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)

-3
votes
0answers
26 views

How to show that if a complex function is analytic then it is infinitely many times differentiable, in an intuitive way?

I am going through the theorem which proves that if a complex valued function is analytic than it is infinitely many times differentiable. But I am not sure how to explain this intuitively without ...
2
votes
1answer
19 views

Physical or geometric menaing of complex derivative?

As in real, derivative of a function at a point is a slope of a function at that point. What is the physical or geometric meaning of complex derivative of a function at a point? Any help is ...
1
vote
0answers
15 views

Partial Derivatives [on hold]

What are the following partial derivatives, given: $$\small p=\frac{\exp\left(\frac{\rho-1}{2}\log(x^2+y^2)-\sigma\arctan\frac{y}{x}\right)}{(e^{-x}\cos y-1)^2+(e^{-x}\sin ...
1
vote
0answers
19 views

Is this right way to prove Riemann mapping theorem?

We can solve laplace equation $\Delta u=0$ with Dirichlet boundary condition $u(x,y)=f\in C(\partial D)$ for the unit disk $D$ $\subset$ $R^2$ . If $f$ is a continuous function on the boundary ...
3
votes
1answer
111 views

The number of solutions of $z^5+2z^3-z^2+z=a$ for $a\in \mathbb{R}$

How we can calculate the number of solutions of $$z^5+2z^3-z^2+z=a\;\;,\;\;a\in \mathbb{R}$$ in the half-plane $\mathfrak {Re}(z)\ge 0$. Any hint would be appreciated.
2
votes
2answers
30 views

Value of $z$ so that the series converges

$$ \begin{align} \sum_{n=0}^\infty \frac{1}{n^2}\left(z^n+\frac{1}{z^n}\right) \end{align} $$ Detrmine the value of $z$ so that the series converges I believe that the series converges when ...
1
vote
1answer
33 views

'Identity theorem' for Meromorphic functions

If $f_1,f_2$ are meromorphic functions in $D$ and there exists a sequence of pairwise distinct points $z_n \in D$ such that $z_n \to z_o \in D$ and $f_1(z_n)=f_2(z_n),$ then $f_{1} \equiv f_2$ on $D.$ ...
1
vote
1answer
28 views

The range of $\arccos$

My question is whether or not the function $\arccos$ takes complex numbers to complex numbers? Specifically, if we identify $\mathbb{R}$ with the subset of the complex numbers which have zero ...
5
votes
2answers
124 views

Integral $\int_0^1 \frac{x\log x+1-x}{x \log^2 x}\log(1+x)\, dx=\log\frac{4}{\pi}$

Hi I am trying to prove this $$ I:=\int_{0}^{1} {x\log\left(\,x\,\right) + 1 - x \over x\log^{2}\left(\,x\,\right)}\, \log\left(\,1 + x\,\right)\,{\rm d}x=\log\left(\,4 \over \pi\,\right). $$ Thanks. ...
2
votes
1answer
17 views

Radius of convergence of sum of complex power series

Could anyone advise me on how to find radius of convergence of $\sum^{\infty}_{n=1} [\frac{1}{n^2}+(-2)^n]z^n \ ?$ Thank you. My attempt: radius of convergence of $\sum^{\infty}_{n=1} ...
2
votes
0answers
25 views

Calculating Euler's Numbers

I've derived the finite series with binomial coefficients for Euler's numbers, as requested in John Conway's Functions of One Complex Variable, about p. 76, by deriving the expansion sec(z). But get ...
3
votes
0answers
71 views

Odd form of controlling derivatves

In Muscalu, Schlag - Classical and Multilinear Harmonic Analysis (Cambridge Universitv Press 2013), page 299 there is a rather odd estimate for wich I cannot find any justification: Functions used: ...
1
vote
1answer
20 views

Radius of convergence of powerseries containing $(\log n)^n$

$$ \begin{align} \sum_{n=2}^\infty (\log n)^n(z+1)^{n^2} \end{align} $$ What is the radius of convergence of this power-series? I tried applying the root test and the ratio test , but I couldn't ...
1
vote
0answers
22 views

Using an induction on $q$.

