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}}...
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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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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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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 ...
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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 ...
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2
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58
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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
$$\sum_{n=0}^\infty 2^{\...
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Uniqueness of Two Series in an Intersection
I've been working on some problems involving series and have found myself applying the identity principle to show that two representations of a series will lead them being unique under some conditions....
- 415
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34
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$\sum_{n=0}^{+\infty} a_n z^n$ is divergent on the Convergence circumference.
Let $f(z)=\sum_{n=0}^{+\infty} a_n z^n$ has only one pole of order $1$ on the Convergence circumference. Prove that $\sum_{n=0}^{+\infty} a_n z^n$ is divergent on the Convergence circumference.
Let $\...
- 3,603
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Entire function such that $f(\sin z)=\sin(f(z))$
Find all the entire functions $f$ such that
$$f(\sin z)=\sin(f(z)),\quad z\in\mathbb C.\tag {*}$$
The motivation to ask this question is an old post Find all real polynomials $p(x)$ that satisfy $\...
- 8,455
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Residue of $ze^{\frac{1}{z}}$
I am trying to calculate the residue of $ze^{\frac{1}{z}}$, here's what I got:
We have a singularity at $z=0$.
We know that $e^w=\sum_{n=0}^\infty \frac{w^n}{n!}$ so $e^{\frac{1}{z}}=\sum_{n=0}^\infty ...
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1
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Understanding the Laurent expansion of a meromorphic function about $\infty$.
Suppose $f:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ were meromorphic, and suppose $f$ has a pole at $\infty$. I'm trying to understand the Laurent series of $f$ about $\infty$. By definition, $f$ has a ...
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Laurent Series question for Exponentials
I must find the Laurent series for $f(z) = \frac{e^z}{z^2}$ in powers of $z$ for the annulus $ |z| > 0$.
I wrote $f(z) = \frac{1}{z^2} \sum_{n=0}^{\infty} \frac{z^n}{n!} = \sum_{n=0}^{\infty} \frac{...
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Determine whether the complex power series converges at a point
I need to determine if a series $$\sum\limits_{n=1}^{\infty} \frac{(z-1+i)^{2n-1}}{5^n(n+1)ln^3(n+1)}$$ converges at a point $z_1 = -1$
After substituting the point, I got: $$ \sum\limits_{n=1}^{\...
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Understanding the conditions for the Lagrange Inversion Formula
Lagrange Inversion Formula: Let $A(u) = \sum_{k \ge 0} a_k z^k$ be a power series in $\mathbb{C}[[z]]$ with $a_0 \ne 0$. Then the equation
$$B(z) = zA(B(Z)) \qquad (1)$$
has a unique solution $B(z) \...
- 4,222
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Taylor series expansion of $f(z)=\frac{\log(1+z^2)}{(1-z)\sinh z}$
I need to find the first 3 terms of the Taylor expansion around $z=0$ of the complex function:
$$f(z)=\frac{\log(1+z^2)}{(1-z)\sinh z}$$
I tried the following way (obviously the radius of convergence ...
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A detail in the proof of the Theorem of the Uniqueness of the Laurent expansion
So the theorem is as follows
Let $f$ be holomorphic in the ring $A=\{z\in\mathbb{C}\,|\,r_1<|z-z_0|<r_2\}$. Also assume that
$$
f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n+\sum_{n=1}^\infty \frac{b_n}...
- 41
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Proof of an identity involving infinite series of Bessel functions
Recently, I came across the following identity among first-kind Bessel functions, namely
$$2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left(x\,J_1(x)-J_0(x)\right)$$
It's ...
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Meaning of "$f$ has a power series expansion around $p$"
In Complex Analysis by Donald Marshall (page 29), there is an exercise problem that starts with "Suppose $f$ has a power series expansion at $0$ which converges in all of $\mathbb{C}$. " ...
