Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

0
votes
1answer
18 views

Understanding convergence of an infinite product

I am reading the book Complex Analysis by Gamelin and I am having trouble understanding the following definition: Let $p_j$ be a sequence of non zero complex numbers. We say that $\prod p_j$ ...
0
votes
2answers
26 views

Residue of $1/(e^z+1)$

How do I find the residues of $\frac{1}{1+e^{z}}$. I have calculated that the root of $1+e^{z}=0$ and the answer is $z=i\pi(2k+1)$, but the problem is that I get stucked at $$Re(f,i\pi)=\lim_{z\...
0
votes
0answers
11 views

Area of Convergence for $\sum_{n=0}^\infty \frac{z}{(1+z^2)^n}$

I want to show that the series $$\sum_{n=0}^\infty \frac{z}{(1+z^2)^n}$$ converges for all values of $z\in \mathbb{C}$ outside the region $|1+z^2|=1$. I've tried using ratio test, but I get the ...
1
vote
2answers
32 views

Homework question using Liouville theorem and Cauchy estimate.

The statement is if $f$ is an entire function that satisfies $$ |f(z)| \leq \pi e^{2Rez} $$ Then $\forall z \in \mathbb{C}$ , there is a complex number $a \in \mathbb{C}$ such that: $$f(z) = ae^{2z}$...
0
votes
1answer
16 views

How to use mean value theorem to get the inequality $|\frac{1}{n^s}-\frac{1}{x^s}|\le \frac{|s|}{n^{\sigma+1}}$?

How to use the mean-value theorem to get the inequality $|\frac{1}{n^s}-\frac{1}{x^s}|\le \frac{|s|}{n^{\sigma+1}}$?
0
votes
0answers
15 views

Characterization of summable families

Could somenone help me with to prove the following result? If $C=\{\lambda\in{\Lambda}:x_\lambda\neq{0}\}$ is countable and for every bijective mapping $\tau:\mathbb{N}\longrightarrow{C}$ $\sum_{n\...
1
vote
1answer
55 views

Calculate $\int_{\pi\over 2}^{{\pi\over2}+i} \cos2 z dz$

Calculate $\int_{\pi\over 2}^{{\pi\over2}+i} \cos 2z dz$. I want to verify my answer please. My solution: Because $\cos 2z$ is analytic everywhere, we just need to calculate the integral: $$ \...
0
votes
0answers
40 views

If $f:\mathbb{C}\to\mathbb{R}$ is continuous and $f\ll1$ then $\int_{|z|=1}f\ll4$

Show that if $f:\mathbb{C}\to\mathbb{R}$ is continuous and $f\ll1$ ($a\ll b$ means $|a|\leq |b|$), then $\int_{|z|=1}f\ll4$. Hint: Show first $\int f\ll\int_0^{2\pi} |\sin t|dt$. I'm not sure how to ...
-3
votes
1answer
41 views

Prove for an entire function $\displaystyle{f(z)=\sum_{n=0}^\infty a_nz^{n}}$,

Prove that for an entire function $\displaystyle{f(z)=\sum_{n=0}^\infty a_nz^{n}}$, $$\displaystyle{\frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^2d\theta=\sum_{n=0}^\infty|a_n|^2r^{2n}}$$ for any $r&...
0
votes
1answer
36 views

Let $f(z)=\sum\limits_{n=0}^\infty a_nz^n$ be an entire function and r be a positive real number

Let $f(z)=\displaystyle{\sum_{n=0}^\infty a_nz^n}$ be an entire function and r be a positive real number, Then $\displaystyle{\sum_{n=0}^\infty |a_n|^2r^{2n} \leq \sup_{|z|=r}|f(z)|^2}$, is it true?
1
vote
1answer
19 views

Limiting behavior of a meromorphic function

If $f$ is an entire function and $\lim_{z \rightarrow \infty} f(z)/z = 0$, then show that $f$ is constant. I found this answer online. Can someone explains how can this follows from Morera's theorem; ...
1
vote
0answers
21 views

If a punctured neighborhood of a Riemann surface is biholomorphic to a punctured disk, is the neighborhood biholomorphic to a disk?

