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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)=20ze-t2dt=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)=zt-1e-tdt=E1(z),
8.4.5 Γ(1,z)=e-z,
8.4.6 Γ(12,z2)=2ze-t2dt=πerfc(z).

For n=0,1,2,,

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

where

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

Also

8.4.12 γ*(-n,z)=zn,
8.4.13 Γ(1-n,z)=z1-nEn(z),
8.4.14 Q(n+12,z2)=erfc(z)+e-z2πk=1nz2k-1(12)k,
8.4.15 Γ(-n,z)=(-1)nn!(E1(z)-e-zk=0n-1(-1)kk!zk+1)=(-1)nn!(ψ(n+1)-lnz)-z-nk=0kn(-z)kk!(k-n).