^2+1)(x^2+4)} \mathrm dx. $$ I am not sure how to solve this question. Can anyone help me to approach this problem. Thanks.
1
vote
2answers
217 views

Use the Residue theorem and its application to compute the integral

$$\int_{-\infty}^{\infty} \frac{x^2}{x^4-4x^2+5} dx. $$ I am not sure how to approach this question. Can anyone use the complex variable theory to help me solving the problem please? Thank you very ...
2
votes
2answers
194 views

Find the argument of $ \frac{-1 + \sqrt3 i}{2+2i} $

I rewrite equation $ \frac{-1 + \sqrt3 i}{2+2i} $ as $$ \frac{ \sqrt3 - 1}{4} + \frac{ \sqrt3 + 1}{4} i $$ using the conjugacy technique. And set forward to find the argument of this complex ...
1
vote
1answer
82 views

$\int_{|z|=1} \frac{f(z) }{z-a} \, dz = 0$ for $f(z)=\sin \pi/z$.

Let $f$ be analytic for all $z$ where $0 < |z| < 2$ and $a \in \mathbb{C}$ is in this domain as well. I wish to prove that $$\int_{|z|=1} \frac{f(z) }{z-a} \, dz = 0$$ for $f(z)=\sin \pi/z$. We ...
3
votes
3answers
164 views

How does $\mathrm {e}^z$ and $\log z$ look like as complex functions.

I want to visualize complex functions $\mathrm e^z$ and $\log z$ in $C$, here $z\in\Bbb C$. I want to know their behavior and zeros and singularities. Can anyone explain me in an easy way. Thank you ...
7
votes
1answer
280 views

Möbius transformation in the complex plane.

Assume that $U$ be a line in the complex plane. And assume a Möbius transformation $\phi $ sends $ U $ again to a line. How can I classify all such $\phi$? I want to write my ideas. But, I ...
1
vote
1answer
748 views

Limit point of sequence vs limit point of the set containing all point of the sequence

I need to show that there exist sequences s.t. for fix $\epsilon>0$ there exist $|z_n-\alpha|<\epsilon$ (1) holds for infinitely many $n\in N$ but s.t. $\alpha$ is not a limit point of the set ...
0
votes
1answer
142 views

exp(x) for imaginary numbers

Well, I know how to get the $e^x$ function polynomial expansion, but how do I know that this is also valid for imaginary numbers, like $i\pi$? I know that the ...
1
vote
1answer
52 views

Integration without using parametrization .

I would like to integrate the following line integral without using parametrization . I wanted to integrate the following $$\int_C \frac{1}{z-a} dz$$ , where $C$ is a a curve along $|z-a| =r$ . ...
5
votes
1answer
115 views

A complex problem.

We have a set $S:= \{e^{inr\pi} | n\in\Bbb N\}$. Where r is an irrational number. I wonder whether this set is dense in $\partial D(0,1)$. i.e. I want to see if $\overline S=\partial D(0,1).$ I ...
0
votes
3answers
231 views

If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.

I know I'm going to use Liouville's Theorem, but my main question is why is $1/|f(z)|$ entire as well if $f$ is entire? Is this just a basic property: if $f$ is entire, then $1/f$ is entire? Thanks ...
2
votes
1answer
286 views

Proof that $\frac{1}{2}(c_{n}-d_{n})\pi=1$ if $n$ is odd, for $f(z)=\csc(z)$ and $\{c_{n}\}$ and $\{d_{n}\}$ Laurent coefficients of $f$

Let $f(z)=\csc(z)$ and $\{c_{n}\}$ and $\{d_{n}\}$ Laurent coefficients of $f$ in $\{z\in \mathbb{C}:0<|z|<1\}$ and $\{z\in \mathbb{C}:1<|z|<2\}$ respectively. Proof that ...
2
votes
1answer
62 views
6
votes
4answers
234 views

Expressing $\sum_{n=-\infty}^\infty\dfrac{1}{z^3-n^3}$ in closed form

I want to express $$\sum_{n=-\infty}^\infty\dfrac{1}{z^3-n^3}$$ in closed form. I know that $$\pi z\cot(\pi z)=1+2z^2\sum_{n=1}^\infty\dfrac{1}{z^2-n^2}$$ which looks close, but I don’t know how to ...
1
vote
0answers
30 views

When is meromorphic continuation possible?

