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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3answers
302 views

Solve $\cos(z)=\frac{3}{4}+\frac{i}{4}$

I tried solving this using the definition of $cos(z)=\frac{e^{iz}+e^{-iz}}{2}$ and equating it to $\frac{3}{4}+\frac{i}{4}$ and converting it to a complex quadratic equation through a substitution ...
1
vote
3answers
118 views

Multiplication of analytic functions

Suppose we have $f$ and $g$ and that both are analytic in neighborhood $D$. Is $fg$ also analytic in $D$? (Afaik yes) This is homework question, and to be honest - I am getting quite a mess with my ...
4
votes
1answer
159 views

Integral $\int_{-1}^{1}\frac{\sqrt{(1-x^{2})}}{1+x^{2}}dx$ [duplicate]

Consider $$\int_{-1}^{1}\frac{\sqrt{(1-x^{2})}}{1+x^{2}}dx$$ I have a problem with this integral; the method I know consists in calculating the complex integral of $$f(z) = \left( \frac{z-1}{z+1} ...
2
votes
0answers
68 views

Proof that $p(z)^2=a^2$ always has a nonreal solution.

Let $p(z)$ be a nonconstant integer polynomial of degree $n$ such that $p(0)=0$ and let $a$ be a nonzero real number. It seems that $$p(z)^2=a^2$$ Always has a nonreal solution (in $z$) if ...
2
votes
1answer
628 views

equality of triangle inequality

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda ...
0
votes
1answer
100 views

check if complex function is differentiable

The question is to check where the following complex function is differentiable. $$w=z \left| z\right|$$ $$w=\sqrt{x^2+y^2} (x+i y)$$ $$u = x\sqrt{x^2+y^2}$$ $$v = y\sqrt{x^2+y^2}$$ Using the ...
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0answers
43 views

A problem with cosine function

I try to understand something from number theory and the author gave this as an excersise: Prove that $z\longmapsto 2\sqrt{p}\cos z$ is a bijection of a set ...
1
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1answer
116 views

Conditions implying that image of $f$ contains the unit disc

I'm stuck with this problem from Stein-Shakarchi: Let $f$ be non-constant and holomorphic in an open set containing the closed unit disc. a) Show that if $|f(z)| = 1$ whenever $|z| = 1$, then the ...
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2answers
46 views

Determing if a function is analytic on a set

Let $f$ be analytic on a set $U$. Let $g(z)=\overline{f(\bar{z})}$. Is $g(z)$ analytic on $V=\{z\, : \bar{z} \in U\}$. Is it enough to show this: $$g'(z)=\lim\limits_{z \to ...
3
votes
1answer
206 views

Is Morera's theorem the inverse theorem of Goursat's theorem?

While I'm reading Complex Analysis by Elias M.Stein, I found that there must be some relations between Goursat's theorem and Morera's theorem. According to Stein, the 2 theorems are as following: ...
0
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1answer
62 views

Problem related with cross ratio in complex analysis

I am stuck on the following problem that says : The value of the cross-ratio $(7+i,1,0,\infty)$ is which of the following? The options are: $6+i$ $-6+i$ $6-i$ $-6-i$ ...
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5answers
2k views

Finding real and imaginary part of exponential function

Can someone explain to me how I find the real and the imaginary part of $e^{\theta i}$? I'm learning complex numbers but I don't quite understand how $e$ is intertwined in all this.
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1answer
71 views

Complex root won't work

So I'm trying to get this: http://www.wolframalpha.com/input/?i=%288*sqrt%283%29%29%2F%28z%5E4%2B8%29%3Di And I've calculated $z^4=16 \left( \cos (\frac{- \pi}{3})+ \sin ( \frac{- \pi}{3}) \right)$ ...
-1
votes
2answers
43 views

Complex roots problem [duplicate]

I've got a complex equation with 4 roots that I am solving. In my calculations it seems like I am going through hell and back to find these roots (and I'm not even sure I am doing it right) but if I ...
0
votes
1answer
161 views

Zeros of $f_{\epsilon}(z) = f(z) + \epsilon g(z)$ with $f$ and $g$ holomorphic

I'm stuck with this problem from Stein-Shakarchi: Suppose $f$ and $g$ are holomorphic in a region containing the disc $|z| \leq 1 $. Suppose that $f$ has a simple zero at $z = 0$ and vanishes nowhere ...
0
votes
2answers
61 views

Why are there several roots of complex equations

I'm trying to understand why there are $n$ amount of roots in an equation of the form $z^n=$ complex equations. I understand why there are several answers to a $ \sin(x)=$ equation but I can't wrap ...
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1answer
155 views

Show that f is a polynomial of degree $\le n$

From Basic Complex Analysis 3rd ed. 1.5 #20 Let f be an analytic function on an open connected set A and suppose that $f^{n+1} (z)$ (the n+1st derivative) exists and is zero on A. Show that f is a ...
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1answer
52 views

Fourier Transform Identity

I'm trying to verify the following: $$ \int_{\mathbb{R}} e^{-z\xi^2} \hat{f}(\xi) \; d\xi = \sqrt{\frac{\pi}{z}} \int_{\mathbb{R}} e^{-\pi^2x^2/z} f(x) dx, $$ for $z = \alpha i$ purely imaginary and ...
0
votes
2answers
163 views

What is “growth lemma” in complex analysis?

