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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1answer
24 views

What is the proper way to determine the order of a root?

Find the multiplicity of the root at $z_0$ for these functions i) $$ \begin{align} f(z)= e^{zcos(z)-z}-1, z_0=0 \end{align} $$ Let $z_0$ be a root of a holomorphic function $f$ , and let n be the ...
3
votes
3answers
75 views

Show that $\displaystyle \int_\gamma \frac{f'(z)}{f(z)}=0$ for every closed curve $\gamma$ in $\Omega$

I have just started taking complex analysis course,The following problem is given in my class.Please help me solving it.Thnx in advance. Suppose $f(z)$ is anlytic and satis fies the relation ...
0
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1answer
28 views

Existence of a holomorphic map

Does there exist a holomorphic map $f:\mathbb C^2 \to \mathbb C^2$ whose rank at (0,0) is 0 but at all other points is 2? ($f$ could also be defined on a domain in $\mathbb C^2$)
4
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1answer
75 views

Can these holomorhic functions $f:D(0,1)\to \mathbb{C}$ exist

i) $$ \begin{align} \text{Let }f:D(0,1)\to \mathbb{C} \text{ holomorphic ,$\\$ Show that } f(\frac{1}{n})\ne \frac{1}{n+1} \end{align} $$ for all natural numbers,except maybe for some finite cases. ...
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1answer
46 views

Can there be an entire function such that $f(\frac{1}{2n})=\frac{1}{2n},$ and $f(\frac{1}{3})=\frac{1}{9}$

$$ \begin{align} f(\frac{1}{2n})=\frac{1}{2n}, \text{ and } f(\frac{1}{3})=\frac{1}{9} \end{align} $$ Prove that there can't be an entire function $f$ such that the upper conditions are fulfilled. ...
0
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2answers
46 views

Understanding proof for $\sum_{k=1}^{n-1}{\sin{\frac{2\pi k}{n}}} = 0$

(English is not my native language, so I apologize if I fail to use the right technical terms) I am stuck in proving the following. I'll explain how far I got and maybe someone can help me out by ...
1
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1answer
33 views

What is the generalization for a convolution in $\mathbb C$?

Since the integration range of "the" convolution is $\mathbb R$, what is a sensible generalization in complex numbers? Would one still integrate over $\mathbb R$, or some other path, or over the ...
4
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4answers
88 views

If $f,g$ are entire functions such that $f(g(z))=0, \forall z, $ then $g$ is constant or $f(z) =0, \forall z \ ?$

Let $f,g$ be entire functions such that $f(g(z))=0, \forall z.$ Could anyone advise me on how to prove/disprove: either $g(z)$ is constant or $f(z) =0, \forall z \ ?$ Hints will suffice, thank you.
2
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1answer
38 views

Can use residue theorem for this integral

I need to compute $$I=\int_C \dfrac{e^{\sqrt{1+u}}\cdot\sqrt[4]{1+u}}{\sqrt{u}} \,\mathrm {d}u$$ where $C$ is the unit circle. I am confused about whether I can use the residue theorem to compute it? ...
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1answer
17 views

Can any meromorphic function be represented as a product of zeroes and poles?

Given any meromorphic function, can it be represented as $$c\prod_i (z-z_i)^{n_i} $$ where $ n_i\in\mathbb Z$ and $n_i> 0$ denotes the multiplicity of the zero $ z_i $ and $ n_i <0$ for the ...
2
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2answers
35 views

Show that $f(x+iy)$ is holomorphic if and only if it can be expressed as a polynomial in the single variable $z$

I am very much new in complex analysis.The following question is given in class. Please help me to solve it. I have tried something myself which is also mentioned. Please help me. Thnx in advance. ...
9
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1answer
137 views

$f: \Omega \rightarrow \Omega$ holomorphic, $f(0) = 0$, $f'(0) = 1$ implies $f(z) = z$

Let $\Omega$ be a bounded connected open subset of $\mathbb{C}$ containing $0$. Let $f: \Omega \rightarrow \Omega$ be holomorphic and $f(0) = 0$, $f'(0) = 1$. The problem I am working on is to show ...
3
votes
2answers
81 views

Radius of Convergence of $ \sum\limits_{n=2}^{\infty} \pi(n) z^{n}$

I came across this question: If $\pi(n)$ denote the Euler function. What's the radius of convergence of this power series? $$ \sum\limits_{n=2}^{\infty} \pi(n) z^{n}$$ Any hint would be ...
0
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1answer
49 views

Absolute convergence of series $\sum_{n=1}^\infty \frac{1}{\frac{1}{z^n}-1}$

$$ \begin{align} \sum_{n=1}^\infty \frac{1}{\frac{1}{z^n}-1} \end{align} $$ Determine the values of $z,z\in\mathbb{C}$ so that the series converges absolutely I know that the series converges for ...
4
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1answer
53 views

Polydisc is not biholomorphic to any strictly pseudoconvex domain

I want to prove the poly disc $P=\left\{z\in \mathbb{C}^2 : |z_1|<1,|z_2|<1\right\}$ is not biholomorphic to any strictly pseudo convex domain in $\mathbb{C}^2.$ Can any one provide a hint?
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0answers
41 views

How to show that if a complex function is analytic then it is infinitely many times differentiable geometrically? [duplicate]

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 geometrically without ...
5
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2answers
59 views

Physical or geometric meaning of complex derivative?

As in, the 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 ...
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0answers
28 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 ...
2
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2answers
39 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
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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 ...
1
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1answer
46 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 $D.$ ...
2
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1answer
29 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
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1answer
35 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 ...
1
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1answer
23 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 ...
0
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2answers
39 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
20 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 $$\sup_{z \in\ B'}|f(z)-P(z)|<\epsilon.$$ ...
0
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1answer
40 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
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4answers
52 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 ...
6
votes
2answers
55 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
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2answers
37 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 ...
1
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2answers
62 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
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1answer
102 views

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

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) ...
3
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3answers
46 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 ...
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0answers
15 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} ...
1
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2answers
33 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
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1answer
25 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 ...
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0answers
18 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
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1answer
40 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 ...
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0answers
25 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 ...
0
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2answers
38 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 ...
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1answer
30 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 ...
4
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0answers
36 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 ...
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0answers
29 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 ...
3
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1answer
115 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.
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0answers
50 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
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1answer
125 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 ...
0
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0answers
24 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 ...
2
votes
2answers
63 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 ...
2
votes
1answer
23 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 ...
3
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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 $$ \sum_{n=0}^ {\infty} \frac{z^{n}}{4^{n+2}}. $$ ...