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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-1dt,

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

Hankel’s Loop Integral

5.9.2 1Γ(z)=12πi-(0+)ett-zdt,

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-1e-ct2dt,
c>0, z>0,

where the path is the real axis.

5.9.4 Γ(z)=1tz-1e-tdt+k=0(-1)k(z+k)k!,
5.9.5 Γ(z)=0tz-1(e-t-k=0n(-1)ktkk!)dt,
5.9.6 Γ(z)cos(12πz) =0tz-1costdt,
5.9.7 Γ(z)sin(12πz) =0tz-1sintdt,
5.9.8 Γ(1+1n)cos(π2n)=0cos(tn)dt,
5.9.9 Γ(1+1n)sin(π2n)=0sin(tn)dt,

Binet’s Formula

5.9.10 LnΓ(z)=(z-12)lnz-z+12ln(2π)+20arctan(t/z)e2πt-1dt,

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

5.9.11 LnΓ(z+1)=-γz-12πi-c-i-c+iπz-sssin(πs)ζ(-s)ds,

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(e-tt-e-zt1-e-t)dt,
5.9.13 ψ(z)=lnz+0(1t-11-e-t)e-tzdt,
5.9.14 ψ(z)=0(e-t-1(1+t)z)dtt,
5.9.15 ψ(z)=lnz-12z-20tdt(t2+z2)(e2πt-1).
5.9.16 ψ(z)+γ=0e-t-e-zt1-e-tdt=011-tz-11-tdt.
5.9.17 ψ(z+1)=-γ+12πi-c-i-c+iπz-s-1sin(πs)ζ(-s)ds,

where |phz|π-δ(<π) and 1<c<2.

5.9.18 γ=-0e-tlntdt=0(11+t-e-t)dtt=01(1-e-t)dtt-1e-tdtt=0(e-t1-e-t-e-tt)dt.
5.9.19 Γ(n)(z)=0(lnt)ne-ttz-1dt,
n0, z>0.