Log - Gamma

q[N_,ν_,τ_]:=(((ν^τ) (((Log[(1/N)]))^(τ-1))(N^(ν-1)))/(Gamma[τ]))

Plot3D[q[N,1/((2^(1/τ))-1),τ]//Evaluate,{N,0.01,0.99},{τ,0.3,20},PlotRange->All,Mesh->False,PlotPoints->50];

q[0.5,.3,3]

(∫_0^1)(((ν^τ) (((Log[(1/N)]))^(τ-1))(N^(ν-1)))/(Gamma[τ]))\[DifferentialD]N//Re[ν]>0&&Re[τ]>0

(∫_0^1)(((ν^τ) (((Log[(1/N)]))^(τ-1))(N^(ν-1)))/(Gamma[τ]))N \[DifferentialD]N//Re[ν]>0&&Re[τ]>0

Variance

(∫_0^1)(((ν^τ) (((Log[(1/N)]))^(τ-1))(N^(ν-1)))/(Gamma[τ]))(N^2) \[DifferentialD]N - ((((ν/(1+ν))))^(2τ))//Re[ν]>0&&Re[τ]>0//PowerExpand//FullSimplify

(ν^τ) (-(((1+ν))^(-τ))+(1/(Gamma[τ]))(((2+ν))^(-τ)) Gamma[τ])//PowerExpand//FullSimplify

((ν^τ) (-(((1+ν))^(-τ))+(((2+ν))^(-τ))))((((((1+ν))/ν)))^(2τ))//PowerExpand//FullSimplify

Plot3D[(ν^τ) (-(((1+ν))^(-τ))+(((2+ν))^(-τ))),{ν,0.01,9},{τ,0.3,20},PlotRange->All,PlotPoints->50];