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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1answer
72 views

Mandelbrot and Julia Set

Consider a dynamical system $$z_{n+1}=\frac{\alpha+z_n}{1+z_{n-1}}$$ for $n=0,1,2,\dots$ In other words the system is $$z_{n+1}=f_{\alpha}(z_n,z_{n-1})$$ where $f_{\alpha}$ is defined from ...
0
votes
1answer
23 views

Limit a complex contour integral

Let $z_{0}$ be a simple pole of $f$ and $\gamma_{r}$ an arc of circle centered on $z_{0}$, of the radius $r$ and angle $\alpha$, i.e., $\gamma_{r}=z_{0}+re^{it}$, with $t\in ...
2
votes
1answer
37 views

Dirichlet Series and Asymptotic Expansions

Consider the Dirichlet series $\tilde{f}(s)= \sum_{n=1}^{\infty} f(n) n^{-s}$. In the page "Zeta Function Regularization" of Wikipedia http://en.wikipedia.org/wiki/Zeta_function_regularization I ...
0
votes
2answers
76 views

How do I calculate the residue of $\sin(z+1/z)$?

How do I do this about $\displaystyle z=0$ ?. I tried creating a Laurent expansion and extracting it from there but I wasn't sure how to isolate the $\displaystyle 1/z$ expression. $$ \mbox{I ...
4
votes
1answer
47 views

How to show $\int_0^1\frac{e^{e^{2\pi it}}}{e^{2\pi it}}dt=1$

I was trying to integrate the contour integral $$\int_\gamma \frac{\vert z \vert e^z}{z^2}$$ where $\gamma$ parametrizes the unit circle counterclockwise. I cannot use the Generalized Cauchy Integral ...
1
vote
2answers
84 views

Proving that $\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2$

Given $f$ entire show that $$ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2 $$ I've come close to getting the exact ...
1
vote
3answers
71 views

How to use complex analysis to find the integral $\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$?

How can I use complex analysis to solve the following: $$\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$$
0
votes
1answer
28 views

Find the transformation that maps real axis to itself and imaginary axis to the circle $|w-\frac{1}{2}|=\frac{1}{2}$

Find the transformation that maps real axis to itself and imaginary axis to the circle $|w-\frac{1}{2}|=\frac{1}{2}$ What I did: $$z_{1}=0,z_{2}=i,z_{3}=\infty ...
0
votes
0answers
32 views

Chain rule (derivative) for for complex data

I found some difficulties in extending the chain rule for complex data. Any suggestion will be appreciated, thanks. In the complex domain, for example, we have a function ...
4
votes
2answers
90 views

Calculate $I_m = \int_{-\infty}^\infty \frac{dx}{1+x+x^2+\cdots+x^{2m}}$ using complex variables

I have come as far as deducing that the denominator can be written as a geometric series. Hence, for $m=2$, \begin{align*} \int_{-\infty}^\infty \frac{1-x}{1-x^5} dx &= 2 \pi i ( B_1 + B_2 ) - ...
0
votes
1answer
19 views

$\left|(1+R^2e^{2i\theta})^2\right| \geqslant (R^2-1)^2$ in complex integration

I need to prove: $$\lim_{R\to +\infty} \left|\int_0^\pi \frac{e^{iaR(\cos\theta+i\sin\theta)}}{(1+R^2e^{2i\theta})^2}iRe^{i\theta} d\theta\right| =0$$ Could someone give me some pointers? A ...
4
votes
2answers
39 views

Analytic continuation of holomorphic function along clockwise/counterclockwise path

"Write down (say, as a power series) a holomorphic function $f(z)$ on $D(1, 1)$ which satisfies $f(z)^5 = z$ and $f(1) = 1$. What is the result of analytically continuing $f$ along a path which ...
3
votes
1answer
28 views

Is i holomorphic over the whole complex plane?

That is, is i entire? I know that it's derivative with respect to z bar is 0, so I would think that the answer is yes, although I'm not sure.
0
votes
2answers
73 views

Direct evaluation of a series from Euler's identity.

Is there a direct way to evaluate: $$ \sum_{k=0}^{\infty} (-1)^k \dfrac{\pi^{2k}}{(2k)!}=-1 $$ Note that this follows from Euler's identity.
0
votes
1answer
262 views

A change of variables in the euler equation

If someone could help me with the proposed change of variables, it would be greatly appreciated. Consider Euler's equation: $$z^2w'' + \alpha zw' + \beta w = 0$$ where $w$ is a function of $z$ and ...
3
votes
1answer
143 views

Evaluating $\int_0^\infty \frac{\log t}{1+t^2}\,\mathrm dt$ using residues

I want to integrate $$\int_0^\infty \dfrac{\log t}{1+t^2}\,\mathrm dt$$ using the residue theorem. The poles are at $i,-i$. If the integral were from $-\infty$ to $\infty$, I would consider ...
2
votes
2answers
54 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then a)both the series converge ...
1
vote
1answer
20 views

Proving the asymptotic behavior of the prime counting function (Prop 2.1 in Ch.7 Princeton Lectures in Analysis-Complex Analysis)

