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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.4 Special Values

For erf(z), erfc(z), and F(z), see §§7.2(i), 7.2(ii). For En(z) see §8.19(i).

8.4.1 γ(12,z2)=20zet2dt=πerf(z),
8.4.2 γ*(a,0) =1Γ(a+1),
8.4.3 γ*(12,z2) =2ez2zπF(z).
8.4.4 Γ(0,z)=zt1etdt=E1(z),
8.4.5 Γ(1,z)=ez,
8.4.6 Γ(12,z2)=2zet2dt=πerfc(z).

For n=0,1,2,,

8.4.7 γ(n+1,z) =n!(1ezen(z)),
8.4.8 Γ(n+1,z) =n!ezen(z),
8.4.9 P(n+1,z) =1ezen(z),
8.4.10 Q(n+1,z) =ezen(z),

where

8.4.11 en(z)=k=0nzkk!.

Also

8.4.12 γ*(n,z)=zn,
8.4.13 Γ(1n,z)=z1nEn(z),
8.4.14 Q(n+12,z2)=erfc(z)+ez2πk=1nz2k1(12)k,
8.4.15 Γ(n,z)=(1)nn!(E1(z)ezk=0n1(1)kk!zk+1)=(1)nn!(ψ(n+1)lnz)znk=0kn(z)kk!(kn).