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5
questions
6
votes
1
answer
145
views
closed form for $\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{dx}{(b^2+x^2)}$
closed form for :
$$\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{\mathrm{dx}}{(b^2+x^2)}$$
where $\beta$ is beta function
I tried with the definition of beta and i got
$$I=\...
- 5,913
9
votes
1
answer
1k
views
Closed form for $\int_1^\infty\frac{\operatorname dx}{\operatorname \Gamma(x)}$
Is a closed form for $$\int\limits_1^{+\infty}\frac{\operatorname dx}{\operatorname \Gamma(x)}$$known?
I tried to find it, but all well-known integrals involving gamma-function (such as of $\log\...
- 4,301
5
votes
1
answer
137
views
Expressing $\int_{-\infty}^\infty dx/(x^2+1)^n$ in terms of Gamma function
How to prove this identity for $n>1/2$?
$$\int_{-\infty}^{\infty}\frac{dx}{(x^2+1)^n}=\frac{\sqrt{\pi}\cdot \Gamma(n-\frac{1}{2}) }{\Gamma (n)}$$
- 51
3
votes
4
answers
174
views
How to compute $\int^{\infty}_{0} t^{(\frac1n-1)}\cos t \,\mathrm{d}t$?
How to calculate the below integral?
$$
\int^{\infty}_{0} \frac{\cos t}{t^{1-\frac{1}{n}}} \textrm{d}t = \frac{\pi}{2\sin(\frac{\pi}{2n})\Gamma(1-\frac{1}{n})}
$$
where $n\in \mathbb{N}$.
- 31
15
votes
1
answer
655
views
Closed form for $\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}$
Consider the following integral:
$$\mathcal{I}=\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}.$$
It can be represented as
$$\mathcal{I}=\int_1^\...
- 47.3k