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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38 views
Is a series (summation) of continuous functions automatically continuous?
I'm being asked to show that a given series (of rational functions) converges uniformly on a given disc, and then and asked to use this fact to show that integrating its limit function (i.e. a ...
2
votes
2answers
52 views
Rouché Theorem to calculate the number of zeros
How can I calculate the number of zeros of $\cos z+3z^3$ using the Rouché Theorem?
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0answers
25 views
Spectrum of a unitary
I have a unitary element $v$ in $C(S^{1}, \mathbb{C})$ with full spectrum (the whole circle). Is it possible to construct another unitary $u$ in $C(S^{1}, \mathbb{C})$ out of $v$ such that the ...
4
votes
2answers
44 views
Does the Weierstrass M-test show analyticity?
I'm trying to show (textbook exercise) that the riemann-zeta function is analytic. The solution is here:
Why does the proof say that the zeta series converges to an analytic function? Doesn't the ...
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2answers
34 views
Is it possible for a function to be analytic anywhere outside the circle of convergence of its power series expansion?
Is it possible for a function to be analytic anywhere outside the circle of convergence of its power series expansion? I'm referring to analytic fuctions of course (i.e. those with power series ...
1
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1answer
42 views
Is $e^z\sum_{k=0}^\infty\frac{k^3}{3^k}z^k$ analytic inside $|z|=3$?
Am I correct that the following function is analytic at least inside $|z|=3$? (I used the ratio test.) The solutions manual says that the function is analytic on and inside |z|=1, so I wonder if I'm ...
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1answer
29 views
Two problems on analytic function and Mapping of elementary functions
Let $G$ be a region and let $f$ and $g$ be analytic functions on $G$ such that $f(z)g(z)=0$ for all $z \in G$. Show that either $f$ or $g$ is identically zero on $G$.
Here is how I do it: Assume $f$ ...
3
votes
1answer
40 views
number of zeros of function $\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)-1$
$$f(z)=\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)-1$$
How many zeros does the above function have in $\Bbb{C}$?
6
votes
2answers
63 views
Is a curve homologous to zero according to Ahlfors actually homologous to zero?
The presentation of the homology version of Cauchy's theorem in Ahlfors is slick, but sweeps a lot of the topology under the rug using clever arguments. This question is an attempt to reconcile ...
2
votes
2answers
116 views
Diagonalizability in $\mathbb{R}$ and $\mathbb{C}$
Give an example of a matrix $A\in M_{n\times n}(\mathbb{R})$ that is not diagonalizable, but A is diagonalizable viewed as a matrix over the field of complex numbers $\mathbb{C}.$
2
votes
1answer
62 views
Why do injective holomorphic functions have nonzero derivative
For some open sets $U$, $V$ in the complex plane, let $f:U\rightarrow V$ be an injective holomorphic function. Then $f'(z) \ne 0$ for $z \in U$.
Now I don't understand the proof, but here it is ...
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1answer
21 views
Mapping of a Lens-shaped region by a Möbius Transformation
Consider the 'lens' described by $\{z:|z-i|<\sqrt{2}\ \text{and}\ |z+i|<\sqrt{2} \}$ . We want to map this to the upper right quadrant using a Möbius transformation.
The two circles meet at ...
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votes
0answers
49 views
Vanishing of Dirichlet Series
Suppose the function
$\sum_{n=1}^{\infty}{a_{n}n^{-s}}$ is $0$ on some open set $U\subset\mathbb{C}$. (Can assume the sum converges absolutely on $U$.)
Is it true that $a_{n}=0$ for all $n$?
(This ...
2
votes
1answer
34 views
Constructing a conformal map from $\mathbb{D}$ to a cut plane
Source: Oxford Exam $A2 \ 1999$
We want to construct a conformal map $F$ from the unit disc $\mathbb{D}=\{z:|z|<1\}$ to $\mathbb{C} \setminus S$ where $S$ is the half-line $\{x+i:x \in (-\infty,0] ...
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0answers
27 views
Harmonic Function Transformation Help
Consider the harmonic function $$u(x,y)=1-y+\frac{x}{x^2+y^2}$$ on the upper half plane $y>0$.
What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
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2answers
50 views
Complex number question
For any complex numbers $z_1, z_2$, is the quantity $S$: $$
S = 4 \left(| z_1|^6 + |z_2 |^6\right ) + 4 |z_1|^3 |z_2 |^3 + \left(2 |z_1|^2\times \overline{z_1}^2\times z_2^2\right) + \left(2 ...
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votes
2answers
67 views
Plotting in the Complex Plane
I just wonder how do you plot a function on the complex plane? For example,$$f(z)=\left|\dfrac{1}{z}\right|$$
What is the difference plotting this function in the complex plane or real plane?
