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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.

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    You wouldn't happen to know about [Cauchy's formula](http://en.wikipedia.org/wiki/Cauchy%27s_integral_formula) or the [Rodrigues formula](http://functions.wolfram.com/Polynomials/LaguerreL/07/02/0001/) for Laguerre polynomials, would you?2012-04-20
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    If you can believe it, my professor said using Rodrigues' formula was _not_ a viable option...2012-04-20
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    I see... but you understand that your formula for Laguerre is the Rodrigues formula in Cauchy disguise, yes?2012-04-20
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    I do, I have the passage between them in my notes. I'm sorry I don't see what your getting at...2012-04-20
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    That's why I don't understand your professor's remark...2012-04-20
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    This can help you http://mathworld.wolfram.com/LaguerreDifferentialEquation.html2012-04-20

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