All Questions
Tagged with complex-analysis power-series
1,470
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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 ...
- 711
1
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1
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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)$$
It's ...
1
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1
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Meaning of "$f$ has a power series expansion around $p$"
In Complex Analysis by Donald Marshall (page 29), there is an exercise problem that starts with "Suppose $f$ has a power series expansion at $0$ which converges in all of $\mathbb{C}$. " ...
- 1,268
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If $f$ is analytic on $B(z_0, R)$, does that imply $f$ has a power series expansion centered at $z_0$ with radius of convergence $R$? [duplicate]
I know that if a power series $f:=\sum_{n=0}^{\infty}a_n(z-z_0)^n$ has radius of convergence $R$, then $f$ is analytic on $B(z_0,R)$. I wonder if the converse of this statement is true. That is, ...
- 1,268
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If $f\colon\mathbb{C}\to\mathbb{C}$ is continuous and analytic off $[-1,1]$ then $f$ is entire.
I know this question had already been asked and answered here.
If $f: \mathbb{C} \to \mathbb{C}$ is continuous and analytic off $[-1,1]$ then is entire.
However I was trying power series approach.
By ...
1
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1
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Fast way to find the __regular__ part of the Laurent expansion of a function at a pole?
To evaluate a certain limit, I need to calculate the first term in the regular part of the Laurent expansion of the function
$$\frac{\pi}{\sin \frac{\pi(s+1)}{2}}$$
around -1 (should be the same thing ...
- 1,197
2
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104
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Power Series with digits of $\pi$
Sorry if this has already been asked, but I haven't found a post.
Can anything be said about the function
$$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
2
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1
answer
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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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2
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146
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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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1
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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^\...
- 174
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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 ...
- 63
1
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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 \...
- 1,242
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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\...
