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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5
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1answer
176 views

Sum of the series $\sum_{k=1}^\infty (-1)^k \frac{2z}{k^2 \pi^2-z^2}\cos kt$

From the relation: $$\csc z=\frac{1}{z}+\sum_{k=1}^\infty (-1)^k \frac{2z}{z^2-k^2 \pi^2}$$ can we obtain the sum of following series? $$\sum_{k=1}^\infty (-1)^k \frac{2z}{k^2 \pi^2-z^2}\cos kt$$ ...
2
votes
2answers
73 views

Finding singularities of a projective curve

For $w \in \mathbb{C}$ we define the projective curve $$p(x,y,z):= x^3+y^3+z^3+wxyz.$$ Now I have to find all $w \in \mathbb{C}$ for which the projective curve $p(x,y,z)$ is singular and show that ...
4
votes
1answer
287 views

functions orthogonal to the exponential Bell polynomials

Consider the single variable Bell polynomials $\phi_{n}(x)$ given by: $$\phi_{n}(x)=e^{-x}\sum_{k=0}^{\infty}\frac{k^{n}x^{k}}{k!}$$ I am looking for a set of functions $\tilde{\phi}_{n}(x)$ such ...
0
votes
1answer
43 views

Prove statement about complex series

The problem statement Let $(a_n)_{n\geq o}$, $(Z_n)_{n\geq 0}$ sequences of complex numbers such that $(a_nZ_n)_{n\geq 0}$ converges. Show that $\sum_{n=0}^{\infty} (a_n-a_{n+1})Z_n$ converges if ...
1
vote
2answers
78 views

why $\infty$ +$\infty$, $\infty$ - $\infty$ and 0⋅$\infty$ are left undefined.

I'm reading http://en.wikipedia.org/wiki/Riemann_sphere, and having the following question. 1. What's the mean of symbol $\infty$?Is it a surreal number? 2. they write note that ∞ + ∞, ∞ - ...
1
vote
1answer
149 views

How to show $\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$

For $z\in\mathbb{R}$ it's very easy to show that it holds $$\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$$ But how do we show the same thing for $z\in\mathbb{C}$
0
votes
2answers
136 views

Map a half sliced unit disk to upper half plane

"half sliced unit disk" Can somebody tell me how to map this conformally to the upper half plane? I think the symmetry principle should be applied here but stuck on that for hours. Pardon my hasty ...
2
votes
1answer
61 views

Find maximum of a complex function $f(z)$

I am trying to find the following maximum, whose existence is justified by the compactness of the close ball $\Delta$ of $\mathbb C$ and continuity of the function $$f(z)=\sum_{k=1}^\infty ...
0
votes
1answer
46 views

Residue of Function

I need help finding the residue of the following function at $z=\pi$: $$\exp\left(\frac{2}{z-\pi}\right)$$ I have put the function into a series expansion about z=pi by using the expansion of $e^x$ ...
1
vote
0answers
163 views

Calculate $\int_{0}^{2\pi} \frac{\cos((2n+1)t)}{\cos(t)}dt$

Calculate the following integral for $n \in \mathbb{Z}$ with the residue theorem $$\int_{0}^{2\pi} \frac{\cos((2n+1)t)}{\cos(t)}dt$$ So far I have tried two approaches. Firsty, for $n\geq 0$: ...
0
votes
1answer
33 views

Calculating a line integral

Calculate $\int_\gamma f(t) dt$ where $f(t) = t^2$ and $\gamma $ is the semi circle from $i$ to $-i$ (counter clockwise) I set $\gamma : \left[\frac{\pi}{2},\frac{3\pi}{2}\right] \rightarrow ...
1
vote
0answers
55 views

Algebraic vs. logarithmic and transcendental branch points

I understand that if you take the function $z\mapsto z^{1/x}$ for integer $x$, there is an algebraic branch point at the origin for positive integer $x$. My question is what happens if $x$ is positive ...
0
votes
1answer
121 views

