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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29 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
3answers
37 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 ...
2
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0answers
28 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 ...
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0answers
20 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 ...
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1answer
59 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
$$\int_\Gamma \frac{e^{tz}}{z^2 + p^2}dz + \int_{a - ...
7
votes
1answer
217 views
interpolating the primorial $p_{n}\#$
The primorial $p_{n}\#$ is given by the product $p_n\# = \prod_{k=1}^n p_k$ (where $p_{k}$ is the $k$th prime) -- is there a natural (a la the gamma function $\Gamma(z)$) way of interpolating it for ...
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2answers
31 views
Does having multiple limit values at a point imply essential discontinuity?
In Complex Analysis, do "jump discontinuities" exist?
If I find that a function of $z$ approaches two different values as z is approached from two different directions, can I immediately conclude ...
3
votes
2answers
73 views
limit of $e^z$ at $\infty$
What's the limit of $e^z$ as $z$ approaches infinity?
I am given that the answer is "There is no such limit."
Is this correct, and if so, am I correct to demonstrate this by showing that as $y$ ...
2
votes
2answers
206 views
Möbius transform which completely preserves circles (how to map a circle?)
(remmert theory of complex function)
I am trying to solve this exercise, however it seems impossible because I don't know how to map a circle, and I will be very thankful if somebody points out to ...
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1answer
20 views
What is the inverse z transform of 1/(z-1)^2?
I'd like to know how to calculate the inverse z transform of $\frac{1}{(z-1)^2}$ and the general case $\frac{1}{(z-a)^2}$
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1answer
123 views
Graphing the complex function
I'm looking for some software that my help me to graph some complex functions on unit circle. I.e. let say if I have $\ f(z)=1/(1-z)$ I want to see to give an input an image with unit circle and want ...
4
votes
3answers
92 views
Finding the Fourier Series of $\sin(x)^2\cos(x)^3$
I'm currently struggling at calculation the Fourier series of the given function
$$\sin(x)^2 \cos(x)^3$$
Given Euler's identity, I thought that using the exponential approach would be the easiest ...
4
votes
1answer
29 views
Complex Analysis problem related to T/F statements
I am stuck on the following problem:
For each of the statements below, indicate whether they are true or false. If true, give a
proof. If false, give a counter example or explain why it cannot ...
6
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1answer
85 views
Integrating: $\int_0^\infty \frac{\sin (ax)}{e^x + 1}dx$
I am trying to evaluate the following integral using the method of contour which I am not being able to. Can anyone point out what mistake I am making?
$$\int_0^\infty \frac{\sin ax}{e^x + 1}dx$$
I ...
2
votes
1answer
49 views
Show that F is a normal family
G is a domain in $C$, where $C$ is the complex plane and $G \neq C$ and $M>0$ we have that:
$F=\{f\in H(G): \int_{G}|f(z)|^2 dm_2(z)\leq M\}$
How do I show that F is a normal family? - Some of the ...
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votes
2answers
13 views
Show coth is a conformal mapping of the horizontal strip
I want to show that $\coth=\frac{e^{2z}+1}{e^{2z}-1}$ is a conformal mapping of the horizontal strip $S=\{z\in C: \pi/4<\text{Im}(z)<3\pi/4\}$ onto the unit disc U, but I can't seem to get the ...
2
votes
2answers
21 views
Problem involving the computation of the following integral
I was solving the past exam papers and stuck on the following problem:
Compute the integral $\displaystyle \oint_{C_1(0)} {e^{1/z}\over z} dz$,where $C_1(0)$ is the circle of radius $1$ around ...
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votes
2answers
32 views
Computing real integrals using the Residue Theorem where singularities are on the real line
How would you compute, for $a>0$ the integral $$\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 ...
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1answer
55 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 ...
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1answer
98 views
An issue with approximations of a recurrence sequence
By trying to give an approximation to a given recurrence sequence I encountered a problem.
To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
3
votes
1answer
329 views
Why is every holomorphic bijection of the Riemann sphere a Möbius transformation?
Just based on some reading, I know that every Möbius transformation is a bijection from the Riemann sphere to itself.
I'm curious about the converse. For any holomorphic bijection on the sphere, why ...
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votes
3answers
36 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 ...
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1answer
72 views
Problem with calculating a winding number
I have a problem with calculating the winding number $n\left ( \gamma ,\frac{1}{3} \right )$ of the curve $\gamma :\left [ 0,2\pi \right ]\rightarrow \mathbb{C}, t \mapsto \sin(2t)+i\sin(3t)$.
...
2
votes
1answer
26 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 ...
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vote
1answer
120 views
Lipschitz Domain
Let $U$ denote the open unit ball in the complex plane and $G=${$(z(1)+z(2),z(1)z(2))$ belongs to $C^2:z(1),z(2)$ belongs to U}.Is the boundary of $G$ Lipschitz?Justify.
Thanks for any help.
16
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1answer
388 views
What is wrong with this fake proof $e^i = 1$?
$$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$
Obviously, one of my algebraic manipulations is not valid.
5
votes
2answers
65 views
If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$
I'm going over old exam problems and I got stuck on this one.
Suppose that $\mathbb{f}\colon \mathbb{D} \to \mathbb{C}$ is analytic and bounded. Let $\{a_n\}_{n=1}^\infty$ be
the non-zero zeros of ...
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3answers
199 views
Riemann sphere and Maps
Could somebody please clarify the following for me?
