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, ...

learn more… | top users | synonyms (2)

10
votes
1answer
144 views

Application of the residue theorem

I want to prove that $$\sum_{j=1}^n \frac{1}{\left[\text{cos} \left( \frac{j \pi}{2n+1} \right)\right]^4}=\frac{8n(n+1)(n^2+n+1)}{3}$$ for $n \in \mathbb{N}$ by using the residue theorem. Which ...
1
vote
2answers
97 views

Möbius transformations lines and circles

I am looking for a basic outline of a proof I know that all MT's are of the form $\frac{ax+b}{cx+d}$ For $c=0$, I know that lines/circles are preserved because translations and dilations do not ...
0
votes
1answer
52 views

Analytic continuation of Zeta type function

Can one analytically continue the function (Not equal to the Zeta function) $$Z(s)=\prod_{p}\frac{1}{1+p^{-s}}=\sum_{k=1}^{\infty}\frac{(-1)^{\Omega(k)}}{k^s}$$ Where $\Omega(k)$ is the number of ...
4
votes
2answers
74 views

Orientation on Riemann surfaces

$\mathcal{X}$ is a Riemann surface and $\mathcal{E}^{(2)}(\mathcal{X})$ is the $\mathbb{C}$-Vector space of all differentiable $2$-forms on $\mathcal{X}$. I want to define the orientation of ...
1
vote
1answer
90 views

Prove that $\sum_{n=1}^{\infty} \frac{a_n}{n^z}$ converges absolutely and uniformly

Let $(a_n)$ be a sequence in $\mathbb{C}$. Assume that $\sum_{n=1}^{\infty} \frac{a_n}{n^z}$ converges absolutely for some $z= z_0 \in \mathbb{C}$. Prove that the series converges absolutely ...
3
votes
0answers
90 views

Subharmonic function and holomorphically parametrized integrals

Let $f_\lambda$ be a family of $L^1$ functions (say on $\mathbb{C}$) such that for all $z$ the map $\lambda \mapsto f_\lambda(z)$ is holomorphic. Consider the map $N(\lambda)=\log \int |f_\lambda(z)| ...
1
vote
2answers
575 views

how to find a complex integral when the singular point is on the given curve

how to evaluate ∮1/(z-2) dz around the square with vertices 2±2i ,-2±2i the function is not analytic on z=2.but z=2 is on the given curve.so that can't apply cauchy,s integral formula.how can i do ...
1
vote
0answers
47 views

Calculate all the local automorphisms

The Kohn - Nirenberg domain $\Omega_{KN}$ defined by $$\Omega_{KN}=\left\{(z,w)\in \Bbb C^2:\text{Re}\ w+|zw|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6<0\right\}$$ How to compute all the ...
0
votes
1answer
456 views

Find the Laurent series of $\sin z/z^2$ using Laurent's theorem

I have the function $f(z)=\frac{\sin z}{z^2}$, wich is analytic over $\Bbb C\setminus\{0\}$, I want to find the Laurent series of $f$ valid for $0<|z|<R\le\infty$. Using Laurent's theorem we ...
1
vote
0answers
25 views

show that there exists $n_0$ such that for $n\geq n_0$ $f_n(z)$ has $k$ roots.

The situations is as follows. We have sequence of holomorphic functions $\{f(n)\}$ which is defined on a open $U$ in $\overline{B(0,1)}$. Suppose now that this sequence converges uniformly to a ...
1
vote
0answers
9 views

affine curve. analysis

Can anyone explain what is written in book? $\Gamma(w,E)\equiv w^{2}-E^{n}+\sum_{i>0,j\ge 0, ni+2j<2n} g_{ij}w^{i}E^{j}=0, $ where $g_{1,0}$ doesn't equal to zero. "At infinity it compactified ...
1
vote
2answers
474 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 ...
1
vote
0answers
56 views

Calculate integral when I have a pole on the unit circle

I'm having trouble integrating the following integral $$\int_0^{2\pi} \frac{d\theta}{3 -2\cos\theta + \sin\theta}$$ I make the substitutions $z= e^{i\theta}$ , $\sin\theta = ...
2
votes
3answers
151 views

Prove that there are no entire function satisfying $|f(z)|\ge |\cos(z)|+|\sin(z)|$ for all $z\in \Bbb C$

Hi. I need to prove that there are no entire function satisfing $$|f(z)|\ge|\cos(z)|+|\sin(z)| \\\forall z\in\mathbb{C}.$$ I think I need to use the Liouville theorem. Appriciate any help, Thanks!
2
votes
1answer
39 views

Determine all entire functions $f$ with $f(z)\not\in (-\infty,0]$.

