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
26 views
Proving convergence of a Hilbert modular theta function $\vartheta(z):= \sum\limits_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)}$
I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\infty}^{\infty} e^{2 \pi i ...
10
votes
4answers
325 views
Why is $2\pi i \neq 0?$ [duplicate]
We know that $e^{\pi i} = -1$ because of de Moivre's formula. ($e^{\pi i} = \cos \pi + i\sin \pi = -1).$
Suppose we square both sides and get $e^{2\pi i} = 1$(which you also get from de Moivre's ...
1
vote
3answers
34 views
Laurent expansion problem
Expand the function $$f(z)=\frac{z^2 -2z +5}{(z-2)(z^2+1)} $$ on the ring $$ 1 < |z| < 2 $$
I used partial fractions to get the following $$f(z)=\frac{1}{(z-2)} +\frac{-2}{(z^2+1)} $$
then
...
3
votes
1answer
32 views
Application of the Identity Theorem to $|x|^3$ for $-1<x<1$
Oxford Exam $2602$ $1997$ $Q3$
We want to show that there is no function $f$ which is holomorphic in $D(0;1)$ and such that $f(x)=|x|^3$ for $-1<x<1$.
Here are my thoughts thus far:
Suppose ...
1
vote
1answer
21 views
Results following from Analyticity on a domain
This is part of an old Oxford exam paper (1997 2602 Q2) I'm working on for revision.
Suppose we have a function $f$ which is holomorphic on the disc radius $R$ about $0$. We want to show that there ...
1
vote
2answers
39 views
Integrate: $\int_0^{\pi} \log ( 1 - 2 r \cos \theta + r^2)d\theta$
If $r \in \Bbb R$ how to integrate $\displaystyle \int_0^{\pi} \log ( 1 - 2 r \cos \theta + r^2)d\theta$?
I need some hints. Special case, if $r = 1$ then I know the above integral is zero.
Here ...
1
vote
3answers
33 views
Series of $\int_0^z \zeta^{-1} \sin \zeta d \zeta$
This is a homeworkquestion so I would appreciate some good hints. I have $f(z) = \int_0^z \zeta^{-1} \sin \zeta d \zeta$. Can this be written as a power-series in $\mathbb C$ around $z = 0$?
1
vote
2answers
46 views
having trouble intuiting analyticity
My textbook seems to suggest that the analytic functions are precisely the functions that can be written in terms of $z$ alone (no $x$ or $y$ or conjugate-$z$).
Am I inferring correctly?
Does this ...
0
votes
0answers
41 views
Integrate: $\int_0^1 \frac{1}{\sqrt[3]{x^2 - x^3}}dx$
How to integrate using Residue theorem.
$$\int_0^1 \frac{1}{\sqrt[3]{x^2 - x^3}}dx$$
How do I choose my branch-cut particularly? I was reading this article on wikiepdia and I think it is related. ...
1
vote
2answers
42 views
The harmonic conjugate of $\Im e^{z^2}$?
It is obvious that $e^{z^2}$ is analytic, right? So the harmonic conjugate of $\Im e^{z^2}$ is $\Re e^{z^2}$, isnt' it?
However, the solutions manual I'm consulting gives the answer as $\Im ...
10
votes
2answers
182 views
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
1
vote
0answers
17 views
The covering space of a region contained in complex plane delete two points.
We all know that C \ {0,1} can be given the Poincare hyperbolic metric, so that a region W in it is an embedded manifold of negative constant curvature. Hence the covering space of W is a hyperbolic ...
2
votes
1answer
43 views
Infinite Series Problem Using Residues [duplicate]
Show that $$\sum_{n=0}^{\infty}\frac{1}{n^2+a^2}=\frac{\pi}{2a}\coth\pi a+\frac{1}{2a^2}, a>0$$
I know I must use summation theorem and I calculated the residue which is:
...
4
votes
1answer
51 views
Evaluating $\int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx$
Q: Suppose $\alpha>0$ and $|\beta|<\pi/2$, show that
\begin{align*}
\textbf{(1)} \; \int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx &= \frac 1 2 ...
4
votes
1answer
68 views
How does it follow $s\int_1^{\infty}\frac{\psi(x)}{x^{s+1}}dx$?
I have two relations:
1)$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{1}^{\infty}\frac{\Lambda(n)}{n^s}$.
2)$\psi(x)=\sum_{n\leq x}\Lambda(n)$.
From these two how does it follow that ...
4
votes
1answer
80 views
Integrate: $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$
Can this integral be solved with contour integral or by some application of Residue theorem?
$$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catlan constant}$$
It has two ...
1
vote
1answer
50 views
$e^z$ is entire yet has an essential singularity (at $\infty$)
Is there no inconsistency? Or does the property of being entire exclude the point $z=\infty$?
p.s. following up from my previous question limit of $e^z$ at $\infty$
1
vote
1answer
33 views
need to show image of $f$ contains the unit disk.
