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.

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How to calculate $ \int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4} $?

I would like to calculate $$\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a half-circle part ($\alpha_r$) and a path connecting the…
13
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Inverse Laplace Transform of $e^{-\sqrt{s^2+s}}$.

I've come across this problem trying to find an integral representation to a PDE (damped wave equation with initial conditions). What I would like to do is compute $$f(t) = \mathcal{L}^{-1}\left[e^{-\sqrt{s^2+s}}\right](t)$$ via the bromwich…
13
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Closed form of an integral involving Lambert function

I'm trying to compute the following integral explicitly. $$I=\int_{0}^{+\infty} dx \left(1+\frac{1}{x}\right) \frac{\sqrt{x}}{e^{-1}+xe^x}$$ The best I managed to do is to do a change of variable $x=W(y)$, where W is the Lambert function. The…
13
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Residue integral: $\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx$ with $0 \lt a \lt 1$.

I'm self studying complex analysis. I've encountered the following integral: $$\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx \text{ with } a \in \mathbb{R},\ 0 \lt a \lt 1. $$ I've done the substitution $e^x = y$. What kind of contour can I…
WLOG
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Evaluate $\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx$ (solution verification)

I tried to find the integral $$I=\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx \tag1$$ I substituted $x=t^2, 2tdt=dx$ and chose $\log x$ and $\sqrt{x}$ to be principal values. We have $$\int_0^{\infty} \frac{\log x}{(x-1)\sqrt{x}}dx=2…
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Contour Integration $\int_0^1\frac1{\sqrt[n]{1-x^n}}dx$

I want to compute: $$\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the $n$th roots of unity (singularities in this case). The…
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Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$

I'm trying to compute $$\int_{-\infty}^\infty \frac{\sinh(kx)}{\sinh(x)}e^{-i\omega x} \ dx$$ i.e. the Fourier transform of $x\mapsto \frac{\sinh(kx)}{\sinh(x)}$, where $0
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$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus

I'm trying to compute $\displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$, $(0
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How to show $\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})$? ($a\ge0$)

$$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a}), \ a\ge0$$ I tried to solve but came up with $\pi(2-e^{-a}) $. Could you tell me where did I do the mistake? if $x=z$ then $dz=dx$ $$\int_\gamma \frac{e^{iaz}}{z(z^2+1)}\quad…
12
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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
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Evaluation by methods of complex analysis $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm{dx}$

How would we evaluate the below integral by methods of complex analysis? $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm{dx}$$ I asked the question a while ago, but at that time I didn't specify this requirement. Now, I'm only interested in a complex…
user 1591719
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A Ramanujan-type identity: $11\sum_{n=1}^{\infty}\frac{n^3}{e^{2n\pi}-1}-16\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{1}{48}$

Out of curiosity, why it is these sums yield a rational answer? $$11\sum_{n=1}^{\infty}\frac{n^3}{e^{2n\pi}-1}-16\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{1}{48}$$ I found this identity during the observing ramanujan identiy via a wolfram sum…
user335850
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2 answers

Integrating $\int_{-\infty}^\infty \frac{1}{1 + x^4}dx$ with the residue theorem

Calculate integral $$\int\limits_{-\infty}^{\infty}\frac{1}{x^4+1} dx$$ with residue theorem. Can I evaluate $\frac 12\int_C \dfrac{1}{z^4+1} dz$ where $C$ is simple closed contour of the upper half of unit circle like this? And find the roots of…
ELEC
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Solve $\int _{x=0}^{\infty }\int _{t=-\infty }^{\infty }\exp \left(\frac{-a t^2+i b t}{3 t^2+1}+i t x\right)\frac{x}{3 t^2+1}\mathrm{d}t\mathrm{d}x$

How to solve this double integral? $$f(a,b)=\int _{x=0}^{\infty }\int _{t=-\infty }^{\infty }\exp \left(\frac{-a t^2+i b t}{3 t^2+1}+i t x\right)\frac{x}{3 t^2+1}\mathrm{d}t\mathrm{d}x$$ $$\text{with }a>0,b\in \mathbb{R},i^2=-1$$ Known special…
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Calculus of residue of function around poles of fractional order (complex analysis)

The complex function $f(z)=\frac{1}{\sqrt{z^2+r_0z}}$ with $r_0>0$ has two poles (at $z=0$ and $z=-r_0$). But they are not simple poles. They are poles of fractional order. Am I right? How I can calculate residue of the function at the poles?…
Vahid
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