Let $R$ be a Riemann surface, $P_0\in R$ a point, and $D\subset R$ an open neighborhood of $P_0$. Suppose $D - P_0$ is biholomorphic to the punctured unit disk $\{z : 0 < |z| < 1\}\subset\mathbb{...
1
vote
1answer
20 views

Construct Discrete Sequence in Complex space

Consider an arbitrary open subset $U \subset \mathbb{C}$. I intend to construct a sequence $(a_i)_{i \in \mathbb{N}}$ contained in $U$ with following two properties: for every rational point $q=q_R+...
1
vote
1answer
28 views

Change of variable theorem for the curvilinear integral

I have to proof the change of variable theorem for the curvilinear integral, but I have no idea about how to do that, and I haven't found information in Google. The theorem states: Let be $\Omega$ $\...
1
vote
0answers
19 views

Criteria for an analytic function when the real and imaginary parts are $n$ dimensional

Say we have two functions $$ f,g:\mathbb{R}^n\rightarrow\mathbb{R} $$ and we define some complex function $\mathbb{R}^n\rightarrow\mathbb{C}$ as $$ h(\vec x) = f(\vec x) + ig(\vec x) $$ If $n=2$, ...
0
votes
0answers
32 views

How to solve this contour integral?

I was reading this where I encountered the following contour integral as given in equation (2.4) of the same. $$S = -i\int_{-\infty}^{+\infty} d\omega \log(\omega^2 + m^2 + E)$$ where $m,E \in \...
0
votes
0answers
41 views

find taylor series but the center is not analytic

I want to find a Taylor Series for complex function $$f(z)=\dfrac{z^2}{2+z},$$ centered at $z=-2$. I have find the taylor series $f(z)$ centered at $z=z_0$, $$f(z)=(z-2)+\dfrac{4}{z+2}=(z-2)+4\sum\...
3
votes
0answers
18 views

Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$ z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)! $$ ...
0
votes
3answers
43 views

Find the Laurent series expansion of the following function and hence find the integral over the unit circle centred at i

We are going to be examined on using the Laurent series expansion to find integrals along simple closed curves. But the notes and lectures barely covered it and we only have 2 examples given. Can ...
0
votes
3answers
25 views

Problem of complex equations and Cauchy-Riemann

I have the following problem: $$u (x, y) = \sin x \sinh y.$$ I need to check that the function is harmonic in the whole plane. But I do not know where to start, I would appreciate your help. Thanks ...
2
votes
4answers
79 views

Is it possible to calculate $\int_{0}^{\pi}(a+\cos{\theta})^nd\theta$, where $a$ is a nonzero integer?

I have tried to answer by taking change the variable $\theta$ to $\theta/2$, so the integration is now over unit circle, then I have taken $z=e^{i\theta}$. Now I tried to use residue formula for ...
0
votes
1answer
20 views

Mobius transformations between two sets

I am doing some revision of complex analysis, and am stuck on this question. I am looking for A mobius mapping sending the set {z: |z+1|<$\sqrt{2}$}, |z-1|<$\sqrt2$} onto the sector {z:3pi/4< ...
0
votes
1answer
25 views

A complex integral with boundary on the norm

I was asked this question. I couldn't even manage to find a starting point on the question. Any help is welcomed. P.S: I know it is from a book but I do not know from which one. If you have any idea ...
0
votes
1answer
36 views

Path dependency of complex integrals

This is a basic question. But it’s my first time really dealing with this topic. When we have a integral along the real line we have: $$\int^{b}_{a} f(x)dx$$ Which obviously has no path dependency. ...
0
votes
0answers
36 views

Compute this integral over $\mathbb C$?

