3
votes
2answers
104 views

Improper Integral of $x^2/\cosh(x)$

I need to compute the improper integral $$ \int_{-\infty}^{\infty}{\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x} $$ using contour integration and possibly principal values. Trying to approach this as ...
0
votes
1answer
67 views

Clarification of Cauchy Principal Value and use of Contour Integration

I am evaluating the improper integral $\int_{-\infty}^\infty{\frac{\sin^3 x}{x^3}dx}$; I am also told to show that this is equal to its principal value, and use this fact to evaluate the integral. I ...
4
votes
2answers
173 views

How to compute $I_n=\int_{-\infty}^{+\infty}\mathrm{d}x\frac{x^{2n}}{\cosh^2 x}$?

I'd like to compute: $$ I_n = \int_{-\infty}^{+\infty}\mathrm{d}x\frac{x^{2n}}{\cosh^2 x}. $$ We have, quite easily: $$ I_0 = \int_{-\infty}^{+\infty}\mathrm{d}x\frac{1}{\cosh^2 x}=\left[\tanh ...
10
votes
1answer
243 views

integral $\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$

Here is a seemingly challenging integral some may try their hand at. $$\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$$ It appears to be ...
2
votes
0answers
112 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 ...
2
votes
3answers
65 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 ...
1
vote
2answers
65 views

Does $ \int x^{-2} \, \mathrm{d}{x} $ have a singularity?

How do you integrate $ \dfrac{1}{x^{2}} $ from $ 0 $ to, say, $ a $? Can you get a principal value? What is the divergence: $ + \infty $ or $ - \infty $?
3
votes
0answers
77 views

Integral $\int_{0}^\infty\frac {(1-{{e}^{-i (q-p)t}})ln(|p^2-p_0^2|)}{(q-p)({{ p}}^{2}-{{p_1}}^{2})({{p}}^{2}-{{p_2} }^{2})}dp$

I am trying to get a closed form analytic result for the integral $$\int _{0}^{\infty }\!{\frac {\left(1-{{\rm e}^{-i \left( {q}-{p} \right) t}}\right){\rm ln}(|p^2-p_0^2|)}{ ( {q}-{p} ) \left( {{ ...
5
votes
1answer
115 views

Prove $\sin a=\int_{-\infty}^{\infty}\cos(ax^2)\frac{\sinh(2ax)}{\sinh(\pi x)} \operatorname dx$

Derive the integral representation $$\sin a=\int_{-\infty}^{\infty}\cos(ax^2)\frac{\sinh(2ax)}{\sinh(\pi x)}dx$$ for $|a|\le \pi/2$.
1
vote
1answer
161 views

Integrals involving Hermite Polynomials

Could you please tell me, How to evaluate this integral which involve hermite polynomials, $\int_{-\infty}^\infty e^{-ax^2}x^{2q}H_m(x)H_n(x)\,dx=?$ where $H_n$ is the $n$-th Hermite polynomial ...
1
vote
2answers
139 views

Integral through Fourier Transform and Parseval's Identity

$$ \int_{-\infty}^{\infty}{\rm sinc}^{4}\left(\pi t\right)\,{\rm d}t\,. $$ Can you help me evaluate this integral with the help of Fourier Transform and Parseval Identity. I could not see how it is ...
5
votes
3answers
127 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 ...
1
vote
1answer
131 views

Evaluate an Integral involving Gaussian divided by square root of a quartic polynomial

Could you please tell me, How to evaluate the integral, $\int_{-\infty}^\infty\int_{-\infty}^\infty\dfrac{e^{-a(x^2+y^2)}}{\sqrt{k^2+\beta^2(x^2-y^2)^2}}dx~dy$ I already have obtained a series ...
3
votes
3answers
194 views

Are the integration contours of this improper integral properly selected?

