Tagged Questions
5
votes
1answer
66 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
3answers
75 views
Cauchy principal value of $\int_{\infty}^{-\infty}e^{-ax^2}\cos(2abx) \,dx$
How do I find out the Cauchy Principal value of $\int_{-\infty}^{\infty}e^{-ax^2}\cos(2abx) \,dx\,\,\,\,\,\,\,\,a,b>0$ using complex integration? The answer is $\sqrt{\frac{\pi}{a}}e^{-ab^2}$, and ...
1
vote
0answers
62 views
Improper integral equal to -pi with square root and Cauchy principal value
I'd like to know if the following proof for the value of $I$ is correct, and if there is a simpler solution to it. Also, I will probably encounter more improper integrals like this in the future, and ...
3
votes
3answers
134 views
Integral $\int_0^\infty \exp(ia/x^2+ibx^2)dx$
Compute the integral:
\begin{equation}
\int_0^\infty \exp\left(\frac{ia}{x^2}+ibx^2\right)\,dx
\end{equation}
for $a$, $b$ real and positive. I tried complex variables, but don't really know how to ...
4
votes
1answer
59 views
evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$ using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du$
evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du = \dfrac{\pi}{\sin( \pi p)}$. I am having trouble finding what is $p$. I set $u = x^4$, I figure $du = 4x^3 dx$, ...
6
votes
3answers
200 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:
...
4
votes
1answer
98 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 ...
2
votes
1answer
58 views
Residue Calculus Integral computation
I ran into this problem when I was doing some residue computations.
For real $a\neq0$, compute,
$$I=\int_{-\infty}^{+\infty} \frac{e^{iax}}{(x+i)^3} $$
Be sure to treat both cases when $a<0, ...
1
vote
3answers
202 views
Evalulate $\int_{-\infty}^{\infty}\frac{1}{(1+x^{2n})^2}dx$ by using residue theorem
I know the answer of the integral $$\int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx=\frac{\pi}{n\sin\left(\frac{\pi}{2n}\right)}$$where $n\in\mathbb{N}$.
But how to evalulate ...
1
vote
0answers
72 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?
8
votes
1answer
861 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 ...
