2
votes
1answer
65 views

Find the Laurent series of the function and give the residue

I have $f(z)={z^2\over {z^2-1}}$. I want to find the Laurent series of $f$ and the residue at the point $z_0=1$. Can I say that $f$ has a pole of order 2 at $z=1?$ Or is that only used when the ...
4
votes
0answers
89 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
1
vote
2answers
97 views

Infinite sums and integrals using residues

I have no idea how to solve these two, any help? $\mathtt{i)}$ $$\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}\frac{e^{tz}}{\sqrt{z+1}}dz$$ $$ a,t\gt0$$ $\mathtt{ii)}$ $$ \sum_{n=1}^\infty ...
0
votes
1answer
43 views

Laurent Seies and Res

Prove that for any Laurent series f(t) one has "Res(f') = 0"? I know for a Laurent series of a complex function f is a representation of that function as a power series which includes terms of ...
7
votes
1answer
139 views

How to find closed form formula for a sum

I am a PhD student in electrical engineering. I need to find a closed form formula for the following series: $$\sum_{k=1}^{\infty}\frac{1}{2}A_k^2e^{-k^2\sigma_m^2}(e^{k^2\sigma_m^2}-1)$$where $A_k= ...
2
votes
2answers
86 views

Show these approximations of $\cos$, $\sin$ and $\tan$ are exact.

A while back I was looking for an approximation to $\cos(x)$ and I constructed a polynomial with zeros in the same places as the first few zeros of $cos(x)$: $$c_n(x) = \frac{\prod_{i=1}^n ...
1
vote
1answer
136 views

Evaluating series by contour integration, the residue theorem, and cotangent

I'm trying to understand this section in Tristan Needham's book Visual Complex Analysis about what he says is a standard method for evaluating series via a contour integral. My specific question is ...
2
votes
1answer
156 views

Calculating residue

I really don't know too much about residue or how to calculate it, but just out of curiosity how would I find the residue of $e^{-1/z^2} $ at $ z=0$? Would it make sense to do so? And would this value ...
5
votes
1answer
111 views

Finding a generalization for $\int_{0}^{\infty}e^{- 3\pi x^{2} }\frac{\sinh(\pi x)}{\sinh(3\pi x)}dx$

$\;\;\;\;$I was reading the introduction of Paul J. Nain's book "Dr. Euler's fabulous formula" where he talks about the sense of beauty in mathematics and quotes the G.N.Watson as saying that a ...
0
votes
1answer
86 views

A improper integral with series

Show that $$\begin{align} & \int_{-\infty }^{+\infty }\frac{\cos \alpha x}{( \beta^2+x^2 )( ( \beta +1 )^2+x^2 )\cdots ( ( \beta +n )^2+x^2 )} \, \text{d}x = 2\pi \sum\limits_{k=0}^n ( -1 )^k ...
6
votes
4answers
374 views

A generalized integral need help

I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: ...
12
votes
2answers
616 views

Sum of the squares of the reciprocals of the fixed points of the tangent function

The sum of the squares of the reciprocals of the positive fixed points of the tangent function is $1/10$. I've seen this proved by means of residues, but I don't remember the details. I've also ...