This questions exists of two parts, I solved the first part. (i) Determine all entire functions $f$ with $Real(f(z))>0$ for all $z$ and $f(0)=1$. It is easy to see that $1/(1+f)$ is bounded and ...
1
vote
0answers
56 views

Show that $f(z)\leq \left|\frac{z-a}{1-\overline{a}z}\right|$

Given is that $|a|<1$ and the transformation $$T\colon z\mapsto\frac{z-a}{1-\overline{a}z}$$ This maps $B(0,1)$ onto $B(0,1)$. Now suppose for some $\epsilon >0$ the function $f$ is holomorphic ...
9
votes
4answers
364 views

Double Integral $\int_0^\infty \int_0^\infty \frac{\log x \log y}{\sqrt {xy}}\cos(x+y)\,dx\,dy=(\gamma+2\log 2)\pi^2$

Hi I am trying to solve this double integral $$ I:=\int_0^\infty \int_0^\infty \frac{\log x \log y}{\sqrt {xy}}\cos(x+y)\,dx\,dy=(\gamma+2\log 2)\pi^2. $$ Thank you. The constant in the result is ...
0
votes
1answer
89 views

Determine whether $\sum_{n=1}^{\infty} \frac{i^n}{\sqrt{n}}$ converges

I am not sure if what I have done is correct: I compare $\frac{i^n}{\sqrt{n}}$ with $\frac{(-1)^n}{\sqrt{n}}$ and have $ 0\le \frac{i^n}{\sqrt{n}} \le \frac{(-1)^n}{\sqrt{n}}$ (However, I am not ...
1
vote
4answers
75 views

Compute the radius of convergence for $\sum_{k=0}^{\infty}k!z^{k!}$

Just working on some practice questions and I'm not too sure what to do with this one. I've never encountered the $n!$ in the exponent of $z$ in these types of questions before. Computing the radius ...
6
votes
3answers
218 views

Integral $\iint \limits_{{x,y \ \in \ [0,1]}} \frac{\log(1-x)\log(1-y)}{1-xy}dx\,dy=\frac{17\pi^4}{360}$

Hi I am trying to integrate $$ \mathcal{I}:=\iint \limits_{{x,y \ \in \ [0,1]}} \frac{\log(1-x)\log(1-y)}{1-xy}dx\,dy=\int_0^1\int_0^1 \frac{\log(1-x)\log(1-y)}{1-xy}dx \,dy $$ A closed form does ...
1
vote
5answers
126 views

Complex integral - exercise

$$ \int\limits_{C(-2, \frac{1}{4})} = \frac{e^z}{z^2-4}dz$$ C is a circle center = -2 and radius = $\frac{1}{4}$ z is a complex number I don't know how to do the exercises like that.
1
vote
1answer
69 views

A particular entire function

Let $$G(z)=\Pi_{n=1}^{\infty}(1+\frac z n)e^{-\frac z n}$$,show that it's an entire function and $G(z-1)=ze^{\gamma}G(z)$ where $$\gamma=\lim_{n\rightarrow \infty}(\sum_{k=1}^n\frac 1k-\log n)$$ I ...
4
votes
1answer
76 views

Complex power series divergent and convergent on dense subsets of the circumference of convergence?

Is it possible to have a complex power series $ \sum a_nz^n $ with radius of convergence R such that the series diverges on a dense subset of the circumference of convergence and converges on another ...
0
votes
1answer
45 views

Question about conformally equivalent domains.