This is taken from Complex Analysis by Elias M. Stein and Rami Shakarchi. $\psi(x) \text{ is Tchebychev’s ψ-function defined by}$ $$\psi(x)=\sum_{p^m\leq x} \text{log }$$ the sum is taken over the ...
43
votes
8answers
7k views

Can someone please explain the Riemann Hypothesis to me… in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
4
votes
1answer
42 views

(Though?)Expression Rearranging

I have the following expression $ 2x+3x^2+e^{5x+x^2}=7 $ which I need rearranged in a form of the type $Ye^Y=Z$ with Y a function of x and Z some constant. I have tried the substitution $y=5x+x^2$, ...
3
votes
1answer
28 views

an analytic function in $\Delta^n$ is bounded in $T^n$, then it is bounded in $\Delta^n$

Is true that if an analytic function in $\Delta^n$ is bounded in $T^n$, then it is bounded in $\Delta^n$? Here $\Delta^n$: polydisc and $T^n$: Torus, distinguished boundary of $\Delta^n$.
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0answers
31 views

Using residue theorem along a branch cut to evaluate the inverse Laplace transform

I am trying to find the inverse Laplace transform of $f(z)$ using the residue theorem. Can you please check to see if what am doing below is correct? I am not really sure about what I am doing. ...
1
vote
1answer
41 views

Meromorphic function with bounded order of zeros and poles

The following problem has been bothering me for a long time; Let $X$ be a compact Riemann surface of genus $g$. Is there a non-zero meromorphic function on $X$ with all of its poles and zeros have ...
0
votes
0answers
36 views

Evaluating the sum $\sum_{n=1}^{\infty}\frac{1}{n^4 + 1}$? [duplicate]

I'm trying to evaluate the sum $$\sum_{n=1}^{\infty}\frac{1}{n^4 + 1}$$ I figure that this has something to do with the Poisson summation formula, which states that $$\sum_{n \in \mathbb{Z}}f(n) = ...
2
votes
1answer
40 views

Proving that a function admits a primitive in a specific set?

I'm trying to show that $$f(z) = \frac{z}{(z^2 - 1)(z^2 - 4)}$$ admits a primitive in the set $\{|z| > 4 \}$ I know that the only singularities of $f(z)$ are poles that occur at the points $z = ...
0
votes
1answer
26 views

Complex Analysis analytic function 1$f(z)=z$ [duplicate]

if$\text{ } f:D(0,1)\longrightarrow D(0,1)$ is analytic such that there exists $a,b\in D(0,1)$ and $\text{ }$$f(a)=a$ , $f(b)=b$ prove that $f(z)=z$ $\forall$ $z\in D(0,1)$
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0answers
22 views

Which one is correct option? [duplicate]

Let $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$. Which of the following are correct? there exists a holomorphic function $f:\mathbb{D}\rightarrow \mathbb{D}$ with $f(0)=0$ and $f'(0)=2$. there exists ...
2
votes
1answer
31 views

Usage of Rouche's theorem?

I'm trying to find the number of zeros for the function $f(z) = z + 2 - e^z$ in the half plane $\{\mathscr{R}z < 0\}$. I know I'm supposed to use Rouche's theorem, which states that if both $f$ ...
5
votes
4answers
624 views

Numerical approximation of a complex integral with a nested exponential

I've been working on a maths problem as part of my Physics PhD; but have been stumped by the following integral. All I need to know is a numeric approximation to the integral (along with an estimation ...
0
votes
0answers
38 views

Evaluate $h(z)=\frac {k}{2\pi} \int_CF(\theta)e^{ikz\cos \theta}\,d\theta$

Suppose this integral $$h(z)=\frac {k}{2\pi} \int_CF(\theta)e^{ikz\cos \theta}\,d\theta$$ $$0\le\theta\le\pi$$ $$|z|\le l$$ We are in complex $\theta$ plane. Assume we have knowledge of $F(\theta)$ ...
2
votes
1answer
19 views

Complex Integral with constant Function

Show $$\dfrac{1}{2\pi i} \oint_{C}\dfrac{f'(z)}{f(z)-f_{0}}dz=N$$ Where $N$ is the number of points "$z$" where $f(z)=f_{0}$(a constant) inside of $C$; $f'(z)$ and $f(z)$ are analytic inside and on ...
2
votes
1answer
32 views

Branch cut and principal value

I do not understand the principal value and it is relation to branch cut. Please tell me about principal value with some examples, then explain the branch cut concept. For instance, what is the ...
0
votes
2answers
35 views

to find radius of convergence of power series.

I have a power series given as: $f(z) =1 + z+ \frac{z^2}{2^2} +\frac{z^3}{3!} + \frac{z^4}{2^4} \frac{z^2}{2^2}+ \frac{z^5}{5!}+ \ldots$ I have to find radius of convergence of above series. My ...
2
votes
2answers
48 views

Best way to evaluate integral with contour integration?

