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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59 views

Existence of an analytic function with nonvanishing derivative mapping the punctured unit disk to the unit disk

Here is an exercise on the page 164 of Conway's book(in the section of Riemann mapping theorem): Show that there exists an analytic function $f$ defined on $G=ann(0;0,1)$ such that $f'$ never ...
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15 views

Univariate residue in multivariate contour integration?

Consider a multivariate contour integral on the complex polydisc: $$\oint \frac{dz_1dz_2...dz_n}{f_1(z_1,z_2,...,z_n)f_2(z_1,z_2,...,z_n)...f_n(z_1,z_2,...,z_n)}F(z_1,z_2,...,z_n)$$ The $f_i$ in ...
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1answer
29 views

Show $\prod_{n=1}^\infty 1 + \frac{-\left( 1 + z \right)}{n^2 + \left( 1 - n^2 \right) z}$ has no analytic extension past the unit disk

I'm studying for a qualifying exam (tomorrow) and I was hoping someone could show me how to finish solving this problem. Let \begin{align} a_n = 1 - \frac{1}{n^2}, && f(z) = ...
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0answers
62 views

Finding an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$

I encountered the following problem in the lecture note in my complex analysis class: Problem. Find an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$ for $n = 1, 2, \cdots$. ...
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1answer
25 views

Cauchy integral formula12

$\int 1/(z^2 + 2i)dz$ I've tried writting it as $1/(z-1+i)*(z+1-i)$ but then it's impossible to find solution. Any help would be great, thanks in advance.
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0answers
20 views

Artin approximation vs implicit function theorem in the class of analytic functions

I am not an algebraist so my question might be stupid. I am doing mainly complex analysis and recently I was informed about the existence of Artin's theorem and it sounded like it could be of interest ...
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0answers
16 views

sketch the set of points in a complex plane

I have two questions in here and absoultely no idea how to approach them. $$0<arg(z-1-i)<\frac{\pi}{3}$$ and $$log|z|=-2arg(z)$$ My approach: In first case since we want the argument to be ...
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1answer
15 views

What to do when there is only one valid value to be used in the Cauchy-Riemann equations

I just did 2 problems where the $u$ part of the C-R equation was $0$. I'll give one as an example. I'm confused as to what conclusions I can correctly arrive at. $$f(z)=Im(z)$$ So I can say that ...
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0answers
14 views

A clarification on an answer on residues and Polya fields

In this very informative and interesting answer about the relation between residues and representation of complex functions as vector fields the author states that the function $$f(z) = \frac{1}{z}$$ ...
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2answers
42 views

Does the inverse of $f(x)=x^3$ have a non-negative domain to have a real output?

I'm not familiar with complex analysis. While playing with Mathematica (a mathematics software), I found that it keeps spitting out unexpected results, and the reason was that it considers differently ...
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1answer
19 views

Where are the following functions differentiable? Where are they holomorphic? Determine their derivatives at points where they are differentiable.

$$ f(z) = e^{−x}e^{−iy}$$ I used the Cauchy Riemann equations to determine that $x=iy-\ln(i)$, but I'm not sure what I'm supposed to conclude. Could I say that the function is differentiable wherever ...
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2answers
38 views

Is it a removable singularity?

In the function: $$ f(z)=2iz\frac{(1-z^{2})^{\frac{1}{2}}}{1-2z^{2}} \qquad \qquad (z \in \mathbb{Z}) \,\, , $$ There is a singularity at the point $z=\pm \sqrt{1/2}$. Is that a removable ...
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2answers
35 views

zeros of $p(z)=z^4+2$

I want to find all zeros of $p(z)=z^4+2$ and I'm not sure if I've done everything correctly. Can you correct this if something is wrong? $$x^4+2=0 \iff x^4=-2=2\cdot(-1)$$ $$\Rightarrow x_k= ...
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0answers
23 views

Let $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ then is $f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$?

