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5 Gamma FunctionProperties

§5.9 Integral Representations

  1. §5.9(i) Gamma Function
  2. §5.9(ii) Psi Function, Euler’s Constant, and Derivatives

§5.9(i) Gamma Function

5.9.1 1μΓ(νμ)1zν/μ=0exp(ztμ)tν1dt,

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+)ettzdt,

where the contour begins at , circles the origin once in the positive direction, and returns to . tz 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
5.9.2_5 1Γ(z)=ezz1z2πππezΦ(t)dt,

where Φ(t)=1tcott+ln(tsint).

5.9.3 czΓ(z)=|t|2z1ect2dt,
c>0, z>0,

where the path is the real axis.

5.9.4 Γ(z)=1tz1etdt+k=0(1)k(z+k)k!,
5.9.5 Γ(z)=0tz1(etk=0n(1)ktkk!)dt,
5.9.6 Γ(z)cos(12πz) =0tz1costdt,
5.9.7 Γ(z)sin(12πz) =0tz1sintdt,
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)=(z12)lnzz+12ln(2π)+20arctan(t/z)e2πt1dt,

where |phz|<π/2 and the inverse tangent has its principal value. Two alternative versions of Binet’s formula are

5.9.10_1 LnΓ(z)=(z12)lnzz+12ln(2π)zπ0ln(1e2πt)t2+z2dt,
5.9.10_2 LnΓ(z)=(z12)lnzz+12ln(2π)+0ezt(1et11t+12)dtt,

where |phz|<π/2.

5.9.11 LnΓ(z+1)=γz12πicic+iπzsssin(πs)ζ(s)ds,

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

5.9.11_1 Γ(z)=112πi0e2πtΓ(teiπ/2)t+izdt+12πi0e2πtΓ(teiπ/2)tizdt,
5.9.11_2 1Γ(z)=112πi0e2πtΓ(teiπ/2)tizdt+12πi0e2πtΓ(teiπ/2)t+izdt,

where |phz|<π/2, and the scaled gamma function Γ(z) is defined in (5.11.3). 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(ettezt1et)dt,
5.9.13 ψ(z)=lnz+0(1t11et)etzdt,
5.9.14 ψ(z)=0(et1(1+t)z)dtt,
5.9.15 ψ(z)=lnz12z20tdt(t2+z2)(e2πt1).
5.9.16 ψ(z)+γ=0etezt1etdt=011tz11tdt.
5.9.17 ψ(z+1)=γ+12πicic+iπzs1sin(πs)ζ(s)ds,

where |phz|πδ and 1<c<2.

5.9.18 γ=0etlntdt=0(11+tet)dtt=01(1et)dtt1etdtt=0(et1etett)dt.
5.9.19 Γ(n)(z)=0(lnt)nettz1dt,
n0, z>0.