Appendix B Derivation of equation (7.5)

[76.3.1.1] By virtue of the symmetry relation (6.5) the integral in (7.4) may be written as

∫-∞∞xσ⁢h⁢x;ϖ,ζ,0,1⁢d⁢x=∫0∞xσ⁢h⁢x;ϖ,ζ,0,1⁢d⁢x+∫0∞xσ⁢h⁢x;ϖ,-ζ,0,1⁢d⁢x

(B.1)

The definition (A.1) implies the general formula [32]

∫0∞xs-1⁢HP⁢Qm⁢n(ax|a1,A1…aP,APb1,B1…bQ,BQ)dx=⁢a-s⁢∏j=1mΓ⁢bj+Bj⁢s⁢∏j=1nΓ⁢1-aj-Aj⁢s∏j=m+1QΓ⁢1-bj-Bj⁢s⁢∏j=n+1PΓ⁢aj+Aj⁢s

(B.2)

by virtue of the Mellin inversion theorem. [76.3.1.2] Specializing to the case at hand

∫0∞xσ⁢h⁢x;ϖ,ζ,0,1⁢d⁢x	=1ϖ∫0∞xσH2211(x\|1-1/ϖ,1/ϖ1-ϱ,ϱ0,11-ϱ,ϱ)dx		(B.3)
	=Γ⁢σ+1⁢Γ⁢-σ/ϖϖ⁢Γ⁢1+ϱ⁢σ⁢Γ⁢-ϱ⁢σ		(B.4)

where ϱ=12-ζϖ+ζ2. [76.3.1.3] Using Γ⁢x⁢Γ⁢-x=-π/x⁢sin⁡π⁢x and the functional equation for the Γ-function gives

∫0∞xσ⁢h⁢x;ϖ,ζ,0,1⁢d⁢x=1π⁢sin⁡π⁢ϱ⁢σ⁢Γ⁢σ⁢Γ⁢1-σϖ

(B.5)

which inserted into (B.1) readily yields the desired result (7.5).