Understanding the slit plane and the complex $\sqrt{z}$

My book (Gamelin's Complex Analysis) talks about the square and square root functions for complex variables. I do not understand the slit plane (from $-\infty$ to $0$) for $\sqrt{z}$, and mapping the ...
1
vote
1answer
158 views

Numerical approximation of trigonometric polynomial

I have the following problem: Let $g$ be a trigonometric polynomial of degree n (there are complex coefficients $c_k$ with $k = -n, ..., n$ such that $g(t) =\sum\limits_{k = -n}^n c_{k}\exp(ikt). $ ...
1
vote
2answers
63 views

Continuous complex function without antiderivative

It's a well-known result that every real continuos function has an antiderivative. Is this theorem still true for a complex function? If not, can someone point out a counter-example (and proof that it ...
0
votes
0answers
138 views

Cauchy-Goursat Theorem Alternative Proof

I have a problem in regards with Cauchy-Goursat Theorem and its proof. The theorem states that any analytic function in a domain bounded by a curve $C$ equals $0$. Now let's take the integral $$ ...
1
vote
0answers
47 views

Plotting in Mathematica

I'm new to mathematica. How would I plot the following functions in the polar complex plane? $$ \phi={-k\over 2\pi r}\cos(\theta_0-\theta) $$ and $$ \psi={-k\over 2\pi r}\sin(\theta_0-\theta) $$ ...
0
votes
1answer
51 views

Radius of convergence for $a_n=(a_{n-1}+a_{n-2})/2$ for $n \ge 2$.

Define a sequence $a_0,a_1,a_2,...$ by setting $a_0=1$, $a_1=2$, $a_n=(a_{n-1}+a_{n-2})/2$ for $n \ge 2$. A) Find the radius of convergence of the series $\sum_{n=0}^{\infty}a_nz^n$. B) Find an ...
2
votes
2answers
82 views

Showing that $\sum_{n=1}^{\infty} \frac{z^n}{1+z^{2n}}$ converges

Studying for an exit exam and it's been years since I work with any series expansions. Here's a past problem: Show that $$\sum_{n=1}^{\infty} \frac{z^n}{1+z^{2n}}$$ converges to an analytic function ...
1
vote
1answer
32 views

Show that $f=0$ on $D.$

Let $f:D=\{z:|z|<1\}\to\mathbb C$ be analytic on $D$ such that $|f(z)|\le1-|z|$ on $D.$ Show that $f=0$ on $D.$ My thought: If possible let $f$ be nonconstant on $D.$ Choose $0<\epsilon<1.$ ...
1
vote
1answer
40 views

$\left(\frac{a_n}{n^k}\right)_n$ is bounded implies $\sum_{n=0}^\infty a_nz^n$ has a radius of convergence $\ge 1$

Let $$\left(\frac{a_n}{n^k}\right)_n\subset\mathbb{C}\;\;\;\;\;(k\in\mathbb{N})$$ be a boundet sequence. I want to show that the power series $$\sum_{n=0}^\infty a_nz^n\;\;\;\;\;(a_n,z\in\mathbb{C})$$ ...
2
votes
1answer
44 views

Question about an extension of an analytic function

Setting: Let $\Omega \subseteq \mathbb{C}$ and suppose that $f$ is analytic on $\Omega' = \Omega - \{a\}$. Suppose that $a \in \Omega$ satisfies the crucial property that $$ \lim_{z \to a} (z - ...
0
votes
1answer
68 views

Proving zeros inside a disk using Maximum Principle?