$f$ be non constant analytic on the closed unit disk,$|f|=1$ if $|z|=1$,we need to show image of $f$ contains the unit disk.
My thoughts:
whenever $|\omega|<1$ if I show that $g(z)=f(z)-\omega$ ...
0
votes
1answer
20 views
need to show antiderivative exist
Let $U$ be a simply connected open set and $z_1,\dots, z_n$ be points of $U$ and let $U^*=U\setminus \{z_1,\dots,z_n\},z_i\in U$ Let $f$ be analytic on $U^*$. Let $\gamma_k$ be a small circle centered ...
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votes
0answers
52 views
Analyticity of a function in $x$ and $y$, without employing the Cauchy-Riemann eqns
Exercise from Saff & Snider's Complex Analysis:
How to determine the analyticity of this function, without using the Cauchy-Riemann equations? I tried to work from first principles (taking the ...
1
vote
1answer
55 views
Showing particular harmonic function is constant
Suppose $u$ is a real valued continuous function on $\overline{\mathbb D}$, harmonic on ${\mathbb D}$\ $\{0\}$ and $u=0$ on $\partial\mathbb D$, show $\mathbb u$ is constant in $\mathbb D$.
I'm going ...
6
votes
3answers
73 views
Series expansion of a meromorphic function
in a theoretical physics book, the author makes the following claim:
$$\frac{1}{e^z + 1} = \frac{1}{2} + \sum_{n=-\infty}^\infty \frac{1}{(2n+1) i\pi - z}$$
and justifies this as
These series ...
4
votes
2answers
68 views
Integrate: $\int_0^{\infty}\frac{\sinh (ax)}{\sinh x} \cos (bx) dx$
Q: If $|a|< 1$ and $b>0$, show that
$$\int_0^{\infty}\frac{\sinh (ax)}{\sinh x} \cos (bx) dx = \frac{\pi \sin (\pi a)}{2 (\cos (\pi a)+\cosh (\pi b))}$$
I need to evaluate the above ...
2
votes
0answers
43 views
Conformal mapping from exterior of closed unit disk onto exterior of horizontal interval.
This is a problem from Bak-Newman's "Complex Analysis", #4 from Chapter 14 "The Riemann Mapping Theorem".
The question is this:
Verify directly that $F(z) = z + \frac{1}{z}$ is the unique conformal ...
1
vote
3answers
50 views
Does having multiple limit values at a point imply essential discontinuity?
In Complex Analysis, do "jump discontinuities" exist?
If I find that a function of $z$ approaches two different values as z is approached from two different directions, can I immediately conclude ...
3
votes
2answers
84 views
limit of $e^z$ at $\infty$
What's the limit of $e^z$ as $z$ approaches infinity?
I am given that the answer is "There is no such limit."
Is this correct, and if so, am I correct to demonstrate this by showing that as $y$ ...
1
vote
2answers
40 views
What is the inverse z transform of 1/(z-1)^2?
I'd like to know how to calculate the inverse z transform of $\frac{1}{(z-1)^2}$ and the general case $\frac{1}{(z-a)^2}$
2
votes
2answers
53 views
Harmonic Function bounded by a linear function
Let $u$ be a harmonic function on $\mathbb C$. Suppose that for each $\epsilon > 0$, there is a constant $C_\epsilon$ such that
$$u(z) \leq C_\epsilon + \epsilon |z| .$$
I am trying to show that ...
4
votes
3answers
133 views
Finding the Fourier Series of $\sin(x)^2\cos(x)^3$
I'm currently struggling at calculation the Fourier series of the given function
$$\sin(x)^2 \cos(x)^3$$
Given Euler's identity, I thought that using the exponential approach would be the easiest ...
4
votes
1answer
36 views
Complex Analysis problem related to T/F statements
I am stuck on the following problem:
For each of the statements below, indicate whether they are true or false. If true, give a
proof. If false, give a counter example or explain why it cannot ...
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votes
2answers
35 views
Show coth is a conformal mapping of the horizontal strip
I want to show that $\coth=\frac{e^{2z}+1}{e^{2z}-1}$ is a conformal mapping of the horizontal strip $S=\{z\in C: \pi/4<\text{Im}(z)<3\pi/4\}$ onto the unit disc U, but I can't seem to get the ...
3
votes
2answers
24 views
Problem involving the computation of the following integral
I was solving the past exam papers and stuck on the following problem:
Compute the integral $\displaystyle \oint_{C_1(0)} {e^{1/z}\over z} dz$,where $C_1(0)$ is the circle of radius $1$ around ...
2
votes
1answer
57 views
Show that F is a normal family
G is a domain in $C$, where $C$ is the complex plane and $G \neq C$ and $M>0$ we have that:
$F=\{f\in H(G): \int_{G}|f(z)|^2 dm_2(z)\leq M\}$
How do I show that F is a normal family? - Some of the ...
6
votes
1answer
93 views
Integrating: $\int_0^\infty \frac{\sin (ax)}{e^x + 1}dx$
I am trying to evaluate the following integral using the method of contour which I am not being able to. Can anyone point out what mistake I am making?
$$\int_0^\infty \frac{\sin ax}{e^x + 1}dx$$
I ...