How can I compute this integral: $$I=\int_{\mathbb C} e^{-|z|^2} \, e^{i\left<z,w\right>} \, e^{a z + b \overline{z}} \, dz,$$ where $w\in \mathbb C$ and $a,b \in \mathbb R$ and where $\left&...
2
votes
0answers
66 views

Complex polynomial [duplicate]

Let $p(z)=z^n+a_{n-1}z^{n-1}+...+a_1z+a_0$ be a complex polynomial with $a_i \in \mathbb C$ such that for $|z| \leq 1$, $|p(z)|\leq 1$, then prove/disprove that $p(z)=z^n$.
0
votes
1answer
27 views

Taylor series of complex function confusion with big O notation

Suppose $u(x,t)$ is a function of two real numbers that outputs a complex number. Usually I would have $u(x,t+k) = u(x,t) + k\frac{\partial u}{\partial t}(x,t) + \frac{1}{2}k^2 \frac{\partial ^2 u}{\...
0
votes
1answer
31 views

Exercise chapter 5 Stein complex analysis

Prove that for every z the product $\prod_{k=1}^{\infty}cos(\frac{z}{2^k}) = \frac{sin(z)}{z}$ [Hint: Use the fact that $sin(2z) = 2*sin(z)*cos(z)$.] Then my idea was the following, I have that: $P_n ...
0
votes
1answer
58 views

Is $f(z) = z^{2}$ an automorphism of $\mathbb{C}$?

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ where $f(z) = z^{2}$. I claim $f$ is an automorphism of $\mathbb{C}$ as a vector space. First $\operatorname{ker}(f) = 0$ and since $\mathbb{C}$ is ...
0
votes
0answers
35 views

uniform convergence on interior

I am reading Gamelin's Complex Analysis book, and stumbled upon a statement while working on a question. the question asks to show that $\sum \frac{z^n}{n}$ is not uniformly convergent for $|z| < ...
1
vote
2answers
57 views

Mapping of unit circle under $f(z) = \frac{1}{2}(z + \frac{1}{z})$

I'm trying to find the mapping of the unit circle centered at the origin $A=\{ z\in \mathbb{C} : |z|=1\}$ and a straight line $B=\{ \arg (z) = \theta_0 \}$, under the mapping $ f(z) = \frac{1}{2}(z + \...
0
votes
1answer
18 views

Obtaining a holomorphic function given a function with no imaginary part

Suppose that $u(x,y) + i v(x,y) = x^{3} − kxy^{2} + 12xy − 12x$ for some constant $k\in Complex$ $plane$.Find all values of $k$ for which $u$ is the real part of a holomorphic function. I know that I ...
-1
votes
0answers
25 views

Prove: $f(z^*)$ is not an analytic function unless it is a constant [on hold]

How do you prove that if $f(z)$ is an analytic function, then $f(z^*)$ is not an analytic function unless it is a constant?
0
votes
0answers
26 views

How to prove $n$-to-$1$ mapping property for Blaschke products?

I have two questions regarding the Blaschke products. 1) I came across the following post Boundary behaviour of finite Blaschke products on the unit circle where it's mentioned that for any $\...
0
votes
1answer
50 views

Residue at infinity of $(e^{2\pi z}-1)/(z(z^2+1)^2)$

I have trouble to proove that the residue at inifinity of this function is zero (I found that the sum of the residues at finite is zero). I tried to expand in series at $z=0$ the function $f(1/w^2)/w^...
0
votes
1answer
17 views

Laurent Series of rational function in z

Im attempting to solve a problem of defining the types of singularities of a complex function. I found that if the function has a Laurent series expansion with a finite number of terms in the ...
0
votes
1answer
22 views

If $D\subset\mathbb C$ is a bounded domain, and $f,g:D\to\mathbb C$ are holomorphic such that $|f|\le|g|$ on $\partial D$, does $|f|\le|g|$ on $D$?

Suppose we have a bounded domain $D\subset\mathbb C$ with smooth boundary, and holomorphic functions $f,g:D\to\mathbb C$ which are continuous up to the boundary, and such that $|f|\le|g|$ on $\partial ...
4
votes
1answer
40 views

When do a holomorphic function's zeroes occur in conjugate pairs?

I have the following proof that a holomorphic function's zeroes occur in conjugate pairs when its derivatives evaluated at $0$ lie on a line through $0$: Let $f:\mathbb{C}\mapsto\mathbb{C}$ be ...
1
vote
1answer
30 views

(Proof verification) $\pi: X\to Y$ birational with $X$ smooth implies $\pi_*O_X=O_Y$.

