# Tagged Questions

1answer
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### Is Cauchy's formula apt for evaluating this integral

I'm trying to evaluate the following. $$\frac{1}{2i}\int_{-\infty}^\infty \frac{s \sin{(sr)}}{(s-k)(s+k)}\mathrm{d}s,$$ with $k$ and $r$ being real constants. The integral could be written as ...
0answers
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### Complex exponential integral: Mathematica and MATLAB give unexpected results

I currently compare analytical vs. numerical evaluation of the complex exponential integral and find mismatches: The imaginary part differs by $\pm \pi$ and the real part has a large error when ...
0answers
73 views

### Residue Integral: $\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x$

Inspired by some of the greats on this site, I've been trying to improve my residue theorem skills. I've come across the integral $$\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x,$$ where ...
1answer
124 views

### How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
0answers
52 views

### Choose appropriate contour for a complex integral

I have a problem to solve integral $$I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}}$$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...
1answer
261 views

### Evaluation of $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

I need some hints, clues for getting the closed form of $$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
0answers
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### Gaussian integral involving $\cos\circ\sin$

I stumbled upon an integral of the form $$\int_{\mathbb R} e^{-x^2/2}\cos(a\sin (bx+ic))\,{\mathrm d}x$$ for some real constant $a,b,c$. Has anybody ever seen such an integral? Mathematica doesn't ...
2answers
62 views

### Cauchy distribution characteristic function

I know that it's easy to calculate integral $\displaystyle\int_{-\infty}^{\infty}\frac{e^{itx}}{\pi(1+x^2)}dx$ using residue theorem. Is there any other way to calculate this integral (for someone who ...
2answers
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3answers
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### Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$ Which function/ contour should I consider ?
4answers
292 views

### Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as ...
1answer
50 views

### Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
1answer
110 views

### Evaluation of a definite integral

I want to find the best way to show $\int_0^\infty\dfrac{x^{2m}}{x^{2n}+1}\,dx=\dfrac{\pi}{2n}\operatorname{csc}\left(\dfrac{2m+1}{2n}\pi\right)$, where $0\leq m<n$. It's easy to verify some ...
0answers
206 views

### Contour integral with branch point

As preparation for my exam I "invented" the following problem as an exercise to see whether I understand how to work with branch points. $f(z) = \frac{z}{\sqrt{z^2+1} (z^2 +a^2)}$ The goal is to ...
3answers
70 views

### Help with a contour integration

I've been trying to derive the following formula $$\int_\mathbb{R} \! \frac{y \, dt}{|1 + (x + iy)t|^2} = \pi$$ for all $x \in \mathbb{R}, y > 0$. I was thinking that the residue formula is the ...
1answer
334 views

### Prove the improper integral of the Gamma function $\Gamma(t)$ converges for $z \in \mathbb C$ with $Re(z) > 0$:

Prove the improper integral of the Gamma function $\Gamma(t)$ converges for $z \in \mathbb C$ with $Re(z) > 0$: The gamma function $\Gamma(t)$ is defined by the following improper integral ...
0answers
66 views

### Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe ...
1answer
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0answers
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### perturbative series expansion of integral via complex integration

Define for real $x>0$ and $\epsilon>0,$ the function $$f(x,\epsilon):= \int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \frac{1}{\sinh^2 s/2} e^{-sx}.$$ Question: is it possible to compute ...
3answers
183 views

### Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx.$$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
1answer
49 views