All Questions
156
questions
1
vote
1
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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
1
vote
1
answer
46
views
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 ...
- 113
2
votes
0
answers
71
views
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
0
votes
1
answer
92
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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
vote
0
answers
51
views
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
vote
0
answers
29
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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 ...
- 57
0
votes
2
answers
73
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Solving Power Series Equations if we introduce Logarithmic Terms
If we have a complex power series equation like
$$
\sum_{n=0}^{\infty} a_n z^n = \sum_{n=0}^{\infty} b_n z^n
$$
then we can conclude $a_n = b_n$. We can see this by viewing $z^n$ as basis elements, or ...
- 351
0
votes
0
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47
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Why this equation holds?
Can someone help me why does this equation holds:
$\sum_{\rho}(1/2 + t -\rho)^{-s}= e^{i\pi s/2}\sum_{k=1}^{\infty}(\tau_k+it)^{-s}+e^{-i\pi s/2}\sum_{k=1}^{\infty}(\tau_k-it)^{-s}$ for $\rho=1/2 \pm ...
- 87
0
votes
1
answer
153
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Is this the expansion for any known function?
The expansion is $$\sum_{n=1}^{\infty}\frac{x^n}{n!(n-1)!}\left[c(1+c)\dots((n-1)^2+c)\right]$$
So the first 3 terms are $cx$, $\dfrac{c(1+c)}{2}x^2$, $\dfrac{c(1+c)(4+c)}{12}x^3$.
- 35
0
votes
1
answer
64
views
Finding the Maclaurin series for the given function.
I am dealing with a problem that says:Find the Maclaurin series for the function $f(x)=\ln(1+x+x^2)$.
I have tried to write the expression inside the brackets exactly, $1+x+x^2$ as a product of two ...
- 5
0
votes
0
answers
41
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When is it necessary to expand out the first term(s) of a power series?
I'm comfortable with the process of finding the Laurent series for a complex function, but in many of the answers from the textbook the first few terms will be expanded from it. Since I'm teaching ...
- 381
1
vote
0
answers
70
views
Is there a "closed form" for the taylor series of $\sqrt{1+f[x]}$?
I would like a simple, computationally viable expansion in $x$ of $\sqrt{1+f[x]}$. If we taylor expand $f[x] $ itself inside the square root to first couple of orders, and expand the square root for ...
- 67
1
vote
1
answer
72
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Expansion of an analytic function in the unit disk
I am trying to solve the following problem:
Suppose that $f(z) = \sum_{k=0}^{\infty} c_kz^k$ is an analytic function in $\mathbb{D} = \{z\in\mathbb{C}:|z|<1\}$. Prove that $F(z)=\sum_{k=0}^{\infty}...
- 973
0
votes
1
answer
738
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Power series expansion of 1/(1+z)^2 around z=1 [duplicate]
I'm struggling with getting the power series/Taylor series expansion of $f(z) = \frac{1}{(1+z)^2}$ around $z_0 = 1$.
Usually, I would do a partial fraction decomposition, and then do some re-arranging ...
- 9
0
votes
1
answer
60
views
Radius of convergence of an infinite series by using a theorem.
Before I ask the question, I will explain a bit of theory and then give the question.
Theorem 3:
The derived series of a power series has the same radius of convergence as the original series.
The ...
- 21