Higher Level Integration Marathon & Questions (2 Viewers)

omegadot

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Re: Extracurricular Integration Marathon



 
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Paradoxica

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Re: Extracurricular Integration Marathon

By inspection, the answer to the first integral is:



Similarly, the answer to the second integral is:



These come from the integral definitions of the Riemann Zeta and Dirichlet Eta functions, respectively, with slight manipulation.
 

Kingom

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Re: Extracurricular Integration Marathon

By inspection, the answer to the first integral is:



Similarly, the answer to the second integral is:



These come from the integral definitions of the Riemann Zeta and Dirichlet Eta functions, respectively, with slight manipulation.
Can you please show working out? I can't really learn from this and what is this "slight manipulation" that you have stated.
 

Paradoxica

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Re: Extracurricular Integration Marathon

Evaluate:



Hence, or otherwise, evaluate:

anyway let's bump this one, been sitting here for a very long time.

the first one is a trivial exercise in contour integration/special functions manipulation

the second is a matter of a straightforward substitution
 

omegadot

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Re: Extracurricular Integration Marathon

Beta Function manipulation is pretty trivial....
Beta function, second derivative with respect to its parameter to get rid of the log squared term, beta function reflection formula is then related to the digamma function (and of course its second derivatives will be needed along the way), then a reflection formula for the digamma function and I guess we are home and hosed but is it not nice to see something different :)
 

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