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
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 ...
1
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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 ...
1
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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
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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 $z=0$?
0
votes
1answer
185 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 ...
1
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)}$ + ...
1
vote
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 ...
1
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
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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 \ $?
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0answers
104 views

Evaluate the contour integral $\int_{\gamma(0,1)}\frac{e^z+e^{-z}}{z^n}dz \hspace{10mm} n=1,2,3,\cdots .$

Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t \leq 2\pi$. Evaluate $$\int_{\gamma(0,1)}\frac{e^z+e^{-z}}{z^n}dz \hspace{10mm} n=1,2,3,\cdots .$$ Using Cauchy's formula: ...
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2answers
400 views

Evaluate the contour integral $\int_{\gamma(0,1)}\frac{\sin(z)}{z^4}dz.$

Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t \leq 2\pi$. Evaluate $$\int_{\gamma(0,1)}\frac{\sin(z)}{z^4}dz.$$ I know that \begin{equation} ...
1
vote
0answers
430 views

The radius of convergence of the hypergeometric function $_2F_1$ (Ahlfors)

In Ahlfors' complex analysis text, page 317 he discusses the radius of convergence of the hypergeometric function $_2F_1(a,b;c;z)$ (which he denotes by $F(a,b,c,z)$): The radius of convergence of ...
3
votes
1answer
135 views

integral to infinity + imaginary constant

A proof I'm reading tries to evaluate the integral (where $i$ is the regular imaginary unit) $$\int_{-\infty}^{\infty} e^{-(x-\alpha i)^2}\mathrm{d}x$$ by doing a substitution $u=x-\alpha i$. ...
0
votes
2answers
87 views

$\int_{0}^{\infty}\frac{\cos2\pi x}{x^4+x^2+1}dx=-\frac{\pi}{2\sqrt{3}}\mathrm{e}^{-\pi\sqrt{3}}$

Can somebody help me out with the following integral? Prove that: $\int_{0}^{\infty}\frac{cos2\pi x}{x^4+x^2+1}dx=\frac{-\pi}{2\sqrt{3}}e^{-\pi\sqrt{3}}$ I have already determined the ...
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4answers
79 views

generalization of a fact about 2D harmonic functions to 3D harmonic functions

Let $U_2,U_3$ be the open unit balls in $\mathbb{R}^2,\mathbb{R}^3$ respectively. Fact 1: (From Rudin's real complex analysis) Let $u:\partial U_2\rightarrow \mathbb{R}$ be continuous, then there ...
3
votes
1answer
53 views

Help with complex integration problem: $\left| \int_{[\sqrt2,\sqrt2i]}\frac{1}{z-(1+i)}dz\right| \le 2(\sqrt2+1)$

I found a math problem (of a 2002 exam) I can't seem to solve, that is simply stated as: Show that $$\left| \int_{\left[\sqrt2,\sqrt2i\right]}\frac{1}{z-(1+i)}dz\right| \le 2(\sqrt2+1)$$ I looked ...
1
vote
0answers
103 views

Determining wether a set of complex numbers is open or closed

Let $S = \{z: arg(z)= \pi/4\}$. Is S open? Is it closed? What?'s its boundary? I graph it in polar and it just seems to be a line from the origin with $\theta = \pi/4$. What is a better way to ...
0
votes
1answer
48 views

estimate Integral

I have to estimate the following integral and then prove that the integral goes to zero if $Re(z_{1,2}) \rightarrow \infty$ : $s\in \mathbb{C}$ $ Re(s)>1$: \begin{align*} \int_{\gamma_{1,2}} ...
2
votes
1answer
79 views

Fubini on path integrals?

I've been given an exercise in more steps where I need to prove that the Riemann $\zeta$ function extends to a meromorphic function on all $\mathbb{C}$ with a single simple pole in $z=1$. To prove it ...
2
votes
1answer
110 views

Example of a certain locally univalent function

I'm looking for an example of a non-quadratic analytic function $f\colon \mathbb{C}\to \mathbb{C}$ (a power series with infinite radius of convergence) that has the following three properties: ...
0
votes
2answers
87 views

Show that $\alpha = (\omega_6)^5$ is a generator for the set of sixth roots of unity

Let $\omega_6 = e^{i{\dfrac{2\pi}{6}}}$. We say that $\omega_6$ is a generator of the set of sixth roots of unity because every sixth root of unity can be written in the form $(\omega_6)^k$ for some ...
2
votes
1answer
122 views

Why $\frac{|1-z|}{1-|z|}\le K$ corresponds to the region defined by the Stolz angle?

In his presentation of Abel's theorem, Ahlfors mentions that for a fixed positive number $K$, the region defined by \begin{equation} \frac{|1-z|}{1-|z|}\le K \end{equation} corresponds to the region ...
0
votes
0answers
186 views

How can I explicitly construct a *nice* conformal mapping from a triangle to a square in MATLAB?