When I'm reading Complex Analysis by Elias M.Stein, I met a question on the proof of the fundamental theorem of algebra. Stein says, Since each term in the parentheses goes to $0$ as $\vert z\vert ...
3
votes
1answer
54 views

To what extent can one imitate holomorphic functions in higher dimensions?

Let $\Omega\subseteq \mathbb{C}$ be open. Let $f:\Omega \rightarrow \mathbb{C}$ be holomorphic. The last statement about $f$ can be restated as: For every $z_0\in \Omega$, there exists $w\in ...
4
votes
1answer
60 views

Quarternionic Analysis

What is/are the current understanding/opinions about Quarternionic Analysis as a generalization of Complex Analysis with respect to a "Quarternionic Residue Calculus" (if such a thing exists)? i.e. ...
2
votes
3answers
69 views

Product of distances from a point in $\mathbb{C}$, Stein-Shakarchi

I 'm trying to do this exercise from Stein - Shakarchi. Let $w_{1} \ldots w_{n}$ be points on the unit circle in $\mathbb{C}$. Prove that there exist a point $z$ on the unit circle such that the ...
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2answers
483 views

What is the conformal equivalence from the half-plane to the unit disc?

Problem: Find a conformal equivalence from the half-plane {z : Re(z) > 1} to the unit disc D. My Attempt: Just a FYI that I am completely new to conformal mapping. Okay, so I know that $$h(z) = ...
3
votes
2answers
204 views

pole, order and residue

I was practising for an exam and I had some trouble with the following excersice: $$f(z)= \frac{1}{z \sin z}$$ a. Find the pole and its order. $$\frac{1}{z(z-z^3/3!+ z^5/5! + \cdots)}= ...
18
votes
1answer
550 views

Integrating $\int_0^\infty\frac{\log (1+z^2)}{e^z-1}dz$ using residue calculus.

I've been looking at how to integrate the following definite integral using the residue calculus, but can't seem to get my thoughts together. I know the $\log$ term is a multivalued function and the ...
2
votes
2answers
177 views

Example of Singularities in Complex Analysis

Let $\displaystyle f(z)=\frac{z-1}{\exp(\frac{2\pi i}{z})-1}$ then, $(1)\ \ f$ has an isolated singularity at $z=0$. $(2)\ \ f$ has a removable singularity at $z=1$. $(3)\ \ f$ has infinitely many ...
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vote
2answers
70 views

Classifying singularity

Having trouble classifying a singularity... $f(z)=$$z^2-1\over z^6+2z^5+z^4$ with $z_0=0$ and $z_0=-1$ The $z_0=0$ is pretty simple, just need to put $z^4$ in evidence. But $z_0=-1$ I can't seem to ...
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0answers
223 views

How to prove that f(z) is constant over an open connected set?

From Basic Complex Analysis 3rd edition 1.5 #16 a) Let $f(z)=u(x,y)+iv(x,y)$ be an analytic function defined on a connected open set A. If $au(x,y)+bv(x,y)=c$ in A, where a,b,c are real constants ...
2
votes
1answer
69 views

$\sum_{1}^{\infty}a_{n}$ converges $\Rightarrow \sum_{1}^{\infty}\mathrm{Log}(1+a_{n})$ converges, $a_{n} \in \mathbb{C}$

Let $a_{n} \in \mathbb{C}$ ; is it true that convergence of $\sum_{1}^{\infty}a_{n}$ implies convergence of $\sum_{1}^{\infty}\mathrm{Log}(1+a_{n})$ ? Why ? Here $\mathrm{Log}$ is the the principal ...
0
votes
2answers
78 views

Complex analysis

Let $f$ be a non-constant entire function.Which of the following properties is possible for each $z \in \mathbb{C}$ $(1) \ \ \mathrm{Re} f(z) =\mathrm{Im} f(z)$ $(2) \ \ |f(z)|<1$ $(3)\ \ ...
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0answers
42 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
2
votes
1answer
42 views

$g(z) = \int_{0}^{2\pi}f(e^{i\theta})\frac{e^{-i\theta}}{e^{-i\theta}-\bar{z}}d\theta$ is antiholomorphic

I 've encountered this fact: if $z \in D(0,1) $ and $f$ is continous on $\partial D(0,1) $ then $$g(z) = \int_{0}^{2\pi}f(e^{i\theta})\frac{e^{-i\theta}}{e^{-i\theta}-\bar{z}}d\theta$$ is ...
2
votes
4answers
140 views

Taylor Series Expansion of $\frac{1}{1+x^2}$ about $x=a$

Let $$f(x)=\frac{1}{1+x^2}$$ Consider its Taylor series expansion about a point $a\in \mathbb{R}$. What is the radius of convergence of this series?? About $x=0$ we could expand it like ...
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vote
2answers
75 views

