# Satisfying a Differential Equation and complex Laguerre

I have the following problem

Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is counterclockwise around the origin.

I have busted through with the x-derivatives, but I'm not sure where to go from there... or if that is even the way to tackle this problem... There was talk about Taylor expanding the integrand and using the properties of contour integration to whittle down the terms to a finite number of contributions inside the integrals. Is this nonsense, or legit?

any help would be great, Thanks.

-
You wouldn't happen to know about Cauchy's formula or the Rodrigues formula for Laguerre polynomials, would you? – J. M. Apr 20 '12 at 2:38
If you can believe it, my professor said using Rodrigues' formula was not a viable option... – kηives Apr 20 '12 at 2:40
I see... but you understand that your formula for Laguerre is the Rodrigues formula in Cauchy disguise, yes? – J. M. Apr 20 '12 at 2:42
I do, I have the passage between them in my notes. I'm sorry I don't see what your getting at... – kηives Apr 20 '12 at 2:44
That's why I don't understand your professor's remark... – J. M. Apr 20 '12 at 2:47