I am a little stuck here, suppose we have some function $$f(z)=f(x+iy)=u(x,y)+iv(x,y)$$ then is $$f'(z)=\frac{\partial u}{\partial x}(x,y)+i\frac{\partial v}{\partial x}(x,y)$$ assuming $f$ is ...
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71 views

Fermat like equation for meromorphic functions.

I found this question in Conway, and really have no idea how to answer it. Can anyone provide any hints? For each integer $n\geq 1$ determine all meromorphic functions on $\mathbb{C}$ $f$ and $g$ ...
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21 views

Trouble understanding singularities and their classification

I know that singularities can be classified into three categories: Pole, removable, and essential. What I don't get is trying to work out what each singularity belongs to. For example, let $f(z) = ...
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1answer
33 views

Bounded non-constant holomorphic function on the complex plane minus the negative real axis

I was recently given this question which I wasn't sure how to answer: Define \begin{align*} \mathbb{R}_- = \left\{x \in \mathbb R \mid x < 0 \right\} \end{align*} and let $D = \mathbb C ...
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6answers
55 views

Complex conjugate of $z$

I am just starting to learn complex analysis, and it appears that you can have a function $f = u+iv$, and thus you can have $$f(z) = u(x,y)+iv(x,y).$$ Am I incorrect in believing this? If this is ...
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1answer
24 views

Find maximal $R$ for which polynomial $f(z)=az^2+bz+c$ is one to one for $|z|<R$

Find maximal $R$ for which polynomial $f(z)=az^2+bz+c$ is one to one for $|z|<R$. I have arrived at an expression and need help with final assumptions or formal ones. If $a=0$, then $R=\infty$. ...
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2answers
43 views

Complex Analysis: Show that $\int_{\gamma}\frac{1}{(z-a)(z-b)}dz=\frac{2\pi i}{a-b}$ [closed]

How can I show that if $|a|<r<|b|$, then $\int_{\gamma}\frac{1}{(z-a)(z-b)}dz=\frac{2\pi i}{a-b}$, where $\gamma$ is the circle with center the origin, radius $r$, and positive orientation? ...
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1answer
66 views

Definite integral of $\int_{-\infty}^{\infty} \frac{e^{ikx}}{x^2 + a^2} \, dx$

I'm doing this integral that showed up in a book in quantum physics and I just want to check if what I did was correct. So this is the integral, which looks like it's a Fourier transform of ...
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1answer
25 views

Extension of Fourier transform to complex analytic function

Let $f(x) \in L^1(\Bbb{R})$ have compact support, say $\operatorname{supp}(f) = [-R,R]$. We have the Fourier transform $$\hat{f}(\xi) := \int_{\Bbb{R}} e^{-ix\xi} f(x) dx = \int_{-R}^R e^{-ix \xi} ...
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1answer
23 views

$z = 1/w$ transformation for parallel lines $y = x + b$

I am supposed to find the image of the family of parallel lines $ y = x + b $ under the transformation $w = \frac 1 z $. Attempt: Replace $x$ and $y$ with $\Re(z)$ and $\Im(z)$, respectively. ...
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1answer
39 views

Writing N-th roots of unity

I have a question regarding roots of unity. In general, we can write the n-th roots of unity as $$e^{2\cdot\pi\cdot i\cdot\frac{k}{n}}$$. However, if we do the following manipulation we get the ...
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1answer
21 views

what are the coordinate vectors and dimension when working with complex numbers?

I am not sure I'm correct here: Let $W$ be a subspace of $\;{\mathbb{C}_{[\mathbb{R}]}}$ If I have for example the vector $$c_1=(2 + 3i)$$ the standart basis is $$E_1=\{1,i\}$$ and ...
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1answer
42 views

In the Set of Extended Complex Numbers, 3/0 = infinity?