A subset of $\mathbb C\times\mathbb C$

I'm trying to think if the space $\{(z,\,i\overline{z})\,:\,z\in\mathbb{C}\}$, where $\overline{z}$ is the complex conjugate of $z$ and $i$ is the imaginary number, is topologically equivalent to ...
2
votes
1answer
198 views

Differentiability vs Analyticity

What makes the crucial difference between the reals and the complex numbers is that the complex numbers are algebraically closed. So while going through all the proofs that "being holomorphic implies ...
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vote
0answers
178 views

Contour integral (inverse Laplace transform) with arctan

I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help. The integral itself is, with τ, λ, and k all real and ...
2
votes
1answer
60 views

How come complex numbers represent coordinates?

I'm wondering why complex numbers represent coordinates without being on the form of a tuple (a,b). The complex numbers come in the form: $a+bi$ where $a$ denotes the real part and $bi$ denotes the ...
0
votes
1answer
99 views

$\lim_{n\rightarrow +\infty} \frac{a_{n}}{a_{n+1}} = z_{0}$ with $z_{0}$ pole [duplicate]

This is an exercise from Stein-Shakarchi. Suppose that $f$ is holomorphic in an open set containing the closed unit disc, except for a pole at $z_{0}$ on the unit circle. Show that if $f(z) = ...
0
votes
1answer
75 views

Determining radius of convergence $f(z)=\frac{\mathrm{e}^z}{z-1}$?

Can somebody help me with determining the radius of convergence of the power series of the following function $$f(z)=\frac{\mathrm{e}^z}{z-1}$$ about $z=0$?
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votes
1answer
184 views

Riemann extension theorem with $Re(g)$ bounded

Let $D$ be an open subset of the complex plane, a point $a$ of $D$ and $g$ a holomorphic function defined on the set $D$ \ ${a}$,if $Re(g)$ is bounded from above,how to show that $g$ can extends to ...
2
votes
2answers
174 views

Extending a holomorphic function defined on a disc

Suppose $f$ is a non-vanishing continous function on $\overline{D(0,1)} $ and holomorphic on ${D(0,1)} $ such that $$|f(z) | = 1$$ whenever $$|z | = 1$$ Then I have to prove that f is constant. We ...
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vote
1answer
100 views

Maclaurin Series Complex Numbers

I'm having trouble getting to the right solution on the function ${z^2\over (1+z)^2}$ ${z^2\over (1+z)^2}$ = ${z^2}$${1\over (1+z)^2}$ = ${z^2}$${1\over (1+z)(1+z)}$ = ${z^2}$${A \over (1+z)}$ + ...
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1answer
44 views

Define a branch of $\frac{z^\alpha}{z^2+1}$

Define a branch of $\frac{z^\alpha}{z^2+1}$. $\alpha$ is considered real and in the interval $(-1,1)$ Sketch the branch cut and the poles in the complex plane. I have that the poles are $z=i, ...
0
votes
1answer
84 views

Problem in harmonic analysis

suppose $p$ be a fixed psitive real number and $f$ is an entire function with $$\lvert f(0) \rvert^p=\int_\mathbb{C}\ \lvert f(z)\exp(-\alpha\lvert z \rvert ^2) \rvert^p dA(z) $$ where $\alpha ...
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vote
2answers
91 views

How to solve $(1+i\sqrt{3})^{-1+i}$??

Good morning, I want to solve this... but I lose my way. I hope somebody help me... I show you my calculus $(1+i\sqrt{3})^{-1+i}=e^{(-1+i)\log(-1+i)}$ $(1+i\sqrt{3})^{-1+i}=e^{(-1+i)(\log ...
0
votes
2answers
63 views

On finding the zeros of a polynomial

What is the zero (real) of the polynomial $$x^{k+1}-2x^{k}+1=0$$ If there is such, how can I find it or what method can I use?
1
vote
4answers
162 views

Multiplicity of zeros

Can you explain me how to get the multiplicity of a zero? In particular, I would ask you how to determine the zeros' multiplicity of $$\cos(\frac{\pi}{2}z)$$ I suppose they are $z = 2k+1, k \in ...
0
votes
4answers
1k views

Find all four roots of the equation $z^4+1 = 0$ and use them to deduce the factorization $z^4+1= (z^2-\sqrt2z+1)(z^2+\sqrt2z+1)$

Find all four roots of the equation $z^4+1 = 0$ and use them to deduce the factorization $z^4+1= (z^2-\sqrt2z+1)(z^2+\sqrt2z+1)$ I got $\displaystyle z=(-1)^{\frac{1}{4}} = ...
5
votes
1answer
127 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
0
votes
1answer
57 views

Series $\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}} \ $, $ z \in \mathbb{C}$

I'm studying the series $$\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}}$$ If $z = x+iy$, what is the behaviour of the series for $-1<x<0 \ $?