This is the well known definition of conformal equivallence between domains in the complex plane: Let $U$ and $U^{\prime}$ be two domains in the complex plane. We say $U$ and $U^{\prime}$ are ...
0
votes
1answer
46 views

Complex Polynomial That is n Times Differentiable: A Concern

I'm looking at a question that asks me to show that: If a function $f$ is known to be $n$-times differentiable in a domain $D$ and if $\forall{z\in{D}}\ \ f^{(n)}(z)=0$, then $f$ is a polynomial ...
1
vote
1answer
69 views

Using Poisson's integral formula

The problem asks to prove the following equality using Poisson's integral formula (or Poisson kernel, if I understood correctly from Wikipedia): $$\int_0^{2\pi} \frac{e^{\cos ...
5
votes
2answers
94 views

How exactly is $i=\sqrt{-1}$ related to $\mathbb{C}$ being a closed algebraic field?

There are many known proofs of why $\mathbb{C}$ (field of complex numbers) is algebraically closed (for example Cauchy's proof ) However: how does introducing the solution to the equation $x^2 + ...
8
votes
3answers
240 views

Integral $\int_0^{\pi/4}\frac{dx}{{\sin 2x}\,(\tan^ax+\cot^ax)}=\frac{\pi}{8a}$

I am trying to prove this interesting integral$$ \mathcal{I}:=\int_0^{\pi/4}\frac{dx}{{\sin 2x}\,(\tan^ax+\cot^ax)}=\frac{\pi}{8a},\qquad \mathcal{Re}(a)\neq 0. $$ This result is breath taking but I ...
7
votes
1answer
151 views

zeros of a polynomial

Given $P(z)=z^6+6z+10$, find how many roots are in each quadrant I have already seen that $P(z)$ has six different roots, and that none of them are real or of the form $ki$, $k\in \Bbb R$. Since ...
1
vote
0answers
39 views

Trying to use the deformation theorem to solve integral

I have this integral: $$\int_{|z|=2}\frac{\cosh z}{(z+1)^3(z-1)}dz$$ Both singularities $z=1,z=-1$ are inside the circle. I have already solve this using partial fractions, and I don't have much ...
-1
votes
2answers
47 views

Complex integration around a singularity [duplicate]

I am trying to integrate the function $f(z)=$$\frac{5}{z^2}$ from -3 to 3 and I am supposed to develop a closed region that avoids the origin and use the analyticity of the function in this region to ...
0
votes
0answers
44 views

Is this complex integral well solved?

I have this exercise: $$\int_{|z|=1}\frac{\cos z}{z^3}dz$$ The way I tried to solve it was: Since we have a singularity in $0$, and it is inside of the curve, lets consider the new curve: ...
1
vote
0answers
117 views

Schwarz–Pick theorem for one inequality with holomorphic function

Let $f(z)$ be holomorphic function in unit disc and $|f(z)| \le 1$. Prove that: $$\left| \frac{f(z)-f(0)}{z} \right| \le \left| 1- \overline{f(z)}f(0) \right|$$ I would use Schwarz–Pick theorem as ...
3
votes
2answers
2k views

Maps circles onto ellipses

Show that the mapping $w=z+\frac1z$ maps circles $|z|=p(p\ne 1)$ onto ellipses $$\frac{u^2}{(p+\frac1p)^2}+\frac{v^2}{(p-\frac1p)^2}=1.$$ I can parametrize the circle by $z(t)=pe^{it}, \ 0 \leq ...
0
votes
1answer
32 views

How do we find a path $\gamma$ with winding number $1$ and $2$ relative to points $1$ and $2$, respectively?

Let $\gamma :[a, b]\to\Omega\subseteq\mathbb{C}$ denote a parametric piecewise continuously differentiable path in $\Omega$ and $$\text{ind}_{\gamma}(z):=\frac{1}{2\pi ...
3
votes
1answer
125 views

Complex contour integral and partial fractions

I'm doing complex integration and I'm trying to evaluate: $$\int_C \frac{\cos{z}}{z^2 + 1} dz$$ Where $C$ is the clockwise boundary of a parallelogram with vertices $3i$, $2$, $-3i$, $-2$ (i.e. a ...
2
votes
1answer
72 views

series of an arbitrary sequence multiplying 1/n

Let $(r_n)_{n=1}^\infty$ be an arbitrary sequence of numbers in $[0,1]$. The series $\sum_{n=1}^\infty\frac{1}{n^2\sqrt{|x-r_n|}}$ converges for almost all $x$ in $[0,1]$. Is it true or not true? I ...
10
votes
1answer
218 views