There's something I don't quite understand about the extended set of complex numbers. Usually, a number $\frac a 0 , a \in R$ is undefined. However, in the set of extended complex numbers, $\frac a ...
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1answer
25 views

Find the limit or explain why it doesn't exist

I'm having trouble starting this problem. I don't understand how I can make the limit apply when it's written with $x$ and $y$. $$\lim_{z \to (1-i)} (x+i(2x+y))$$ If I change $z$ to $(x+iy) \to ...
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24 views

primitive of a complex function

If it exists, find the primitive of $z^2+\frac{1}{z^2}$. Here: I start my solution as follows. Since $z^2+\frac{1}{z^2}$ is holomorphic everywhere except at zero, hence its primitive is ...
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0answers
17 views

Quasi-periodic Meromorphic function

Is there an example of a complex function $f(z)$ that satisfies the following 2 properties? Quasi-periodicity: $f(z+1) = f(z)$ and $f(z+ \tau) = f(z) e^{i \phi}$, where $\phi$ is a real number and ...
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0answers
25 views

Find the harmonic conjugate of the following

I just want to make sure my reasoning is correct. I followed another similar question from this site. Also, is there a method I can use after coming to the conclusion below to check to make sure my ...
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2answers
33 views

Prove that $f(z)=f(re^{i \theta})= \sqrt{r}e^{i \frac{\theta}{2}}$ is discontinuous

I would like to prove thatthe multiform function $f(z)=f(re^{i \theta})= \sqrt{r}e^{i \frac{\theta}{2}}$ is not continuous for $z \in (- \infty, 0)$ and $\theta \in (-\pi, \pi]$. I think I could use ...
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2answers
33 views

Find the Primitive of Complex function [closed]

if it exists, find the primitive of $\mathbb{e}^{z^2} $ on $\mathbb{C}$ How can I start the proof?
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2answers
57 views

Prove Complex analysis inequality [closed]

Prove $ | \mathbb{e^z}-1| \le |z|$ if $Re(z) < 0$. Any hints on how to start?
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1answer
20 views

Prove that for any piecewise smooth curve it is possible to find the parametrisation

Prove that for any piecewise smooth curve it is possible to find the parametrisation $\phi$ that is consistent with its length, ie. length of a curve segment between $\phi(a)$ and $\phi(b)$ is equal ...
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0answers
27 views

Is the composition of an harmonic function with an analytic function an harmonic function in any dimension?

I was wondering if it is true or not that, given a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ and $g:\Omega\subset\mathbb{R}^n\rightarrow \mathbb{R}^n$ such that $f$ is harmonic and $g$ real ...
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2answers
25 views

General way to express holomorphic function in terms of z?

For the holomorphic, complex-valued function f, defined as $f(x + iy) = xy - x + y + i(-(1/2)x^2 + (1/2)y^2 - x - y + c)$ We can express this in terms of $z$ and $\bar z$ by substituting $x = ...
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0answers
22 views

Let $f(z) = e^{z^2}$. Find $\theta \in (-\pi,\pi]$ such that $\lim_{r \to \infty} f(re^{i\theta})=0$

Let $f(z)=\exp(z^2)$, with $z=re^{i\theta}$. Find $\theta \in (-\pi,\pi]$ such that $\lim_{r \to \infty} f(re^{i\theta})=0$. With the identity $e^z=e^x(\cos(y)+i \sin(y))$, I found that ...
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2answers
41 views

If a holomorphic map $f$ has constant real part on some ball $B \subseteq \Omega$, then $f$ is constant on $\Omega$.

I would like to know if my reasoning is correct. I tried to prove the following : Let $\Omega \subseteq \Bbb C$ a connected open set, $f : \Omega \to \mathbb C$ a holomorphic function, such that ...
3
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1answer
29 views

Show that $d(E,F) > 0$ - Is $E \times F$ is a compact set here?