I have a problem: We denote by $[X,\ Y]$ the commutator of $X$ and $Y$ defined by $$[X,\ Y]f(p)=X(Yf)(p)-Y(Xf)(p), \tag{1}$$ for any smooth function $f$ defined on a hypersurface $M$. ...
0
votes
2answers
31 views

Disk of convergence of the series $ \sum\limits_{n=1}^\infty n!\,(z-i)^{n!} $

$$ \sum_{n=1}^\infty n!(z-i)^{n!} $$ Find the disk of convergence of this powerseries. Can I set $n!=k$ and then deal with $\sum_{n=1}^\infty k z^k$ . On another note $\frac{z^{(n+1)!}}{z^{n!}}$ ...
0
votes
1answer
17 views

truncate power series to approximate holomorphic function by polynomial

Fix (open) polydisks $B' \subset B \subset \mathbb{C}^n$ and $\epsilon >0$. If $f$ is holomorphic on $B$, then there exists a polynomial $P$ such that $$\sup_{z \in\ B'}|f(z)-P(z)|<\epsilon.$$ ...
4
votes
0answers
30 views

Finding an analytic function such that real part is the given function.

I am reading the book Complex Analysis by Lars V Ahlfors. In the book he uses a nice method without involving integration to evaluate $f$ given that the real part of the function is $U$. The method ...
0
votes
1answer
38 views

Entire Function Which Tends to Zero At Infinity In All Directions

Say we have an entire function in the complex plane which tends to zero in all directions, i.e. $$f(z)\to 0 $$ as $$|z|\to \infty $$ Intuitively, this seems highly unlikely to me. There are many ...
2
votes
4answers
50 views

Writing the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form

Now I can't finish this problem: Express the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form, where $0 < \alpha < \frac{\pi}{2}$. So the goal is to determine both ...
17
votes
5answers
838 views

A closed form for the sum $\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$

How can I find a closed form for the following sum? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
1
vote
2answers
54 views

When can an infinite sum and complex integral be interchanged?

Are there some conditions under which the following two are equal? $$\displaystyle \oint_C \sum f_n(z)= \sum \oint_C f_n(z)$$ In the case of real valued functions, the condition $f_n(z) \geq 0$ ...
5
votes
1answer
42 views

Geometric interpretation of complex path integral

Let's say that we want to make sense of integrating a function $f: \mathbb{C}\rightarrow\mathbb{C}$ over some path $\gamma$. I can imagine two reasonable ways of doing it. First, there's the way ...
0
votes
0answers
27 views

Complex analysis, cutoff integration

The diff-invariant distance between $z'$ and $z$ is (for short distances) $e^{w(z)}|z'-z|$, so a diff-invaraint cutoff would be at $|z'-z|=\epsilon e^{-w(z)}$. Then $ \int ...
9
votes
1answer
266 views

Integral$\int_1^\infty \log \log \left(x\right)\frac{dx}{1-x+x^2}=\frac{2\pi}{\sqrt 3}\left(\frac{5}{6}\log (2\pi)-\log \Gamma \frac{1}{6}\right)$

UPDATED Hi I am trying to prove the following$$ I:=\int_1^\infty \log \log \left(x\right)\frac{dx}{1-x+x^2}=\frac{2\pi}{\sqrt 3}\left(\frac{5}{6}\log (2\pi)-\log \Gamma \big(\frac{1}{6}\big)\right). ...
6
votes
2answers
141 views

A strange answer for $\int _{-1}^1 \log x\; dx$

I typed $\int _{-1}^1 \log x\; dx$ on Wolfram Alpha. It is giving the answer to be $-2+i\pi$. Can someone please explain what is happening?
0
votes
2answers
35 views

