Integral representation of the Hurwitz-zeta function
We show the proof of the following result posted by @integralsbot: ∫10xnlna−11x1−xdx=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 a∈R is well known, so here we show a proof for a∈N using the Feynman's trick just for the sake to give a different proof.
Proof
For a∈N
∫10xnlna−11x1−xdx=∫10xn(da−1dta−1|t=0+1xt)1−xdx=da−1dta−1|t=0+∫10xn−t1−xdx=da−1dta−1|t=0+∫10∞∑k=0xkxn−tdx=da−1dta−1|t=0+∞∑k=0∫10xk+n−tdx=da−1dta−1|t=0+∞∑k=01(k+n+1−t)=∞∑k=0[da−1dta−1|t=0+1(k+n+1−t)]=∞∑k=0[limt→0+(a−1)!1(k+n+1−t)a]=(a−1)!∞∑k=01(k+n+1)a=(a−1)!ζ(a,n+1)=Γ(a)(ζ(a)−H(a)n) ∫10xnlna−11x1−xdx=Γ(a)(ζ(a)−H(a)n)
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