Thank ...
1
vote
2answers
79 views
Must a complex power series *fail* to be convergent somewhere on its circle of convergence?
My textbook asserts so, but I can't find its proof of the claim. On the other hand, a lecture slide I'm cross-referencing claims that a power series is allowed to be convergent at ALL points of its ...
1
vote
0answers
30 views
show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle [duplicate]
I need to show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle except the point $z=1$, well at $z=1$ we get our known divergent harmonic series, but I am not able to show easily ...
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2answers
33 views
Understanding the (Partial) Converse to Cauchy-Riemann
We have that for a function $f$ defined on some open subset $U \subset \mathbb{C}$ then the following if true:
Suppose $u=\mathrm{Re}(f), v=\mathrm{Im}(f)$ and that all partial derivatives ...
3
votes
1answer
32 views
Proving convergence of a Hilbert modular theta function $\vartheta(z):= \sum\limits_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)}$
I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\infty}^{\infty} e^{2 \pi i ...
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votes
4answers
338 views
Why is $2\pi i \neq 0?$ [duplicate]
We know that $e^{\pi i} = -1$ because of de Moivre's formula. ($e^{\pi i} = \cos \pi + i\sin \pi = -1).$
Suppose we square both sides and get $e^{2\pi i} = 1$(which you also get from de Moivre's ...
9
votes
1answer
108 views
How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?
I need to solve the to following integral:
$$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx.$$
I tried this integral in Mathematica, but it was not able to solve it. ...
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vote
3answers
40 views
Laurent expansion problem
Expand the function $$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)} $$
then
...
3
votes
1answer
34 views
Application of the Identity Theorem to $|x|^3$ for $-1<x<1$
Oxford Exam $2602$ $1997$ $Q3$
We want to show that there is no function $f$ which is holomorphic in $D(0;1)$ and such that $f(x)=|x|^3$ for $-1<x<1$.
Here are my thoughts thus far:
Suppose ...
1
vote
1answer
21 views
Results following from Analyticity on a domain
This is part of an old Oxford exam paper (1997 2602 Q2) I'm working on for revision.
Suppose we have a function $f$ which is holomorphic on the disc radius $R$ about $0$. We want to show that there ...
2
votes
2answers
47 views
Integrate: $\int_0^{\pi} \log ( 1 - 2 r \cos \theta + r^2)d\theta$
If $r \in \Bbb R$ how to integrate $\displaystyle \int_0^{\pi} \log ( 1 - 2 r \cos \theta + r^2)d\theta$?
I need some hints. Special case, if $r = 1$ then I know the above integral is zero.
Here ...
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vote
3answers
33 views
Series of $\int_0^z \zeta^{-1} \sin \zeta d \zeta$
This is a homeworkquestion so I would appreciate some good hints. I have $f(z) = \int_0^z \zeta^{-1} \sin \zeta d \zeta$. Can this be written as a power-series in $\mathbb C$ around $z = 0$?
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2answers
48 views
having trouble intuiting analyticity
My textbook seems to suggest that the analytic functions are precisely the functions that can be written in terms of $z$ alone (no $x$ or $y$ or conjugate-$z$).
Am I inferring correctly?
Does this ...
3
votes
1answer
71 views
Integrate: $\int_0^1 \frac{1}{\sqrt[3]{x^2 - x^3}}dx$
How to integrate using Residue theorem.
$$\int_0^1 \frac{1}{\sqrt[3]{x^2 - x^3}}dx$$
How do I choose my branch-cut particularly? I was reading this article on wikiepdia and I think it is related. ...
1
vote
2answers
43 views
The harmonic conjugate of $\Im e^{z^2}$?
It is obvious that $e^{z^2}$ is analytic, right? So the harmonic conjugate of $\Im e^{z^2}$ is $\Re e^{z^2}$, isnt' it?
However, the solutions manual I'm consulting gives the answer as $\Im ...
10
votes
2answers
189 views
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
1
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0answers
17 views
The covering space of a region contained in complex plane delete two points.
We all know that C \ {0,1} can be given the Poincare hyperbolic metric, so that a region W in it is an embedded manifold of negative constant curvature. Hence the covering space of W is a hyperbolic ...
2
votes
1answer
44 views
Infinite Series Problem Using Residues [duplicate]
Show that $$\sum_{n=0}^{\infty}\frac{1}{n^2+a^2}=\frac{\pi}{2a}\coth\pi a+\frac{1}{2a^2}, a>0$$
I know I must use summation theorem and I calculated the residue which is:
...
