All Questions
18
questions
2
votes
1
answer
60
views
Confusion regarding the use of the residue theorem in complex analysis.
I have been studying about the residue theorem and I am a little confused about the way the theorem is used.For example considering the function: $$f(z)=\frac {1}{1-z}+\frac{2}{2-z} $$ this function ...
- 306
0
votes
1
answer
109
views
Show recursive formula for coefficients in power series, and use them to evaluate a contour integral
This is a two part question. First I'm being tasked with the following:
Given $f(z)=\frac{e^{iz}}{z^2+2}$, show that the coefficients for the Taylor expansion with center 0 follow the recursive ...
- 314
11
votes
2
answers
681
views
Ramanujan's Master Theorem relation to Analytic Continuation
$\DeclareMathOperator{Re}{Re}$
To provide some background, this is a question based on establishing the identity $$\int_0^\infty \frac{v^{s-1}}{1+v}\,dv=\frac{\pi}{\sin \pi s},\qquad 0<\Re s<1$$...
- 24.8k
0
votes
1
answer
672
views
What's the power series of $\log$ at $=1$?
A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka
(Exer 7.28(a)) Find a power series for $\frac1z$ centred at $z_0=1$. --> I got this.
(...
- 13.7k
0
votes
2
answers
94
views
Find the integral of $(z^5+3z^3+4z^2+1)*e^{\frac1{z^2}}$dz
I first take the power series of $e^{\frac{1}{z^2}}$ as $\sum\limits_{n=0}^{\infty} \frac{\left(\frac{1}{z^2}\right)^n}{n!}$, multiple by that by ($z^5+3z^3+4z^2+1$) and end up with $\int_{|z|=1}\sum_{...
- 97
3
votes
0
answers
175
views
Verifying my proof of Liouville's theorem?
Is my proof to Liouville's Theorem, as formally discussed in $\text{Proposition (1.0)}$ correct ?.
$\text{Proposition (1.0)}$
A bounded holomorphic function $f$ (i.e $|f(z)| \leq M$) on $C$ ...
- 1,061
1
vote
2
answers
163
views
$f(z)=\sum_{n=0}^\infty a_n(z-a)^n$ converges for $|z-a|<s$ , then for $0<r<s$ , $\int_0^{2\pi} |f(a+re^{it})|^2 dt=2\pi\sum |a_n|^2r^{2n}$? [duplicate]
Suppose $f(z)=\sum_{n=0}^\infty a_n(z-a)^n$ converges for $|z-a|<s$ .
Then how to show that for $0<r<s$ , $\dfrac 1{2\pi}\int_0^{2\pi} |f(a+re^{it})|^2 dt=\sum_{n=0}^\infty |a_n|^2r^{2n}$ ?
user228168
5
votes
0
answers
359
views
Deriving a Series Representation of the Bessel Function of the First Kind
I've tried to use an integral representation of the Bessel Function of the First Kind $J_n(x)$ to derive a series representation of the function. My end result is pretty close to the answer that it ...
- 3,309
6
votes
2
answers
309
views
How does one prove $\int_0^\infty \frac{\log(x)}{1 + e^{ax}} \, dx = -\frac{\log(2)(2\log(a) + \log(2))}{2a}$ for $a > 0$?
Link to WolframAlpha's assertion. Here's my attempt. Using the substitution $t = ax$, we can show the integral is equal to
$$ \frac{1}{a} \int_0^\infty \frac{\log(t)}{1 + e^t}\, dt -\frac{\log(a)}{a} ...
- 181
5
votes
3
answers
599
views
Interesting Series with Zeta Function
I was trying to find another representation for the value of an integral when I found the following series:
$$f (z)=\sum_{n \in \Bbb N} (-z)^{n-1}\frac {(2^n-1)}{2^n}\zeta (n+1) $$
For $|z|<1$ and ...
- 153
1
vote
1
answer
67
views
Prove that $f^{(k)}(0) = \frac{k!}{2πi} \int_{|z|=1} \frac{f(z)}{z^{k+1}} dz$
Let $f(z)$ be a convergent power series with convergent radius greater than 1. Prove that $$f^{(k)}(0) = \frac{k!}{2πi} \int_{|z|=1} \frac{f(z)}{z^{k+1}} dz$$
Since $f(z)$ is a convergent power ...
- 361
1
vote
1
answer
85
views
Finding the coefficients of $h(z)$ laurent series
Consider:
$$h(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$
Find the coefficients $a_n$ of the Laurent Series of $h(z)$ centered at $z=-2$
I got this from the approach here: Infinite sum complex analysis
...
- 89
2
votes
1
answer
112
views
Showing integral on contour tends to zero
I'm trying to prove:
$$\int \frac{e^{t(z+\frac{1}{z})}}{z^2} = \sum_0 ^{\infty} b_m t^{2m+1}$$
Where the integral is over a contour centre the origin, radius R, and the $b_m$ are some coefficients.
...
- 3,805
2
votes
2
answers
106
views
If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero
If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero.
How can I able to prove the above problem without ...
- 71
2
votes
1
answer
873
views
Evaluate a complex integral using power series expansions
Using power series expansions, evaluate the integral
$$\int_{\gamma_r}\sin\left(\frac{1}{z}\right)dz.$$
where $\gamma_r:[0,2\pi]\rightarrow \mathbb C$ is given by $\gamma_r(t)=r(\cos t + i\sin t)$...