I am not too clear about the relationship between the Riemann sphere and Mobius maps. I know that we can through projection make some Mobius maps ...
3
votes
1answer
33 views
A sequence of analytic functions with converging path integrals converges how?
Suppose you have a sequence $\{f_n\}_{n \in \mathbb Z^+}$ of analytic functions on a compact set $K$ such that for every smooth curve $\gamma$ in $K$ the sequence $\{\int_\gamma f_n\}_{n \in \mathbb ...
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votes
2answers
138 views
+50
Integrate: $\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)}dx$
How to evaluate the following
$$\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)} dx $$
Given hints says to construct a rectangle $0\to R\to R+i\to i \to 0$ and consider $\displaystyle ...
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2answers
37 views
Largest disc around which this complex function is one-to-one?
How would you determine explicitly the largest disc round the origin on which the function $f(z) = z^2 +z$ is one-to-one?
Is there a general method to do this for functions of this type?
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vote
2answers
46 views
Finding all $z$ such that the modulus of $f(z)=e^{(z+1)/(z-1)}$ is equal to/at most $1$
I was solving a previous exam paper and there I got stuck on the following problem:
Let $f(z)=e^{\frac{z+1}{z-1}}$. Then find all $z \in \Bbb C$ for which
$|f(z)|=1$,
$|f(z)|\le ...
1
vote
1answer
137 views
Sequence of entire functions with some conditions
Let $\{f_{n}(z)\}$ be a sequence of analytic functions in the upper half plane (in a Hilbert space $H$) and continuous on the real axis, such that
(1) $0<|f_{n}(x)|\leq 1$ for all $x\in \mathbb ...
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votes
0answers
78 views
Why is Every Elliptic Function of Order $2$ the Möbius Tranform of a $\wp$-function?
I'm trying to prove that every elliptic function of order $2$ has the form
$$f(z)=\frac{a\wp(z-z_0)+b}{c\wp(z-z_0)+d}$$
I've got the following so far. Let $f$ be an elliptic function of order 2. ...
1
vote
2answers
63 views
Calculate the following integral
I was thinking about the following problem:
Calculate $\displaystyle \oint_{C}(\bar z)^2 \mathrm{dz}$ where $C:|z-1|=1$ is oriented counter clock-wise.
My Attempt: I take $z-1=e^{i\theta}$ ...
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votes
4answers
125 views
Why is it meaningless for a closed set to be polygonal path connected?
My textbook (Complex Analysis by Saff & Snider) defines connectedness for open sets; the given definition of a connected open set is: a set in which every pair of points can be joined by a ...
2
votes
2answers
81 views
Laurent series, complex analysis.
I am working on a complex analysis exercice. I need to find the Laurent series of the function:
$$f(z) = \frac{e^z}{z + 1}$$
about $z = −1$.
I know that the result is
...
2
votes
3answers
36 views
When speaking of neighbourhoods in complex analysis, are we always referring to circular neighbourhoods?
In Complex Analysis, does "neighbourhood" automatically mean "circular neighbourhood", or do non-circular ones exist?
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1answer
152 views
Solution of a differential matrix equation
Given a differential matrix equation, ie $X'=A(z)X+B(z)$ where both $A$ and $B$ are matrix of size $n\times n$ with coefficients that are holomorfic functions in a convex open set $\Omega$ and ...
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votes
4answers
83 views
what is the relation of smooth compact supported funtions and real analytic function?
What is the major difference between real analytic and test function (smooth compact supported functions). Can we find a real analytic function $f$ on $R^n$ which is also smooth compact supported? If ...
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vote
1answer
20 views
Extension of real functions to Riemann surface
Let $f:\mathbb{R}^*_+\to \mathbb{R}$ be a function that is locally the restriction of an holomorphic function.
Notating $R$ the Riemann surface of the complex logarithm, with coordinates ...
2
votes
1answer
34 views
Show $5z^n=e^z$ has a finite number of zero in $\{a<\Im z < v\}$ and $\{a < \Re z < b \}$
Show that the number of roots $N$ of the function
$$
h(z)= f(z)-g(z)=5z^n - e^z, n\ge 1,
$$
is at most finite in any horizontal strip
$$
\alpha=\{z:a<\Im z<b\},
$$
and any vertical strip,
$$
...
1
vote
1answer
62 views
To show that $\sqrt{z^2-1}=\exp(\frac{1}{2} \log(z^2-1))$ is analytic in the plane minus [-1,1]
By the definition of logarithm branch, we set that $\sqrt{z^2-1}=\exp(\frac{1}{2} \log(z^2-1))$
However to show that the $\sqrt{z^2-1}$ is analytic in the entire plane minus the interval [-1,1], it ...
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votes
1answer
25 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 ...
5
votes
3answers
69 views
A boundary version of Cauchy's theorem
I am looking for a reference for the following theorem (or something like it) that is not Kodaira's book.
Let $D$ be a domain and $\overline{D}$ be it's closure. Suppose that $f:\overline{D} ...
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vote
2answers
39 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 ...
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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 ...
7
votes
2answers
249 views
True/False Questions for Complex Analysis
I am studying for my (introductory) complex analysis final exam tomorrow. I am practicing an old final exam, which unfortunately has no answer key. Here is a link:
...
2
votes
2answers
39 views
Application of Open Mapping Theorem
This was stated without proof in the complex analysis text I am reading (Complex Made Simple by Ulrich, page 107). I'm sure it's easy, but I'm tired and need a little help.
Let $f$ be nonconstant and ...