0
votes
2answers
35 views
Computing real integrals using the Residue Theorem where singularities are on the real line
How would you compute, for $a>0$ the integral $$\int_0^\infty \frac{\sin x}{x(x^2 + a^2)} dx \, \, ?$$
I've computed the residues of the function $$f(z) = \frac{e^{iz}}{z(z^2 + a^2)} $$ which I ...
4
votes
1answer
59 views
Does there exist $g$ s.t $g'=f$?
I have the following homework question:
Let G be the bounded open set shown in gray in this picture, whose
boundary consists of eight line segments. The endpoints of those
segments are, as ...
2
votes
3answers
37 views
Finding the Taylor series of $f(z)=\frac{1}{z}$ around $z=z_{0}$
I was asked the following (homework) question:
For each (different but constant) $z_{0}\in G:=\{z\in\mathbb{C}:\,
z\neq0$} find a power series $\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n}$
whose sum ...
2
votes
1answer
28 views
Evaluate the following contour integral
I was solving old exam papers and I am stuck on the following question:
Evaluate the contour integral $\displaystyle \oint_{C} \frac{dz}{(\bar z-1)^2}$ where $C$ is the semi-circle $|z-1|=1, \Im ...
7
votes
2answers
80 views
If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$
I'm going over old exam problems and I got stuck on this one.
Suppose that $\mathbb{f}\colon \mathbb{D} \to \mathbb{C}$ is analytic and bounded. Let $\{a_n\}_{n=1}^\infty$ be
the non-zero zeros of ...
1
vote
3answers
69 views
+50
Further explanation needed :Finding all $z$ such that the modulus of $f(z)=e^{(z+1)/(z-1)}$ is equal to/at most $1$
I was solving a previous exam paper and there I got stuck on the following problem:
Let $f(z)=e^{\frac{z+1}{z-1}}$. Then find all $z \in \Bbb C$ for which
$|f(z)|=1$,
$|f(z)|\le ...
4
votes
1answer
74 views
Integrate: $\int_{a - i\infty}^{a + i\infty} \frac{e^{tz}}{z^2 + p^2}dz$
Q. Show that : $$\int_{a - i\infty}^{a + i\infty} \frac{e^{tz}}{z^2 + p^2}dz = \frac{\sin pt}{p}$$
I considered the following contour
$$\int_\Gamma \frac{e^{tz}}{z^2 + p^2}dz + \int_{a - ...
1
vote
2answers
63 views
Calculate the following integral
I was thinking about the following problem:
Calculate $\displaystyle \oint_{C}(\bar z)^2 \mathrm{dz}$ where $C:|z-1|=1$ is oriented counter clock-wise.
My Attempt: I take $z-1=e^{i\theta}$ ...
1
vote
2answers
38 views
Largest disc around which this complex function is one-to-one?
How would you determine explicitly the largest disc round the origin on which the function $f(z) = z^2 +z$ is one-to-one?
Is there a general method to do this for functions of this type?
3
votes
3answers
38 views
When speaking of neighbourhoods in complex analysis, are we always referring to circular neighbourhoods?
In Complex Analysis, does "neighbourhood" automatically mean "circular neighbourhood", or do non-circular ones exist?
5
votes
4answers
128 views
Why is it meaningless for a closed set to be polygonal path connected?
My textbook (Complex Analysis by Saff & Snider) defines connectedness for open sets; the given definition of a connected open set is: a set in which every pair of points can be joined by a ...
0
votes
1answer
26 views
Impossibility of polynomial approximation
This is exercise 12.6 in David Ullrich's Complex Made Simple. He has discussed many ways to prove the existence of polynomial approximations to functions in the complex plane, but not how to show such ...
1
vote
2answers
41 views
Contour Integral: $\int^{\infty}_{0}(1+z^n)^{-1}dz$
I'm working through Priestley's Complex Analysis (really good book by the way) and this Ex 20.2:
Evaluate $\int^{\infty}_{0}(1+z^n)^{-1}dz$ round a suitable sector of angle $\frac{2\pi}{n}$ for ...
3
votes
1answer
38 views
Dual of holomorphic functions (with the $L^1$ topology)
Let $\Omega$ be a connected domain of the complex plane, and let $E$ be the vector space of integrable holomorphic functions on $\Omega$. Then it can be checked that $E$ is a closed subspace of ...
0
votes
1answer
24 views
Conformal Map from Vertical Strip to Unit Disc
I haven't found a similar question on here, though I suspect the question may be rather well-covered.
I want to find a conformal map from the vertical strip $\{z:-1<Re(z)<1\}$ onto the unit ...
2
votes
2answers
52 views
value of fraction
Can we find out the value of
$$\frac{1+i}{1-i}$$
I have tried to solve it by multiplying $(1+i)$ to both sides and in the end see that the result is still $i$. Am I correct,or is there a different ...