Radius of convergence of powerseries $\sum_{n=1}^\infty \frac{(-1^n)}{n!}z^n$

$$ \begin{align} \sum_{n=1}^\infty \frac{(-1)^n}{n!}z^n \end{align} $$ Find the radius of convergence of this powerseries. To determine the radius of convergence should I split it into two separate ...
14
votes
2answers
867 views

Definite Integral $\int_0^{\pi/2} \frac{\log(\cos x)}{x^2+\log^2(\cos x)}dx = \frac{\pi}{2}\left(1-\frac{1}{\log 2}\right)$

I want to prove that $$\int_0^{\pi/2} \frac{\log(\cos x)}{x^2+\log^2(\cos x)}dx = \frac{\pi}{2}\left(1-\frac{1}{\log 2}\right)$$
0
votes
2answers
37 views

Complex Analysis - Uniform Convergence

Question State The Weierstrass M-test, and use it to prove that if $\rho$ is a positive real number then the series $$\sum_{n=1}^\infty \frac n{e^{nz}}$$ is uniformely convergent on $\{x + iy ...
1
vote
1answer
28 views

Analysis of a Holomorphic function $f$ given $1 \geq |f '(z)|$.

Since $f$ is holomorphic we can use Cauchy's inequality. Thus for $n = 1$ we have $ |f'(z)|\leq \frac{M}{R} $ where is $M$ is the max value of $|f(z)|$ and $R$ is the radius of a random region. We ...
3
votes
3answers
45 views

radius of convergence for $\sum_{n=1}^{\infty} \frac{z^{n} n^{n}}{n!}$ and $\sum_{n=1}^{\infty} z^{n!}$

Exercise 4:10 in John D'Angelo's text is to find the radius of convergence for : A) $\sum_{n=1}^\infty \frac{z^n n^n}{n!}$ and B) $\sum_{n=1}^\infty z^{n!}$ I got half of an answer for A) which I ...
4
votes
0answers
53 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) \end{equation} ...
1
vote
2answers
28 views

rationalize the complex number multiplication rule

For a middle school student without previous knowledge of complex number, how do one introduce the multiplication rules of complex number? i.e., if we have two real number pairs of $(a,b)$ and ...
0
votes
0answers
14 views

Convergence of Series of Complex Numbers with Decreasing Modulo (non-zero imaginary part)

Let $(a_n)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}$, $\theta$ not a multiple of $2\pi$, the series $\sum_{n\geq1} ...
0
votes
0answers
33 views

Zeros of a complex function

Consider the function $$f(x)= \sum_{j=1}^n b_j e^{i a_jx},$$ where $a_j,b_j$ are algebraic numbers. Denote $A=\{f(x)| x\in \mathbb{R}_{\geq 0}\}$, i.e., $A=f([0,\infty))$. Does this hold? $ 0\in ...
1
vote
1answer
75 views

Proving convergence of real and imaginary parts

I am trying to prove that a complex sequence $(z_n)$ converges if and only if $(\Re(z_n))$ and $(\Im(z_n))$ converge. Now $\impliedby$ was straightforward, but I got a bit stuck with $\implies$: ...
0
votes
1answer
24 views

Complex Analysis D shaped contour

Hi there. I am stuck on c. I proved (b) using Rouches theorem. To calculate the integral in c, I was not sure what to do. I am guessing you use the result in (b) somehow, but I thought that ...
0
votes
0answers
17 views

Complex Analysis Dirichlet Problem

I have managed to answer (a) and (b). But so not know how to do the questions thereafter. For (c) could I tried to solve, with $argz=\frac{\pi}{2}$ and $\phi=\pi$, but that did not satisfy the ...
2
votes
1answer
32 views

Is it possible to write the function $f(x) = i \textrm{erf} (ix)$ (with $i$ imaginary unit) in a way that doesn't involve complex numbers?

