Mathematics Stack Exchange
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Questions tagged [residue-calculus]

Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory. The method consists of using a closed contour on the complex plane to evaluate complex or real integrals.

2742 questions
15
votes
2 answers

Calculate residue at essential singularity

I know you can calculate a residue at an essential singularity by just writing down the Laurent series and look at the coefficient of the $z^{-1}$ term, but what can you do if this isn't so easy? For instance (a friend came up with this function):…
Willem Beek
  • 447
6
votes
1 answer

Variation of residue theorem?

Residue theorem can be stated informally as $$\oint_C f(z)dz=2\pi i\sum a_{-1}$$ A contour integral sums up all the $-1$ coefficients inside. Then, one would naturally ask: Is there something like $$\text{something of }f(z)=\sum a_{-2}$$ where…
Szeto
  • 11,159
6
votes
1 answer

Integrate by the method of residue

I want to integrate $$\int_{0}^{\infty}\frac{1}{(1+x)^5}dx$$ by the method of residue, but I have done only problems of simple poles, but this is a problem of fifth order pole. So I am stuck in it. Also, why the value of this integral is 0 if the…
Roshan Shrestha
  • 609
5
votes
1 answer

what are the residues at poles of $\frac{1}{1+\cosh{z}}$?

Consider the function $f(z)=\frac{1}{1+\cosh{z}}$. It has poles of order 2 at odd multiples of $\pi i$, but what are the residues at the poles? I've tried using $\frac{d}{dz} \Big((z-a)^2 f(z)\Big)$ for the residue at $a$, but get the answer to be…
Harry Macpherson
  • 1,824
4
votes
0 answers

What exactly is Complex Residue?

I know how to calculate residues of several functions, but I really don't know what residue is. Is there any way to really explain what residue is?
AstroFox
  • 343
3
votes
3 answers

What is the residue of $f(z)=\tan{z}$ at any of its pole ? Is the solution correct?

The residue of $$f(z)=\tan{z}$$ at any of its pole is, $$f(z)=\tan{z}=\frac{(z-\frac{\pi}{2})(\tan{z})}{(z-\frac{\pi}{2})}$$ $$\begin{align} \left({\operatorname{Res} {f(z)=\tan{z}; z=\frac{\pi}{2}}}\right)&=\lim_{z\to…
HOLYBIBLETHE
  • 2,770
3
votes
1 answer

Integral using residue theorem: $ \int_{-\infty}^\infty \frac{\cos(t)^2}{t^4 + 5 t^2 + 4} \, \mathrm dt$

We have the following problem given: $$ \int_{-\infty}^\infty \frac{\cos(t)^2}{t^4 + 5 t^2 + 4} \, \mathrm dt. $$ I thought that I could solve it using the residue theorem and by arguing that the integral along the great upper semicircle…
Martin Ueding
  • 4,523
3
votes
1 answer

residue of $\frac{1}{z^{2n}} \pi \cot(\pi z)$ at $z=0$

how to calculate the residue of $$\frac{1}{z^{2n}} \pi \cot(\pi z)$$ at $z=0$ I know the answer is $$(2\pi i)^{2n} \frac{B_{2n}}{(2n)!}$$ but I dont know how I saw an answer using "the coefficient extraction operator" but I dont know any thing…
mnsh
  • 5,875
3
votes
1 answer

Possible values of the integration $\frac{1}{2\pi i}\int_\gamma \frac{2i}{z^2 + 1}dz$

Possible values of $I := $ $\frac{1}{2\pi i}\int_\gamma \frac{2i}{z^2 + 1}dz$ where $\gamma $ is any closed curve in upper half plane not passing through $i$. My approach: There are two cases possible: Case I: $i$ lies inside $\gamma$, then using…
henceproved
  • 499
3
votes
2 answers

How the calculate $\int_0^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x$?

Just as the title say, consider the integral: $$I=\int_0^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x=\frac{1}{2}\int_{-\infty}^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x,$$ how to apply the residue theorem to get the answer?
van abel
  • 1,461
3
votes
1 answer

Rational Fresnel -type integral.

I have a question about an integral that looks like a great candidate for residues. $\displaystyle \int_{0}^{\infty}\frac{\cos(x^{2})}{x^{4}+1}dx-\int_{0}^{\infty}\frac{\sin(x^{2})}{x^{4}+1}dx=\frac{\pi\sqrt(2)}{4e}$. My difficulty arises in…
Cody
  • 14,054
2
votes
1 answer

Compute $\int_0^{\pi/2} \frac{1}{(\sinh t)^{2}+(\sin{\theta)^{2}}} d\theta$ by residues

We want to compute $\int_0^{\pi/2} \frac{1}{(\sinh t)^{2}+(\sin{\theta)^{2}}} d\theta$ with $t>0$ using residues. The first thing I want to do is using $z=e^{i\theta}$ to transform the integral to an integral over the unit circle. But I don't know…
DeMerlo
  • 301
2
votes
1 answer

Computing real definite integral using residue theorem

I am trying to solve integral $\int_{-∞}^\infty \frac{cos(x)dx}{(x^2+a^2)(x^2+b^2)}$ using residue theorem. I found example of that with very similair integral, which is $\int_{0}^\infty \frac{cos(x)dx}{x^2+b^2}$. Everything would be clear, except…
Vytautas Sirautas
  • 23
2
votes
2 answers

How do I find the residue of a function with a huge exponent?

How would I find the remainder of a function that has a huge exponent that would take ages to work out? Say I have something like this: $\frac{5x^{110} + x^4 - 7x^2 - 6}{x-1}$ I honestly don't know how to do division like this other than manually…
Cozen
  • 533
2
votes
2 answers

How to evaluate poles of fractional order (that do not lie on branch cuts!)

I would appreciate some guidance. I am trying to evaluate the residue of a function that looks like this: $f(z)=\frac{g(z)}{p-\sqrt z}$ at $z=p^2$ in the complex plane. $p>0$ is real, so I do not believe this pole lies on the branch cut, which I…
Daniel Cherno
  • 23
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