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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.7 Series Expansions

For the functions en(z), in(1)(z), and Ln(α)(x) see (8.4.11), §§10.47(ii), and 18.3, respectively.

8.7.1 γ*(a,z)=ezk=0zkΓ(a+k+1)=1Γ(a)k=0(z)kk!(a+k).
8.7.2 γ(a,x+y)γ(a,x)=Γ(a,x)Γ(a,x+y)=exxa1n=0(1a)n(x)n(1eyen(y)),
|y|<|x|.
8.7.3 Γ(a,z)=Γ(a)k=0(1)kza+kk!(a+k)=Γ(a)(1zaezk=0zkΓ(a+k+1)),
a0,1,2,.
8.7.4 γ(a,x)=Γ(a)x12aexn=0en(1)x12nIn+a(2x1/2),
a0,1,2,.
8.7.5 γ*(a,z)=e12zn=0(1a)nΓ(n+a+1)(2n+1)in(1)(12z).
8.7.6 Γ(a,x)=xaexn=0Ln(a)(x)n+1,
x>0.

For an expansion for γ(a,ix) in series of Bessel functions Jn(x) that converges rapidly when a>0 and x (0) is small or moderate in magnitude see Barakat (1961).