Studying a physical problem I crashed into this differential equation (condition: $\lim_{x \to 0} = 0$) \begin{equation*} y' + A x y + B x^4 = 0 \end{equation*} where $x,A,B \in \mathbb{R}^+$. With ...
0
votes
0answers
23 views

why the numbers of poles and zeros of meromorphic function on the riemann sphere is finite?

why the numbers of poles and zeros of Meromorphic function on the Riemann sphere is finite? Can I use two statement below to conclude above question? if $f$ be a meromorphic function on ...
2
votes
1answer
327 views

complex main branch of a logarithmic function holomorphic correct

Let $z:= x+iy$ It will now be shown that $$f(z)= f(x+iy) = \frac{1}{2} \log(x^2 + y^2) + i \arctan\left(\frac{y}{x}\right); (z\in \mathbb{C}, x = \operatorname{Re} z \ne 0),$$ where $arctan$ ...
5
votes
3answers
83 views

Is entire function constant when $ |f(z)|\le \log|z|,\ |z|>1$.

Let $ f : \mathbb{C} \to \mathbb{C} ,$ entire and $|f(z)|\le \log|z|,\ |z|>1. $ Show that $f$ is constant. What first comes to mind is Louville's theorem, but log 's problems with analyticity ...
0
votes
0answers
46 views

Zero moment of arc length measure

Suppose $\gamma$ is a simple smooth closed curve and is not a circle. Does there exist a monomial $z^n$ so that $\int_{\gamma}z^n ds(z)=0$ for some positive integer $n$? (In here, $ds$ is the arc ...
4
votes
1answer
123 views

Can the winding number be infinite?

Let $z$ be a point in the complex plane, and $\gamma$ be a closed curve. Is it possible that $$n(\gamma,z) = \frac{1}{2\pi i}\int_\gamma \frac{dw}{w-z}$$ becomes unbounded? In other words, is it ...
2
votes
2answers
58 views

Poles of $\large e^{f(z)}$

$\fbox{1}$ If $z_0$ is a pole of $$f:U \subset \mathbb{C}\longrightarrow \mathbb{C}$$how to prove that $z_0$ can not be a pole of $\large e^{f(z)}$. $\fbox{2}$ If $z_0$ is an essential singularity of ...
0
votes
0answers
22 views

Holomorphic and meromorphic functions on Riemann surfaces

On any domain $\Omega\subset \mathbb{C}$, the set of all holomorphic functions form an integral domain. Its field of quotient is the set of all meromorphic functions on $\Omega$. However this is not ...
1
vote
1answer
22 views

Showing that $\tan(\pi z) = z$ has exactly three solutions in the strip $|\Re(z)| < 1$

We can't use Rouche's theorem here directly, so we have to apply the argument principle. If $f(z) = \tan(\pi z) - z$ , then $f'(z) = \pi \sec^2(\pi z) - 1$. Choose the rectangle $\Gamma$ with ...
0
votes
1answer
22 views

maximum modulus principle question

Suppose that f is analytic on a domain D which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$ then either f is constant or f has a zero inside ...
10
votes
1answer
97 views

Show that the set of one-to-one holomorphic maps $\Bbb{C}\setminus\{a,b,c\} \to \Bbb{C}\setminus\{a,b,c\}$ forms a finite group.

Let $\Omega = \mathbb{C}\setminus\{a, b, c\}$ be the complement of three distinct points in the complex plane. Show that the set of one-to-one holomorphic maps $f : \Omega \to \Omega$ forms a ...
3
votes
0answers
45 views

Check my answer for find a formula for $\sum_{n=0}^{\infty} \frac{z^{n}}{4^{n+2}}$

The next question in John D'Angelo's text is exercise 4.9. I got an answer but wanted to check it because there's no solution manual: Find a formula $$ \sum_{n=0}^ {\infty} \frac{z^{n}}{4^{n+2}}. $$ ...
1
vote
1answer
34 views

Finding an explicit mapping

Here is a question from an old prelim exam in complex analysis that I am stuck on: Let $f: \mathbb{D} \rightarrow \mathbb{D}$ be analytic and satisfy $f(\frac{1}{2})= \frac{1}{2}$ and ...