All Questions
Tagged with complex-analysis power-series
1,470
questions
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Find the Laurent expansion of $f(z) = \sin{\frac{1}{z(z-1)}}$ in $0<|z-1|<1$
Here is my idea:
$\sin{\frac{1}{z(z-1)}} = \sin{\left( \frac{-1}{z} + \frac{1}{z-1}\right)} = \sin{\left(\frac{1}{z-1} - \frac{1}{z}\right)} = \sin{\frac{1}{z-1}}\cos{\frac{1}{z}} - \cos{\frac{1}{z-1}}...
- 9,399
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1
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Counter example to the identity theorem for two generating functions
I want to give an example of two generating functions $\psi_{X_+}$ and $\psi_{X_-}$ for random variables $X_+$ and $X_-$ with values in $\mathbb{N}_0$ which coincide on infinitely many points $x_i\in(...
- 1,279
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Zeta Lerch function. Proof of functional equation.
so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following.
In the article "Note sur la function" by Mr. Mathias Lerch, a ...
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3
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Value of $\displaystyle\sum_{k=1}^{+\infty} \log(1+\dfrac{z}{k})-\dfrac{z}{k}$
I'm trying to get an explicit value of this series but I'm only able to show that it converges and that we can for $|z|\leq 1$ after expanding the log into its power series and applying Fubini's ...
- 33.2k
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Problem 6.2.10 From Complex Analysis by Jihuai Shi
Definition: let $f$ be holomorphic on a region $G$ (here region means a non-empty connected open set). For $\xi\in \partial G$, if there exists a ball $B(\xi,\delta)$ and a holomorphic function $g \in ...
- 10.1k
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Maximizing the radius of convergence around a point for an analytic function
Let $f:\Omega\longrightarrow \mathbb{C}$ be analytic, and let $z_0\in\Omega$ s.t.
$$f (z)=\sum_{n\geq 0} a_n(z-z_0)^n,\quad\forall \left|z-z_0\right|<r,$$
for some $r>0$, and some complex-valued ...
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Taylor-Laurent series expansions
I'm having some issues finding how to series expand some complex functions that my professor gave past years in exams.
For example, in this exercise, it is asked to find the first two terms of the ...
- 2,126
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Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$
This question involves the following homework problem:
PROBLEM
Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form
$$f(z)=\sum_{-\...
- 27.1k
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Using Ratio Test for a Series with odd and even indexed coefficients
The problem I have is the following:
Suppose the radius of convergence of the series $\sum_{n=0}^\infty a_n z^n$ is equal to $2$. Find the radius of convergence of the series
$$\sum_{n=0}^\infty 2^{\...
- 47.4k
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Uniqueness of Two Series in an Intersection
I've been working on some problems involving series and have found myself applying the identity principle to show that two representations of a series will lead them being unique under some conditions....
- 415
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$\sum_{n=0}^{+\infty} a_n z^n$ is divergent on the Convergence circumference.
Let $f(z)=\sum_{n=0}^{+\infty} a_n z^n$ has only one pole of order $1$ on the Convergence circumference. Prove that $\sum_{n=0}^{+\infty} a_n z^n$ is divergent on the Convergence circumference.
Let $\...
- 3,603
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Entire function such that $f(\sin z)=\sin(f(z))$
Find all the entire functions $f$ such that
$$f(\sin z)=\sin(f(z)),\quad z\in\mathbb C.\tag {*}$$
The motivation to ask this question is an old post Find all real polynomials $p(x)$ that satisfy $\...
- 8,455
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Finding coefficients of laurent series for $\frac 1{1-\cos z}$ about zero
I know I'm wrong, but I fail to see why I'm wrong. My goal is to try and find the terms for the Laurent series of $f(z)=\frac{1}{1-\cos(z)}$ but I'm surely off.
$$\begin{align} f(z)&= \frac{1}{1-\...
- 1,846
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How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?
I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
- 1,846
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Residue of $ze^{\frac{1}{z}}$
I am trying to calculate the residue of $ze^{\frac{1}{z}}$, here's what I got:
We have a singularity at $z=0$.
We know that $e^w=\sum_{n=0}^\infty \frac{w^n}{n!}$ so $e^{\frac{1}{z}}=\sum_{n=0}^\infty ...
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