All Questions
Tagged with complex-analysis power-series
1,470
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Find the Laurent series for the following function [duplicate]
Find the Laurent series for $f(z)=\frac{2z-3}{z^2-3z+2}$ centered in the origin and convergent in the point $z=\frac32$, specifying it's convergence domain.
So I'm having troubles understanding what ...
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Find the power series expansion about $z=3$
Find the power series expansion for
$$f(z)=\frac{z^2}{4-z}$$
about the point $z=3$. So I know
$$\frac{1}{1-z}=\sum_{n=0}^\infty z^n$$
And we can write
$$\frac{z^2}{4-z}=\frac{z^2}{4-z}=\frac{1}{4}\...
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Find the disc of convergence of $\sum_{k=0}^\infty a_{k}z^{k}$, where $a_{0}=1$ and $a_{k+1}=2\sqrt{a_{k}}$
I want to find the convergence disc of the power series:
Find the disc of convergence of $\sum_{k=0}^\infty a_{k}z^{k}$, where $a_{0}=1$ and $a_{k+1}=2\sqrt{a_{k}}, \forall k \geq 0$.
Here is my try, ...
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Absolute convergence on boundary implies continuity of power series
Let $f(z) = \sum_{n=0}^{+\infty}c_nz^n$ be a complex power series with radius of convergence $R=1$. Suppose that the series of coefficients converges absolutely, i.e. $\sum_{n=0}^{+\infty}|c_n| < +\...
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Proof that $e^z=e^xe^{iy}$ [duplicate]
The proof goes like this:
$$
\begin{aligned}
e^z=e^{x+iy}&=\sum_{n=0}^\infty\dfrac{(x+iy)^n}{n!}\\
&=\sum_{n=0}^\infty\sum_{k=0}^n\dfrac{x^k(iy)^{n-k}}{k!(n-k)!}\\
&=\sum_{n=0}^\infty\...
- 2,578
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Can a nonzero complex power series have an uncountable set of complex roots?
Following Can a real power series have an uncountable number of real roots? and this essentially equivalent question about a sort of linear independence of powers of a real function, the natural thing ...
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2
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Prove that tan($Z$) = $\frac{e_1t - e_3t^3 + e_5t^5 - \cdots}{1 - e_2t^2 + e_4t^4 - \cdots}$
Let $e_r$ and $p_r$ denote the $r$-th elementary symmetric
function and power sum, respectively. Let $t$ be a formal variable
and define
$$Z := p_1t-p_3t^3/3+ p_5t^5/5- \cdots $$
Prove that the ...
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Is there such a thing as an intermediate value theorem in complex analysis?
I have an exercise I want to solve, but got stuck in the second part due to not having something like the intermediate value theorem for the complex plane. The exercise is: Let $\Omega$ be an open set ...
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3
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For holomorphic function $f: \Omega \to \mathbb{C}$ with $f^{(k)}(z_0) \in \mathbb{R}$ prove that $f(x) \in \mathbb{R}$ for every $(z_0 - r, z_0 +r)$
I have some difficulties with a question I have come across.
The question goes as follows:
Let $\Omega$ be an open set with $z_0 \in \Omega \cap \mathbb{R}$. Let $f: \Omega \to \mathbb{C}$ be ...
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Holomorphic extension of the power series $\sum_{n \geq 0} z^n $
So the power series $\sum\limits_{n \geq 0} z^n $ converges in the unit disc and is holomorphic in the unit disc as well with the derivative $\sum\limits_{n \geq 0} nz^{n-1}$. I am trying to prove ...
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Series of Analytic Functions is Analytic
Let $0 \in \mathbf{N}$. Let $P_m(x): [0,1] \to \mathbf{C}$ be bounded analytic functions for every $m\in \mathbf{N}$. Formally, define
$$
f(x) = \sum_{m\in \mathbf{N}}c_m P_m(x)\overline{P_m}(x),
$$
...
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To find Radius Of Convergence of Power series
Question
Let $\sum_{n=0}^\infty a_nz^n$ be a convergent series such that
$\lim_{n\to\infty} a_n = L$. Let $P(z)$ be a polynomial of degree $s$. Then what is the radius of convergence of series $\sum_{...
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Formally conjuguate to its linear model
I'm doing this problem in Complex Analysis :
Let $\lambda \in \mathbb C^*$ not be a root of $1$. Consider $F(z) = \lambda z + \sum_{n=2}^\infty a_nz^n$. Show there exists a power series $g(z) = \sum_{...
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Borel-Carathéodory theorem
I'm trying to prove the Borel-Carathéodory theorem. One of the steps given is this:
Let $f(z)=\sum_{n=0}^{\infty}a_nz^n$, $a_n \in \mathbb C$, $A(r)= \sup_{|z|\leq r} \Re(f(z)) $ for $r \in \mathbb ...
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Determining radius of convergence of the series $\displaystyle{\sum_{n=0}^{\infty} \sin(nz^n)}$
I have the following question: for what values of $z\in \mathbb{C}$ does the following series converges:
$\displaystyle{\sum_{n=0}^{\infty} \sin(nz^n)}$
My initial idea was to use the series expansion ...
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