Let $f$ be non-constant analytic in a neighborhood of the closed unit disk such that $|f|=1$ on the unit circle. Show that $f$ has a zero inside the unit disk. It has been suggested to argue this by ...
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votes
0answers
38 views

Complex Analysis Questions Compilation

All of my questions are in relation to Gamelin's Complex Analysis. How was the parametric form of the line from the North Pole on the unit sphere through a point P come to be? It is $$ N + t (P-N) ...
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vote
2answers
73 views

Proving a simple equation with complex numbers

Fix $A \in ℂ$ and $B \in ℝ$ Let $z \in ℂ$. Show that the equation $|z^2| + Re(Az) + B = 0$ has solutions iff $|A^2| ≥ 4B$ I have no trouble proving the forward implication, but its the "only if" ...
0
votes
2answers
76 views

Complex Analysis, Integral over a Square

Given that $C$ is the boundary of the square with corners at $\pm4 \pm4i$ (sorry my formatting always seems to be stubborn, but that is plus or minus 4 plus or minus 4i, I am asked to compute $$\int_C ...
2
votes
0answers
60 views

Convergence of $\sum_{n=1}^{\infty}(n+1)z^n$

Consider the series $\sum_{n=1}^{\infty}(n+1)z^n$ A) For which complex numbers $z$ does this series converge? B) For those $z$, let $f(z)$ be the sum of the series and find $f(z)$. C) Evaluate ...
2
votes
2answers
30 views

Finding zeros of a function with an $i$ constant

I want to show that the zeros of $$f(z)=z^4+3iz^2+3$$ lie in the disk $\{z \in \mathbb{C}:|z| \le \sqrt{4} \}$ I'm using Rouche's theorem. Here's where I'm stuck. I let $g(z)=3iz^2$ for $|z|=2$, so ...
0
votes
1answer
25 views

How do I know a function is a fractional linear transformation?

I'm asked to find a nonconstant function from an open unit dist to an open unit disk which analytic on all of $D= \{z \in \mathbb{C}: |z|<1 \}$ with $g(1/2)=0$ and $g(1/3)=0$. Here's what I put: ...
4
votes
2answers
195 views

proving $\csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8$

How can I prove the following identity using complex variables $$ \begin{align*} 1) & \csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8 ...
2
votes
1answer
72 views

Meromorphic Function Question

I'm currently trying to solve this question: Let $f$ be meromorphic on $\mathbb{C}$ whose only pole is at $0$. Suppose that $f$ is bounded outside the unit disk $\Delta_1(0)$. Show that $f(z) = ...
0
votes
1answer
40 views

Find a maximum complex function $ \max_{z \in {\mathbb C},|z| \leq \frac{\pi}{4}} \left|1-\frac{\sin z}{z}\right|$

I am trying to find the following maximum, whose existence is justified by the compactness of the close ball $\Delta$ of $\mathbb C$ and continuity of the function $f:z \mapsto \left|1-\frac{\sin ...
1
vote
2answers
79 views

Complex Analysis Polar Circle Question

My book (Gamelin ' s Complex Analysis) says that the roots of a complex number are distributed in equal arcs on the circle centered at $0$ with radius $|w|^{1/n} $. Why is the radius centered at zero ...
2
votes
4answers
150 views

Values for $(1+i)^{2/3}$

This question might seem easier than I'm making it seem. But how many values are there for $(1+i)^{2/3}$? Do I let $z=(1+i)^{2/3}$ so that $z^3=2i$? I'm asked to write each in polar coordinates and in ...
1
vote
1answer
98 views

Is every closed set $K\subseteq \mathbb{C}$ the essential range of a measurable function?

