Digital Library of Mathematical Functions
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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)=-zk=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)=-xxa-1n=0(1-a)n(-x)n(1--yen(y)),
|y|<|x|.
8.7.3 Γ(a,z)=Γ(a)-k=0(-1)kza+kk!(a+k)=Γ(a)(1-za-zk=0zkΓ(a+k+1)),
a0,-1,-2,.
8.7.4 γ(a,x)=Γ(a)x12a-xn=0en(-1)x12nIn+a(2x1/2),
a0,-1,-2,.
8.7.5 γ*(a,z)=-12zn=0(1-a)nΓ(n+a+1)(2n+1)in(1)(12z).
8.7.6 Γ(a,x)=xa-xn=0Ln(a)(x)n+1,
x>0.

For an expansion for γ(a,x) 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).