Integral involving the Gamma functon Γ(⋅)
Today we show the proof of this integral posted by @Ali39342137 ∫∞0(√1+x4−x2)nxk−1dx=Γ(n2−k4)Γ(k2)2k2Γ(n2+k4)n2n+kn>k2
This is a generalization for this other integral posted by @integralsbot with n=s and k=1:
∫∞0(√1+x4−x2)sdx=s2s+1√π2Γ(s2−14)Γ(s2+14)
We previously posted another similar generalization here
Proof:
I=∫∞0(√1+x4−x2)nxk−1dx=12∫∞0(√1+w2−w)nwk2−1dw(w↦x2)=12k2+1∫∞0tn−k2−1(1−t2)k2−1(t2+1)(t↦√1+w2−w)=12k2+1[∫∞0tn−k2+1(1−t2)k2−1dt+∫∞0tn−k2−1(1−t2)k2−1dt]=12k2+2[∫∞0sn2−k4(1−s)k2−1ds+∫∞0sn2−k4−1(1−s)k2−1ds](s↦t2)=12k2+2[B(n2−k4+1,k2)+B(n2−k4,k2)]=12k2+2[Γ(n2−k4+1)Γ(k2)Γ(n2+k4+1)+Γ(n2−k4)Γ(k2)Γ(n2+k4)]=12k2+2[(n2−k4)Γ(n2−k4)Γ(k2)(n2+k4)Γ(n2+k4)+Γ(n2−k4)Γ(k2)Γ(n2+k4)]=Γ(n2−k4)Γ(k2)2k2+2Γ(n2+k4)[(n2−k4)(n2+k4)+1]=Γ(n2−k4)Γ(k2)2k2Γ(n2+k4)n2n+k
∫∞0(√1+x4−x2)nxk−1dx=Γ(n2−k4)Γ(k2)2k2Γ(n2+k4)n2n+kn>k2
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