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5 Gamma FunctionProperties

§5.9 Integral Representations


§5.9(i) Gamma Function

5.9.1 1μΓ(νμ)1zν/μ=0exp(-ztμ)tν-1t,

ν>0, μ>0, and z>0. (The fractional powers have their principal values.)

Hankel’s Loop Integral

5.9.2 1Γ(z)=12π-(0+)tt-zt,

where the contour begins at -, circles the origin once in the positive direction, and returns to -. t-z has its principal value where t crosses the positive real axis, and is continuous. See Figure 5.9.1.

See accompanying text
Figure 5.9.1: t-plane. Contour for Hankel’s loop integral. Magnify
5.9.3 c-zΓ(z)=-|t|2z-1-ct2t,
c>0, z>0,

where the path is the real axis.

5.9.4 Γ(z)=1tz-1-tt+k=0(-1)k(z+k)k!,
5.9.5 Γ(z)=0tz-1(-t-k=0n(-1)ktkk!)t,
5.9.6 Γ(z)cos(12πz) =0tz-1costt,
5.9.7 Γ(z)sin(12πz) =0tz-1sintt,
5.9.8 Γ(1+1n)cos(π2n)=0cos(tn)t,
5.9.9 Γ(1+1n)sin(π2n)=0sin(tn)t,

Binet’s Formula

5.9.10 lnΓ(z)=(z-12)lnz-z+12ln(2π)+20arctan(t/z)2πt-1t,

where |phz|<π/2 and the inverse tangent has its principal value.

5.9.11 lnΓ(z+1)=-γz-12π-c--c+πz-sssin(πs)ζ(-s)s,

where |phz|π-δ (<π), 1<c<2, and ζ(s) is as in Chapter 25.

For additional representations see Whittaker and Watson (1927, §§12.31–12.32).

§5.9(ii) Psi Function, Euler’s Constant, and Derivatives

For z>0,

5.9.12 ψ(z)=0(-tt--zt1--t)t,
5.9.13 ψ(z)=lnz+0(1t-11--t)-tzt,
5.9.14 ψ(z)=0(-t-1(1+t)z)tt,
5.9.15 ψ(z)=lnz-12z-20tt(t2+z2)(2πt-1).
5.9.16 ψ(z)+γ=0-t--zt1--tt=011-tz-11-tt.
5.9.17 ψ(z+1)=-γ+12π-c--c+πz-s-1sin(πs)ζ(-s)s,

where |phz|π-δ(<π) and 1<c<2.

5.9.18 γ=-0-tlntt=0(11+t--t)tt=01(1--t)tt-1-ttt=0(-t1--t--tt)t.
5.9.19 Γ(n)(z)=0(lnt)n-ttz-1t,
n0, z>0.