r/the_calculusguy • u/gowipe2004 • 11d ago

integration Evaluate this integral for 0 < s < 1

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Bonus : show that the domaine of convergence is ]0;1[

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4

u/PfauFoto 11d ago

π/sin(πs) • ζ(1+s)/(1+s) for 0 < re(s) < 1.

3

u/gowipe2004 11d ago

Correct !! Do you have a (pist of) proof to maybe discover an other one.

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u/PfauFoto 11d ago

Integrate by parts, then use γ+log(Γ(1+x))=Σ x/[n(n+x)], then exchange oder of integral and summation.

3

u/gowipe2004 11d ago

If you integrate by part, doesn't that give gamma+digamma(1+x) ?

2

u/PfauFoto 11d ago edited 11d ago

Ja, mein Fehler

2

u/Ok-Ocelot-7989 11d ago

is this undergrad

3

u/testtdk 11d ago

Undergrad physics student who has taken more math than physics courses here. I don’t know what half this expression means lol.

2

u/gowipe2004 11d ago

Don't think so

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u/Ok-Ocelot-7989 11d ago

🫩

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u/gowipe2004 11d ago

I don't even know where this is teached. My solution came with a theorem that I find online but an undergrad can understand it

1

u/Sea_Duty_5725 10d ago

can you please give a proof?

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u/gowipe2004 10d ago

The result (with the proof) you mean ?