Integral $\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3$

Hi I am trying to prove this below. $$ I:=\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3 $$ where $$ \zeta_3=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I am ...
7
votes
3answers
254 views

Prove that $\int_{-\infty}^{\infty} \frac{e^{2x}-e^x}{x (1+e^{2x})(1+e^{x})}\mathrm{d}x$ equal $\log 2$

This integral poppep up recently $$\int_{-\infty}^{\infty} \frac{e^{2x}-e^x}{x (1+e^{2x})(1+e^{x})}\mathrm{d}x = \log 2$$ A solution using both real and complex analysis is welcome. I tried ...
3
votes
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 ...
0
votes
1answer
75 views

Contour integral of analytic function with singularity

I am supposed to integrate $f(z)=$$\frac{5}{z}$ from -3 t0 3 but I am having trouble understanding how to do this. I've done the integration the "hard" way by using parametrizations but now I need to ...
5
votes
4answers
277 views

Integral $\int_0^1\frac{dx}{\sqrt{\log \frac{1}{x}}}=\sqrt \pi$

Hi I am trying to prove this result below $$ \mathcal{J}:=\int_0^1\frac{dx}{\sqrt{\log \frac{1}{x}}}=\sqrt \pi. $$ The result is quite interesting however I realized I am not familiar with working ...
1
vote
1answer
64 views

Zeros of analytic function

If an analytic function has a single zero, can it only be a linear function? This is obviously correct for finite order polynomials yet is it true for all analytic functions?
0
votes
0answers
60 views

Complex analysis winding number

We have that $f:\mathbb S^1 \rightarrow \mathbb S^1 $ and $f(z)=f(1)\widehat{\phi}(z)$ with $\widehat{\phi}(\exp{2\pi it}) = \exp(2 \pi i \phi(t)),$ where $\phi:I \rightarrow \mathbb{R} $ is a ...
0
votes
1answer
31 views

Dual isogenies of complex tori in Birkenhake-Lange

Let $f: X\to Y$ be an isogeny of complex tori, of degree $n$. On page 13 of Complex Abelian Varieties, Birkenhake-Lange show that there is a dual isogeny $g: Y\to X$. Basically, they show ...
2
votes
2answers
65 views

Find all holomorphic functions on $\mathbb{C}$, except for some singularities, such that $|f(z)|\leq C(|z|^{3/2}+|z-1|^{-3/2}), z\in\mathbb{C}-\{1\}$

First I wrote the Laurent series of $f(z)$ around $z=1$: $$ f(z)=\sum_{n=-\infty}^{-1}c_n(z-1)^n+\sum_{n=0}^{\infty}c_n(z-1)^n. $$ Now if $|z|$ becomes very large, the first sum with the negatives ...
4
votes
1answer
298 views

Proving that a Function is Analytic Given that it is Equal to the Complex Conjugate of an analytic Function

So I'm working a problem that states: A function $f$ is analytic in an open set $U$. Define $g$ by $g(z)=\overline{f(\overline{z})}$ (just because the notation can be hard to read, this is the the ...
1
vote
0answers
94 views

Texts for Complex Analysis

I am interested in reading about complex dynamics, Riemann surfaces, and related subjects, but I lack complex analysis as a prerequisite. I want a text that is rigorous and challenging (e.g. not a ...
3
votes
0answers
23 views

Caculating $\log(z^{p(z)})$ along a contour around the origin.

Let $R$ be a rectangle with vertices $1+i, -1+i, -1-i, 1-i$ and $f(z) = \log(z^{p(z)})$ where $p(z)$ is a polynomial. What is the argument change of $f(z)$ around the contour?
0
votes
1answer
59 views

Proving that $\Re(\frac{r}{r-c})\geq 1$ is equivalent to $|c-\frac{r}{2}|\leq \frac{1}{2}$.

For $r, c \in \mathbb{C}$ such that $|r|=1$ and $|c|\leq 1$, how can we show that $\Re(\frac{r}{r-c})\geq 1$ is equivalent to $|c-\frac{r}{2}|\leq \frac{1}{2}$? I was working through the proof of a ...