- 1,268
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If $f$ is analytic on $B(z_0, R)$, does that imply $f$ has a power series expansion centered at $z_0$ with radius of convergence $R$? [duplicate]
I know that if a power series $f:=\sum_{n=0}^{\infty}a_n(z-z_0)^n$ has radius of convergence $R$, then $f$ is analytic on $B(z_0,R)$. I wonder if the converse of this statement is true. That is, ...
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103
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If $f\colon\mathbb{C}\to\mathbb{C}$ is continuous and analytic off $[-1,1]$ then $f$ is entire.
I know this question had already been asked and answered here.
If $f: \mathbb{C} \to \mathbb{C}$ is continuous and analytic off $[-1,1]$ then is entire.
However I was trying power series approach.
By ...
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1
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Fast way to find the __regular__ part of the Laurent expansion of a function at a pole?
To evaluate a certain limit, I need to calculate the first term in the regular part of the Laurent expansion of the function
$$\frac{\pi}{\sin \frac{\pi(s+1)}{2}}$$
around -1 (should be the same thing ...
- 1,197
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104
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Power Series with digits of $\pi$
Sorry if this has already been asked, but I haven't found a post.
Can anything be said about the function
$$f(z)=\sum_{n=0}^\infty a_n z^n$$
where the $a_n$'s are the digits of $\pi= 3.14159...$, so $$...
- 6,277
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A confusion about the radius of convergence in Ahlfors' "Complex Analysis"?
On the third edition of Ahlfors' Complex Analysis, page 39 Theorem 2 it states: The derived series $\sum_{1}^{\infty}na_n z^{n-1}$ has the same radius of convergence, because $\sqrt[n]n \rightarrow 1$....
- 174
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I find a "mistake" on p.40 in Ahlfors' "Complex Analysis"?
On the third edition of Ahlfors' Complex Analysis, page 40 Theorem 2 it states: we conclude that
\begin{equation*}
\left|\frac{R_n(z)-R_n(z_0)}{z-z_0}\right| \leq \sum_{k=n}^{\infty} k|a_k|\rho^{k-1}
\...
- 174
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How to prove $f(z)=a_0+a_1z+\cdots+a_nz^n+\cdots$, which has infinite items, is an analytic function? [duplicate]
It is easy to prove the following four statements.
(i) If $f$, $g$ are analytic functions and $f^{\prime}$, $g^{\prime}$are continuous then $(f+g)(z)$ is analytic and $(f+g)^\prime(z)=f^\prime(z)+g^\...
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Series expansion of $z^{1/3}$ at z=1
Obtain the series expansion of $f(z)=z^{1/3}$ at z=1 such that $1^{1/3}=\frac{-1+i\sqrt{3}}{2}$
The way I've done it is the following:
I need $1^{1/3}=e^\frac{i\arg{1}}{3}=e^{i2\pi/3}$, so any branch ...
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2
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Prove that If singular part of Laurent series has infinite many terms, then $\lim_{z\to z_0}(z-z_0)^mf(z)$ doesn't exist for all nautural number $m$.
Given $f$ an analytic function in open $D \subset \mathbb C$, $z_0$ is an isolated singularity defined as $B(z_0;r)\backslash\{z_0\} \subset D$, then know that $f$ can be written as an expansion of ...
user1194243
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Union of two circles is star domain with center any of the circles center
I'm trying to prove that a power series $f(z)=\sum_{i=0}^\infty a_n (z-a)^n$ with a positive finite radius of convergence $r$, then $\exists z_0, |z_0-a|=r$ such that $\nexists \Omega, D(a,r) \subset \...
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Power Series Expansion of $\frac{z}{e^{z} - 1}$ at $z = 0$ and $z = 1$.
I'm trying to find the Power Series expansion and Radius of Convergence for the function
$$f(z) = \frac{z}{e^{z} - 1}$$ at points $z = 0$ and $z = 1$.
When it comes to the radii of convergence, $e^{i\...