As the title suggests, I am looking for a confirmation or falsification of my proof of the following Let $\pi: X\to Y$ be a birational map of algebraic varieties over the complex numbers with $X$ ...
6
votes
3answers
218 views

Will these geometric means always converge to $1/e$?

Let $p_n$ be the $n$-th prime and $F_n$ be the $n$-th Fibonacci number. We have $$ \lim_{n \to \infty}\frac{(p_1 p_2 \ldots p_n)^{1/n}}{p_n} = \lim_{n \to \infty}\frac{\{\log(F_3)\log(F_4)\ldots \...
1
vote
2answers
36 views

How do I evaluate $\oint_{|z|=1} \dfrac{1}{z^{2} \sin z} dz$ by means of the Cauchy residue theorem?

Evaluate the integral by means of the Cauchy residue theorem. $$\oint_{|z|=1} \dfrac{1}{z^{2} \sin z} dz$$. So Cauchy's Residue Theorem states: If $\Gamma$ is a simple closed positively oriented ...
0
votes
1answer
19 views

Contour integration with the contour $\sigma=[0,1]+[1,i]$

$\sigma=[0,1]+[1,i]$ is a contour. I am asked to sketch the contour, and evaluate $\int_\sigma Re(z)$. Firstly, I am not sure how to visualise this contour, since there are two parts. What does it ...
0
votes
1answer
56 views

Proof of $ \ \vert \log(z) \vert \lt \log \frac{1}{1-r} \ \ $ given $ \forall z ,\ \vert z-1 \vert \leq r \lt 1$?? [on hold]

Let $\ \vert z-1 \vert \leq r \lt 1$ and let $\log(z)$ be the principal branch of the logarithmic function. How can you prove $$ \vert \log(z) \vert \lt \log \frac{1}{1-r} \ \ ?$$ Also, how can you ...
-3
votes
0answers
37 views

Find the imaginary part of z [on hold]

If $z = (a+ib)^r \times e^{x+iy}$, find the imaginary part of $z$, (note that r may not be an integer).
0
votes
2answers
68 views

Is $f(x)=\sqrt{x}$ continuous in $\mathbb{C}$?

This is probably a super naïve question, but it seems like the only reason $f(x)=\sqrt{x}$ isn't continuous on $\mathbb{R}$ is because it pops over $\mathbb{C}$ when $x<0$. So, seeing as $\mathbb{R}...
1
vote
0answers
27 views

The Laplacian operator is invariant to $SL_2(\mathbb{R})$

I am reading Iwaniec's book on the spectral analysis of automorphic forms, where I bumped into the following statement in p.20 section 1.6. Given a function $f:\mathbb{H}\longrightarrow \mathbb{C}$, ...
0
votes
1answer
25 views

Equation in $\mathbb{C}$

I state that I am not studying the resolution of equations in the complex field, but now I should solve the following for other reasons. Since I haven't solved it for a long time, I've forgotten the ...
0
votes
2answers
33 views

Möbius transformations mapping non-unit circle to non-unit circle

I have a problem in which I need to find a möbius transformation which has as one of the criterion to map the circle $|z−2+i| = \sqrt5$ onto the circle $|w+2| = 2$, I dont really understand how to ...
1
vote
2answers
77 views

Complex function only differentiable in $y=x^2$

Find a function $F: \mathbb{C} \rightarrow \mathbb{C}$ that is differentiable in the parabola $y=x^2$ and not differentiable in the rest of the complex plane. Let $F(x,y) = u(x,y) + i v(x,y)$. If $F$ ...
1
vote
1answer
44 views

$\{g_{n_k}\}_{k \in \Bbb N}$ normally convergent $\implies \{g^2_{n_k}\}_{k \in \Bbb N}$ normally convergent (as meromorphic functions)?

Let $\mathscr F$ be a family of one-to-one holomorphic functions on a simply-connected domain $D \subset \Bbb C$ such that $\mathscr F$ omits 0. Show that $\mathscr F$ is a normal family (when ...