10
votes
3answers
160 views

Scary contour integral, but is also an integral representation for $\Gamma$-function

This problem is supposed to be from an old Acta Mathematica volume I circa 1880's, and is attributed to Bourguet. By using a parabola with its focus on the origin as a contour, show that: ...
0
votes
1answer
26 views

Integral / Gamma Expectation

I would like to solve the following integral, $\int_{0}^{\infty}\frac{\phi}{a+b\phi} \phi^{c-1}e^{-d\phi}d\phi$. Note $\phi \sim Ga(c,d)$ is a gamma distributed random variable and the integral can ...
1
vote
1answer
47 views

complex integral of z to the power alpha

I would like to perform an inverse laplace and at some point of the calculation I have to compute this integral $$\int_{\gamma-i\infty}^{\gamma+i\infty} z^{(1+n)\alpha-1}e^{z} \frac{dz}{2\pi i}$$ ...
2
votes
1answer
98 views

Gamma Function Contour Integration

So, I've been trying to prove the following integral related to the gamma function, and I'm really banging my head against the wall over this: ...
8
votes
2answers
179 views

Ramanujan's 'well known' integral, $\int_\frac{-\pi}{2}^\frac{\pi}{2} (\cos x)^m e^{in x}dx$.

$$ \int_{-\pi/2}^{\pi/2}\cos^m\left(x\right){\rm e}^{{\rm i}n x}\,{\rm d}x ={\pi \over 2^{m}}\, {\Gamma\left(1 + m\right) \over \Gamma\left( 1 + \left[m + n\right]/2\right)\ \Gamma\left( 1 + ...
1
vote
1answer
167 views

Generating Laguerre polynomials using gamma functions

An exercise given by my complex analysis assistant goes as follows: For $n \in \mathbb{N}$ and $x>0$ we define $$P_n(x) = \frac{1}{2\pi i} \oint_\Sigma ...
1
vote
1answer
270 views

half-line Fourier transform of $x^{z-1}$ w.r.t. $x$?

Can someone help me evaluate $G_g(z)=\int_0^{\infty}x^{z-1}e^{igx}dx$, where $g$ is real and $z$ is complex? By closing the contour in the upper half plane, I've managed to prove that if ...
0
votes
2answers
59 views

Sequential Limit of Line Integral Is The Same As The Usual Limit of Line Integral? (Gamma Function Related)

Let $\epsilon > 0$, and $ n \in \mathbb{Z}^{+} $. Let $C_{n}$ be a positively oriented polygonal line that is from $-n + 1/2 - i \epsilon$ to $ 1/2 - i \epsilon$ and from $ 1/2 - i \epsilon$ to $ ...