4
votes
1answer
52 views
Evaluating $\int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx$
Q: Suppose $\alpha>0$ and $|\beta|<\pi/2$, show that
\begin{align*}
\textbf{(1)} \; \int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx &= \frac 1 2 ...
4
votes
1answer
75 views
How does it follow $s\int_1^{\infty}\frac{\psi(x)}{x^{s+1}}dx$?
I have two relations:
1)$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{1}^{\infty}\frac{\Lambda(n)}{n^s}$.
2)$\psi(x)=\sum_{n\leq x}\Lambda(n)$.
From these two how does it follow that ...
4
votes
1answer
82 views
Integrate: $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$
Can this integral be solved with contour integral or by some application of Residue theorem?
$$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catlan constant}$$
It has two ...
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vote
1answer
50 views
$e^z$ is entire yet has an essential singularity (at $\infty$)
Is there no inconsistency? Or does the property of being entire exclude the point $z=\infty$?
p.s. following up from my previous question limit of $e^z$ at $\infty$
1
vote
1answer
33 views
need to show image of $f$ contains the unit disk.
$f$ be non constant analytic on the closed unit disk,$|f|=1$ if $|z|=1$,we need to show image of $f$ contains the unit disk.
My thoughts:
whenever $|\omega|<1$ if I show that $g(z)=f(z)-\omega$ ...
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votes
1answer
20 views
need to show antiderivative exist
Let $U$ be a simply connected open set and $z_1,\dots, z_n$ be points of $U$ and let $U^*=U\setminus \{z_1,\dots,z_n\},z_i\in U$ Let $f$ be analytic on $U^*$. Let $\gamma_k$ be a small circle centered ...
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0answers
53 views
Analyticity of a function in $x$ and $y$, without employing the Cauchy-Riemann eqns
Exercise from Saff & Snider's Complex Analysis:
How to determine the analyticity of this function, without using the Cauchy-Riemann equations? I tried to work from first principles (taking the ...
1
vote
1answer
56 views
Showing particular harmonic function is constant
Suppose $u$ is a real valued continuous function on $\overline{\mathbb D}$, harmonic on ${\mathbb D}$\ $\{0\}$ and $u=0$ on $\partial\mathbb D$, show $\mathbb u$ is constant in $\mathbb D$.
I'm going ...
6
votes
3answers
73 views
Series expansion of a meromorphic function
in a theoretical physics book, the author makes the following claim:
$$\frac{1}{e^z + 1} = \frac{1}{2} + \sum_{n=-\infty}^\infty \frac{1}{(2n+1) i\pi - z}$$
and justifies this as
These series ...
4
votes
2answers
69 views
Integrate: $\int_0^{\infty}\frac{\sinh (ax)}{\sinh x} \cos (bx) dx$
Q: If $|a|< 1$ and $b>0$, show that
$$\int_0^{\infty}\frac{\sinh (ax)}{\sinh x} \cos (bx) dx = \frac{\pi \sin (\pi a)}{2 (\cos (\pi a)+\cosh (\pi b))}$$
I need to evaluate the above ...
2
votes
0answers
45 views
Conformal mapping from exterior of closed unit disk onto exterior of horizontal interval.
This is a problem from Bak-Newman's "Complex Analysis", #4 from Chapter 14 "The Riemann Mapping Theorem".
The question is this:
Verify directly that $F(z) = z + \frac{1}{z}$ is the unique conformal ...
1
vote
3answers
52 views
Does having multiple limit values at a point imply essential discontinuity?
In Complex Analysis, do "jump discontinuities" exist?
If I find that a function of $z$ approaches two different values as z is approached from two different directions, can I immediately conclude ...
3
votes
2answers
86 views
limit of $e^z$ at $\infty$
What's the limit of $e^z$ as $z$ approaches infinity?
I am given that the answer is "There is no such limit."
Is this correct, and if so, am I correct to demonstrate this by showing that as $y$ ...
1
vote
2answers
47 views
What is the inverse z transform of 1/(z-1)^2?
I'd like to know how to calculate the inverse z transform of $\frac{1}{(z-1)^2}$ and the general case $\frac{1}{(z-a)^2}$
2
votes
2answers
54 views
Harmonic Function bounded by a linear function
Let $u$ be a harmonic function on $\mathbb C$. Suppose that for each $\epsilon > 0$, there is a constant $C_\epsilon$ such that
$$u(z) \leq C_\epsilon + \epsilon |z| .$$
I am trying to show that ...
4
votes
3answers
137 views
Finding the Fourier Series of $\sin(x)^2\cos(x)^3$
I'm currently struggling at calculation the Fourier series of the given function
$$\sin(x)^2 \cos(x)^3$$
Given Euler's identity, I thought that using the exponential approach would be the easiest ...