Let $d(E,F)=\inf\{|z-w| : z \in E, w \in F\}$ where $E \subset \mathbb{C}$ is compact and $F \subset \mathbb{C}$ is closed such that $E \cap F = \emptyset$. Show that $d(E,F) > 0$; namely, ...
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0answers
34 views

Evaluate $\int_{0}^{+\infty}\frac{\sin x}{x}dx$ [duplicate]

$$\int_{0}^{+\infty}\frac{\sin x}{x}dx$$ My start: Zero is a singular point, Let's define $g(z):=(e^{iz})\big/z$ $$\int_{\Gamma}\frac{e^{iz}}{z}dz=\\ ...
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0answers
24 views

Uniform Convergence for Digamma function sum representation

I am dealing with the following summation: $$ \frac{1}{\psi_{(1)}(1)} \sum_{n=0}^{\infty} \frac{1}{(1+n-x)(1+n-y)} = \frac{-1}{\psi_{(1)}(1)}\frac{1}{x-y} \left( \psi_{(0)}(1-x) - \psi_{(0)}(1-y) ...
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0answers
30 views

If $z=2(\cos \theta + i \sin \theta)$, use the triangle inequality to find an upper estimate for $|e^{z^2} + 4\sin(z)|$

If $z=2(\cos \theta + i \sin \theta)$ ($0 \leq \theta \leq 2\pi$), use the triangle inequality to find an upper estimate for $|e^{z^2} + 4\sin(z)|$. Okay so if I write $z$ in the form $x + iy$ I ...
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2answers
48 views

Calculate Ln$(i^i)$

Calculate Ln$(i^i)$ My attempt: Ln$(z)$=$\ln|z|+i\arg z$ $$z=0+i^i=0+i\cdot i$$ $$|z|=\sqrt{0^2+i^2}=i\\ \arg z=\arctan(i/0)$$ $1.$ how it can be that the modulus equal to $i$? $2.$ how ...
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1answer
44 views

Find $\lim_\limits{R\to \infty}{1\over 2\pi i}\int_{{1\over 2}-iR}^{{1\over 2}+iR}{x^s\over s}ds$

$\lim_\limits{R\to \infty}{1\over 2\pi i}\int_{{1\over 2}-iR}^{{1\over 2}+iR}{x^s\over s}ds$ where $x>0$. Split it to cases: $x>1,x=1,0<x<1$. I tried using contour integration but I am ...
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105 views

Mapping in the complex plane

I have the following two circles in the complex plane, $z = x + iy$, which bound a region, $R$. The equations for the circles and a sketch of the region is given as follows: $$ x^2 + (y-1)^2 = 1\\ x^2 ...
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0answers
63 views

Assigning values to a divergent integral?

Question If I can assign the series of the zeta function to: $$ \zeta(-1) \to 1+2+3+\dots$$ why can't we assign the integral $$ \int_{0}^{\infty} x dx \to 0$$ and it still have some physical ...
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2answers
59 views

Laurent series expansion of $f$

Find the Laurent series expansion of $f(z)=\dfrac{1}{2z^2-13z+15}$ about the annulus $\dfrac{3}{2}<|z|<5$. I did like this : $f(z)=\dfrac{2}{7}(\dfrac{3}{3-2z}-\dfrac{1}{z-5})$ Then I took ...
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2answers
17 views

Manipulating Complex Exponentials

I am trying to show that $$ sin(5\pi t) = \frac 12 e^{-j \frac {\pi}2}e^{j5\pi t} - \frac 12 e^{-j \frac {\pi}2}e^{-j5\pi t} $$ I am aware that $$ sin(\theta) = \frac {e^{j\theta} - ...
2
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0answers
40 views

Manifolds or Complex Analysis for Algebraic Geometry? [closed]

I'm an undergraduate and I have one year left to take some courses at the graduate level to prepare myself for graduate school. I go to a quarter school (U. Washington) so I only have time to take two ...
1
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4answers
133 views

Complex number ( prove ) [closed]

Let $$ {x-yi\over{x+yi}}=a+bi\;\;. $$ Prove $a^2+b^2=1$ I don't know how to start prove it, can anyone help me?