I know the basics of the Riemann mapping theorem, SC maps, etc. I can look up formulae for the maps from the half-plane to a triangle or rectangle. But I want a particularly nice explicit map--easily ...
0
votes
2answers
186 views

Proof that $e^{i\bar{z}}=\overline{e^{iz}}$ if and only if $z=k\pi\in Z$

I need to proof that $e^{i\bar{z}}=\overline{e^{iz}}$ if and only if $z=k\pi\in Z$, I will show you my procedure $e^{i\overline{z}}=\overline{e^{iz}}$ $e^{i\overline{x+iy}}=\overline{e^{i(x+iy)}}$ ...
0
votes
1answer
52 views

Can $\mathbb{Z}^2$ be a subgroup of $\mathrm{PSU}(1,1)$?

Is there a subgroup of $\mathrm{PSU}(1,1)$ which is isomorphic to $\mathbb{Z}^2$ (as a group, ignoring the topology)?
2
votes
1answer
107 views

Finding the Laurent series of $f(z)=\frac{1}{(z-1)^2}+\frac{1}{z-2}$?

Can anyone help me out with finding the Laurent series of $f(z)=\dfrac{1}{(z-1)^2}+\dfrac{1}{z-2}$ in $\{z \in \Bbb C: 2<|z-4|<3\}$?
0
votes
1answer
64 views

Residuals and an identity of the cotangent

My girlfriend has a complex analysis problem set where I am stuck. The first part is: Let $$ \cot(z) = \frac{\cos(z)}{\sin(z)} $$ be the cotangent. Show that $\cot$ is homomorphic in $\mathbb ...
2
votes
1answer
63 views

Simple expression for this sum?

Is there any simple expression for the sum: $$ S = \sum_{n = 0}^{N-1} \frac{1}{a + e^{2 \pi i n / N}} $$ where $ N $ is a positive integer and $ a $ is some real number. It feels to me like there ...
2
votes
0answers
273 views

Showing $|z|$ is nowhere differentiable using the limit definition.

I need to show that the complex valued function, $f(z) = |z|$ is nowhere differentiable using the limit definition: \begin{align*} \lim_{\Delta z \rightarrow 0} \frac{f(z+\Delta z) - f(z)}{\Delta z}. ...
1
vote
2answers
210 views

is argument function continuous?

Is argument function defined as $f:\mathbb{C} \rightarrow \mathbb{R}$ $f(z)=Arg z$ continuous? And what about the function $\iota f$? Any hints are appreciated.
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0answers
53 views

To determine a complex function

A question of my book & i find some difficulty in solving it The only singularities of a single valued function f(z) are poles of order 2 & 1 at z=1 & z=2 with residue of these poles 1 ...
4
votes
4answers
637 views

Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as ...
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0answers
37 views

Intuitively what is it if making a modification of a torus?

It is well-known that if we have a equivalence relation in $\mathbb{R}^2$:$(z_1,z_2)\sim (z_1',z_2')$ iff $$\begin{pmatrix} z_1'\\ z_2' \\ \end{pmatrix}=\begin{pmatrix} 1&0\\ 0&1 \\ ...
18
votes
2answers
479 views

Simpler way to evaluate the Fourier transform of $\exp\left(i e^x\right)$?

I have the task to evaluate $|a(k)|^2$ with $$ a(k) = \int_{-\infty}^\infty \!dx\,\exp\left(i k x + i e^{x}\right).\tag{1}$$ The integral in (1) can be evaluated explicitly via the substitution ...
1
vote
3answers
72 views

Determining power series $\mathrm{e}^z$ and $\sin^2z$

Determine the powerseries of the following functions: $f(z)=\mathrm{e}^z$ in $a=5$ $f(z)=\sin^2(z)$ in $a=0$. I don't know if I'm making a mistake, but the first one is a powerseries I already ...
2
votes
1answer
45 views

Let $I=\int_C\dfrac{f(z)}{(z-1)(z-2)}$ where $f(z)=\sin\dfrac{\pi z}{2}+\cos\dfrac{\pi z}{2}$ and $C:|z|=3$ Then the value of $I$ is

Let $I=\int_C\dfrac{f(z)}{(z-1)(z-2)}$ where $f(z)=\sin\dfrac{\pi z}{2}+\cos\dfrac{\pi z}{2}$ and $C:|z|=3$ Then the value of $I$ is $$1. 4\pi i,~2.-4\pi i,~3.0,~4.2\pi i$$ My try: ...
0
votes
2answers
90 views

Show that a continuous complex function $f$ attains a maximum

Let the continuous function $f:\mathbb{C}\rightarrow\mathbb{C}$ satisfy $\lim\limits_{|z|\rightarrow\infty}f(z)=0$. Show that $|f|$ attains a maximum value at some point of the complex plane. I am ...
1
vote
0answers
29 views

How to linearize and solve ODE $\dot{z}_n = \sum_{m} i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}$ for $z_n\approx 0$?

I came across a physical system which obeys the following ODE $$\frac{d z_n}{dt} = \sum_{m=1}^N i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}, \qquad n\in\{1,2,\dots,N\}$$ where $z_n \equiv z_n(t)$ are ...
0
votes
1answer
57 views

Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
0
votes
2answers
243 views

Determining Laurent series $f(z)=\frac{1}{(z-2)(z-3)}$.

I have a question about determining the Laurent series of the function $f(z)=\frac{1}{(z-2)(z-3)}$. I have determined the three rings: $|z|<2$, $2<|z|<3$ and $|z|>3$. I know that in the ...