I'm trying to evaluate the integral: $$\int_{-\infty}^{\infty}\frac{\sin^2{x}}{x^2}dx$$ with contour integration and am not sure if the basic idea of what I'm doing is correct. I know that $$\sin{x} ...
2
votes
0answers
46 views

using Paley-Wiener to get support and then estimate inf sup

Define the function $$ \tilde{f}_n(\omega)=\frac1{\sqrt{2\pi}} \frac{\sin R\omega/2}{R\omega/2} s_n(R\omega/2\pi),$$ where (using the Weierstrass product representation for $\sin$) $$ s_n(w) = ...
-1
votes
0answers
42 views

List of simple, common functions with an incomplete domain or range on $\mathbb{C}$

This may seem like a strange question, but it's an interest of mine and I would appreciate the help of the community in addition to brainstorming on my own. As the question states, I'm looking for ...
0
votes
2answers
25 views

Find the Order of the Zero of the Function [closed]

Determine the order of the zero of the function $z=0$ given a) $e^{\sin(z)}-e^{z}$ b) $(\cos(z)-1)^{3}\sin(z)$ Please, can anyone help me, what should I do?
1
vote
1answer
32 views

Holomorphic function satisfies estimate

Determine whether there exist functions $f$ which are holomorphic in a neighborhood of 0 and satisfy $$n^{-5/2}<|f(1/n)|<2n^{-5/2}$$ for $n\geq 1$. What method should you use?
1
vote
1answer
36 views

Show that $f(z):=\sum a_n (z-z_0)^n$ is continuous whenever $z$ is in disk of convergence.

Consider a power series $\sum a_n(z-z_0)^n$, and assume it has radius of convergence $r$. Then we know that $\forall z\in(z_0 -r,z_0 +r)$, this power series converges absolutely by root test. Thus we ...
1
vote
1answer
53 views

how to solve complex integration problem

While working on complex integration problem I got stuck at the following problem: $\int \frac{|dz|}{|z-2|^2}$ where $|z| = 1$ is the domain. The only idea that I am getting is that we can use the ...
2
votes
1answer
52 views

Complex structures on punctured disks.

Let $X$ be a smooth surface diffeomorphic to the punctured unit disk $\{(x,y)\in \mathbb{R}^2 \ | \ 0<x^2+y^2<1\}$ in the plane. It admits a lot of non equivalent complex structures, for example ...
0
votes
0answers
26 views

Expressing principal value of integral as real/imaginary

How is it that we can express $$ \mathrm{p.v.}\int_{-\infty} ^{\infty} \frac{\cos 3x}{x^2+4}=\Re \ \mathrm{p.v.}\int_{-\infty} ^{\infty} \frac{e^{3xi}}{x^2+4} $$ while we cannot for $$ ...
0
votes
0answers
40 views

Complex Function in the unit disc

If $f$ is a complex valued function which takes the unit disc $U$ to itself and $f(\frac43)=\frac43$ while $f'(\frac23)=\frac43$, how can we find $f$ if it exists?
3
votes
1answer
81 views

integration of $\int \frac{1}{x+ i\:y}\mathrm{d} x$

I can't seem to find where this result comes from $$ \int \frac{1}{x+ i\:y}\mathrm{d} x = \frac{\ln(x^2 +y^2)}{2} - i \: \arctan \left( \frac{x}{y} \right) $$ by my calculation the result should be ...
5
votes
4answers
4k views

What is the analytic continuation of the Riemann Zeta Function

I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other ...
0
votes
3answers
116 views

Using Rouche's Theorem to find the number of zeros of $p(z)=z^8 +10z^3 −50z+1$ in the right halfplane

I'm studying for a complex analysis qualifying exam and was wondering if someone could help me out with this. I am not sure how to apply Rouche's Theorem to this. How many zeros does the polynomial ...
0
votes
0answers
16 views

Inverse transformation of continous transformation is bounded

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with $B \subseteq \mathbb C $ bounded. Now I want to proove that $ A = f^{-1} (B)$ is bounded! How can I proove that this ...
3
votes
1answer
33 views

A question regarding sheaf cohomology

I am trying to understand a statement from http://arxiv.org/abs/1312.1562, saying "... $H^1(\Sigma^x,T\Sigma) = 0$ since $H^1(\Sigma^x,\mathcal{O}) = 0$ and the Mittag-Leffler problem is solvable on ...
0
votes
1answer
39 views

Schwarz Lemma/Conformal mapping problem

Let $F:\mathbb{H}\rightarrow \mathbb{D}$ be holomorphic, where $\mathbb{H}$ is the upper half plane and $\mathbb{D}$ is the unit disc. Show that if $F(i)=0$, then $$|F(z)|\leq ...
1
vote
0answers
47 views

Prove that $\overline{f(z)}=f(\overline z)$ [duplicate]

Let $f:\Bbb C \to \Bbb C$ be a entire function sutch that $f(\Bbb R) \subseteq \Bbb R$, prove that $$\overline{f(z)}=f(\overline z)$$ In the hint of question said, cosider $g:\Bbb C \to \Bbb C$, ...