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}}...
- 172
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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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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
$$\sum_{n=0}^\infty 2^{\...
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Uniqueness of Two Series in an Intersection
I've been working on some problems involving series and have found myself applying the identity principle to show that two representations of a series will lead them being unique under some conditions....
- 379
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$\sum_{n=0}^{+\infty} a_n z^n$ is divergent on the Convergence circumference.
Let $f(z)=\sum_{n=0}^{+\infty} a_n z^n$ has only one pole of order $1$ on the Convergence circumference. Prove that $\sum_{n=0}^{+\infty} a_n z^n$ is divergent on the Convergence circumference.
Let $\...
- 3,573
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Entire function such that $f(\sin z)=\sin(f(z))$
Find all the entire functions $f$ such that
$$f(\sin z)=\sin(f(z)),\quad z\in\mathbb C.\tag {*}$$
The motivation to ask this question is an old post Find all real polynomials $p(x)$ that satisfy $\...
- 8,312
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Residue of $ze^{\frac{1}{z}}$
I am trying to calculate the residue of $ze^{\frac{1}{z}}$, here's what I got:
We have a singularity at $z=0$.
We know that $e^w=\sum_{n=0}^\infty \frac{w^n}{n!}$ so $e^{\frac{1}{z}}=\sum_{n=0}^\infty ...
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Understanding the Laurent expansion of a meromorphic function about $\infty$.
Suppose $f:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ were meromorphic, and suppose $f$ has a pole at $\infty$. I'm trying to understand the Laurent series of $f$ about $\infty$. By definition, $f$ has a ...
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Laurent Series question for Exponentials
I must find the Laurent series for $f(z) = \frac{e^z}{z^2}$ in powers of $z$ for the annulus $ |z| > 0$.
I wrote $f(z) = \frac{1}{z^2} \sum_{n=0}^{\infty} \frac{z^n}{n!} = \sum_{n=0}^{\infty} \frac{...
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Determine whether the complex power series converges at a point
I need to determine if a series $$\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) \...
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