Suppose I have an expression of the form $$f(z) := f_1(z)+f_2(z)$$ ($f,f_1,f_2$ can e.g. be integrals) with $f_1$ convergent in the region $R_1=\{\Re(z)>-1\}$ and $f_2$ convergent in the region ...
0
votes
2answers
35 views

Given $\begin{pmatrix} a & b\\ c & d\end{pmatrix}∈GL_2^+(R)$ , $\beta(w)=\frac{aw+b}{cw+d},\Im(w)>0$.Is $\beta$ bijective?

Given any matrix $A=\begin{pmatrix} a & b\\ c & d\end{pmatrix}∈GL_2^+(R)$, we can define a function $\beta:H\to{ \mathbb{C} }$ by $$\beta(w)=\frac{aw+b}{cw+d},w∈H$$,where $H$ is the upper ...
1
vote
1answer
313 views

Evaluating series by contour integration, the residue theorem, and cotangent

I'm trying to understand this section in Tristan Needham's book Visual Complex Analysis about what he says is a standard method for evaluating series via a contour integral. My specific question is ...
2
votes
2answers
194 views

how to find accumulation point of $z_n=e^{in}$

How can I find accumulation point of a) $z_n=e^{in} $ b)$z_n=i^n$ c)$z_n=(1-\frac{1}{n})+(-1)^ne^{\frac{1}{n}}i$ I tried at b) $\lim_{n\to\infty}(i^2)^{\frac{n}{2}}=(-1)^{\frac{n}{2}}=-1,+1$ at ...
0
votes
2answers
75 views

Fractional exponents and when they commute.

In any elementary algebra class students are taught that if $a$ and $b$ are coprime, $x^{a/b}=(\sqrt[b]{x})^{a}$ or $\sqrt[b]{x^{a}}$. But only after teaching this lesson I realized this isn't always ...
3
votes
1answer
102 views

How to prove $\sum_p {1 \over p^s} = \sum_{n=1}^\infty {\mu(n) \over n} \log \zeta(ns)$?

Problem Prove that for $\operatorname{Re}(s)> 0$, $$ \sum_p {1 \over p^s} = \sum_{n=1}^\infty {\mu(n) \over n} \log \zeta(ns), $$ where the sum extends over all primes $p$. Notes: $\log$ is ...
4
votes
2answers
113 views

A convergent-everywhere expression for $\zeta(s)$ for all $1\ne s\in\Bbb C$ with an accessible proof

I'm looking for a way to define the Riemann zeta function $\zeta(s)=\sum_{n\in\Bbb N_0}n^{-s}$ on the whole complex plane, without having to use analytic continuation, or perhaps more accurately, in a ...
5
votes
3answers
253 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 ...
0
votes
1answer
107 views

Prove the Jordan lemma i.e. $\int e^{-R\sin{\theta}}< \pi/R$

In complex variables my instructor wrote on the board "Jordan's Lemma", and then, somewhat imprecisely, $$\int e^{-R\sin{\theta}}< \pi/R \;\;\;\; \text{ e.g. } \int \frac{s \sin{x}}{x^2 + 2x + ...
0
votes
1answer
39 views

finding $\sum_{n=0}^\infty(\frac{(n+1)}{n})^{n^2}(z-2)^2 $ radius of convergence

find this power serie radius of convergence and the area where it converges. $\sum_{n=0}^\infty(\frac{(n+1)}{n})^{n^2}(z-2)^2 $ my attempt: a) $L=lim sup|an|^\frac{1}{n} \quad $$L=Lim ...
3
votes
1answer
343 views

Elementary bound on the Riemann zeta function

I am currently preparing for a course in analytic number theory and I wanted to get a heads start. In my preparation, I came across the following problem: Show that for $|y|\geq 2$, $|\zeta(1+iy)| ...
0
votes
1answer
37 views

how to show a diameter function exist by using compactness and closeness

Let $A,B \in \Bbb C$. and say $$d(A,B)=\inf{|a-b|:a \in A,b\in B }$$ is distance between $A$ and $B$. if $B=\{b\}$ is with one component then let's show $d(A,\{b\})=d(A,b)$. a) if $A\in\Bbb C$ is ...