Tagged Questions

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 ...
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 ...
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 ...
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 ...
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 $du = 4x^3 dx$, ...
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: ...
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 ...
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, ...
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 ...
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?
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 ...
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