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Wednesday, August 11, 2021

Hurwitz-zeta function

Hurwitz zeta function

Integral representation of the Hurwitz-zeta function


We show the proof of the following result posted by @integralsbot: 10xnlna11x1xdx=Γ(a)(ζ(a)H(a)n) This result is the integral representation of the Hurwitz-zeta function. Recall that: ζ(a,n+1)=(ζ(a)H(a)n) The proof for the case aR is well known, so here we show a proof for aN using the Feynman's trick just for the sake to give a different proof.

Proof

For aN
10xnlna11x1xdx=10xn(da1dta1|t=0+1xt)1xdx=da1dta1|t=0+10xnt1xdx=da1dta1|t=0+10k=0xkxntdx=da1dta1|t=0+k=010xk+ntdx=da1dta1|t=0+k=01(k+n+1t)=k=0[da1dta1|t=0+1(k+n+1t)]=k=0[limt0+(a1)!1(k+n+1t)a]=(a1)!k=01(k+n+1)a=(a1)!ζ(a,n+1)=Γ(a)(ζ(a)H(a)n) 10xnlna11x1xdx=Γ(a)(ζ(a)H(a)n)

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