I have been recently trying to review some topics on improper integrals. The Integral I am trying to solve is: $$ \int_0^\infty {log(x) \over x^2 -1} dx $$ The branch cut of the $log(x)$ is ...
3
votes
3answers
125 views

Evaluating $\int_{0}^{\infty} \frac{1}{1+x^{3/2}}\,\textrm{d}x$

I'm trying to evaluate this integral using contour integration (over a Riemann surface), but I'm stuck at the step where I need to calculate the residues. The roots of $1+z^{3/2}$ are $1$ and ...
3
votes
1answer
175 views

Evaluating $\int_0^\infty \frac{\cos(ax)-e^{-ax}}{x \left(x^4+b^4 \right)}dx$

How can we evaluate $$\int_0^\infty \frac{\cos(ax)-e^{-ax}}{x \left(x^4+b^4\right)}dx \quad a,b>0$$ using Complex Analysis? This problem was given in a Complex Analysis book which I was reading. ...
3
votes
3answers
201 views

Evaluating an improper integral that involves $\exp(-|x|)$

I am trying to prove that the function $f:\mathbb C\setminus\mathbb R\rightarrow\mathbb C$ defined by $$ f(z) := \frac{1}{2\pi i}\int_{-\infty}^\infty\frac{\exp(-|x|)}{x-z}dx $$ is holomorphic. I ...
5
votes
1answer
103 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 ...
4
votes
2answers
379 views

Integrating $\int_0^\infty \sin(1/x^2) \, \operatorname{d}\!x$

How would one compute the following improper integral: $$\int_0^\infty \sin\left(\frac{1}{x^2}\right) \, \operatorname{d}\!x$$ without any knowledge of Fresnel equations? I was thinking of using ...
18
votes
3answers
603 views

Evaluating the integral $\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x}$

Many recent questions have been asked here similar to this integral $$\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x} = 2.39587\dots$$ whose "closed form" I cannot seem to figure out. I have ...
2
votes
1answer
227 views

How does this calculation of $\int_0^\infty(\sin^2x)/x^2\ dx$ work?

I'm having trouble with two steps in a calculation of $$\int_0^\infty\left(\frac{\sin x}{x}\right)^2\ dx$$ in a book. They take the contours $C_R$ composed of upper half-circles ...
5
votes
4answers
416 views

$\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint.

I have to calculate $$\int_0^\infty\frac{\log x dx}{x^2-1},$$ and the hint is to integrate $\frac{\log z}{z^2-1}$ over the boundary of the domain $$\{z\,:\,r<|z|<R,\,\Re (z)>0,\,\Im ...
6
votes
1answer
159 views

Is this a correct way to calculate $\int_{-\infty}^\infty\frac{x\sin(\pi x)}{x^2+2x+5}dx?$

I have this integral to calculate: $$I=\int_{-\infty}^\infty\frac{x\sin(\pi x)}{x^2+2x+5}dx.$$ I think I have done it, but I would like to make sure my solution is correct. I take the function ...
4
votes
1answer
127 views

Integral using residue theorem (maybe)

I came across the following integral in a book (Kato's Perturbation Theory for Linear Operators, $\S$3.5): $\int_{-\infty}^\infty (a^2+x^2)^{-n/2}\,dx$ where $n$ is a non-negative integer and $a$ is ...
5
votes
3answers
253 views

Improper integration involving complex analytic arguments

I am trying to evaluate the following: $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$/ Any help will be much appreciated.
1
vote
0answers
75 views

What is suitable contour shape for $\int_0^\infty\dfrac{b^2+2ab+k}{b(b^2+ab+l)}e^{bx}~db$

$\int_0^\infty\dfrac{b^2+2ab+k}{b(b^2+ab+l)}e^{bx}~db$ . What kind of contour is suitable for this integral?
2
votes
2answers
1k views

Evaluate improper integral $(\cos(2x)-1)/x^2$

Consider the following improper integral: \begin{equation} \int_0^\infty \frac{\cos{2x}-1}{x^2}\;dx \end{equation} I would like to evaluate it via contour integration (the path is a semicircle in ...
0
votes
1answer
666 views

How to calculate the principal part of improper integral?

How to calculate the principal part of this improper integral via contour integration? \begin{equation} P\int_{0}^{+\infty}\frac{dx}{x^2+x-2} \end{equation} I have seen some examples where you ...
9
votes
1answer
1k views

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 $0<k<1$ is fixed. But ...