For a complex-valued function $h$ on a measure space $(S,\Sigma, \mu)$, the $\textit{essential range}$ of $h$ is the set of all $\lambda \in \mathbb{C}$ such that for all $\epsilon >0$ the ...
0
votes
1answer
58 views

How to find the branch points

$w=\operatorname{sech}^{-1}z$ satisfies $\frac{1}{\cosh w}=z$. Prove $\operatorname{sech}^{-1}z=\log\left(\frac{1+(1-z^2)^{1/2}}{z}\right)$ and find the location of its branch points
5
votes
5answers
557 views

Showing that $a$ is a removable singularity if $\mathrm{Im}(f(z))$ is bounded from above

Problem: Suppose $f$ is analytic on the domain $\Omega$ except at the isolated singularity $a \in \Omega$. Show that $a$ is a removable singularity if $\mathrm{Im}(f(z))$ is bounded from ...
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vote
3answers
75 views

Prove with generalized Cauchy integral formula

Show with the help of Generalized Cauchy Integral Formula, if $ f(z)$ is entire with $|f^{(n)}|\le N$ for all z, then $f(z)$ is a polynomial of degree at most n.
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0answers
64 views

Fourier Series to Laurent Series

Given a periodic function $f(\sigma)$ with period $T$, one can compute its Fourier series, $$f(\sigma)=\sum_{n\in\mathbb{Z}} c_n e^{i \omega n\sigma}$$ where $\omega=2\pi/T$ and the coefficients of ...
0
votes
2answers
59 views

Does the logarithm inequality extend to the complex plane?

For estimates, the inequality $\log(y)\le y-1,$ $y>0$ is often helpful. Is there any sort of upper bound for the logarithm function in the complex plane? Specifically, $|\log(z)|\le$ something for ...
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vote
0answers
27 views

Criteria for a parameter-dependent integral to be squared integrable

I'm currently stumbling over a criteria of integrability. Let's suppose we have a real-valued function $h$ defined on the positive real axis which is integrable. We then define the function $K ...
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0answers
57 views

A theoretical question regarding Frobenius method

The following is a theoretical question regarding Frobenius method. Let $b(x),c(x)$ be real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. ...
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votes
0answers
36 views

How do I convert a function $f(x)$ to $f(\frac{x}{a})$?

I have a pretty simple problem but I kinda am confused. I have the function $$f(x)_a = \sqrt{\frac{2}{\pi}} \cdot \frac{Aa}{x^2 + a^2}$$ And especially: $$f(x=0)_a = \sqrt{\frac{2}{\pi}}\cdot ...
0
votes
2answers
32 views

Complex Analysis Polar Representation Questions

My text (Gamelin's Complex Analysis) talks about the polar representation of a complex number. At one point the equation $z^n = w$ is given.The polar representation $w = \rho e^{i \phi}$ is then ...
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0answers
44 views

evaluate $e^{ie^i}$

Using the idea that $e^i=e^{i\cdot 1}$, i think I can rewrite this using Euler's equation as $$e^{i(\cos{1}+i\sin{1})}$$ And then one more time to give me ...
4
votes
2answers
206 views

Holomorphic function is bijective if neutral fixed point

I have a question that asks If $S \subset \mathbb{C}$ is a bounded domain and $f : S \to S$ is a holomorphic map such that $f(p) = p$ and $|f'(p)| = 1$ for some $p \in S$, then $f$ is bijective. ...
0
votes
2answers
41 views

Factorization of a Complex Polynomial

My book (Complex Analysis by Gamelin) states that a complex polynomial $$p(z) = a_nz^n + a_{n-1}+\cdots +a_1z + a_0$$, where $z$ is an element of the complex numbers, can be factored as a product of ...
0
votes
1answer
44 views

holomorphic, bounded function

Let $f(z)$ be holomorphic for |z| <= R, let $f(0) = 0$, and let $|f(z)| <= M$ for $|z| = R$. Prove that $|f(z)|<(M/R)|z|$ for $0<|z|<R$, unless $f(z) = cz$ for some constant $c$.
1
vote
1answer
477 views

Evaluation of definite integral using complex analysis

I want to evaluate the following indefinite integral $$ \int_0^{\infty} x^{p - 1} \cos (ax) dx$$ where $0 < p < 1$ and $a > 0$. I was considering the function $f(z) = z^{p - 1} e^{iaz}$ and ...
3
votes
1answer
96 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...