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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.2 Definitions and Basic Properties

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§8.2(i) Definitions

The general values of the incomplete gamma functions γ(a,z) and Γ(a,z) are defined by

8.2.1 γ(a,z) =0zta-1e-tdt,
a>0,
8.2.2 Γ(a,z) =zta-1e-tdt,

without restrictions on the integration paths. However, when the integration paths do not cross the negative real axis, and in the case of (8.2.2) exclude the origin, γ(a,z) and Γ(a,z) take their principal values; compare §4.2(i). Except where indicated otherwise in the DLMF these principal values are assumed. For example,

8.2.3 γ(a,z)+Γ(a,z)=Γ(a),
a0,-1,-2,.

Normalized functions are:

8.2.4 P(a,z) =γ(a,z)Γ(a),
Q(a,z) =Γ(a,z)Γ(a),
8.2.5 P(a,z)+Q(a,z)=1.

In addition,

8.2.6 γ*(a,z)=z-aP(a,z)=z-aΓ(a)γ(a,z).
8.2.7 γ*(a,z)=1Γ(a)01ta-1e-ztdt,
a>0.

§8.2(ii) Analytic Continuation

In this subsection the functions γ and Γ have their general values.

The function γ*(a,z) is entire in z and a. When z0, Γ(a,z) is an entire function of a, and γ(a,z) is meromorphic with simple poles at a=-n, n=0,1,2,, with residue (-1)n/n!.

For m,

8.2.8 γ(a,ze2πmi) =e2πmiaγ(a,z),
a0,-1,-2,,
8.2.9 Γ(a,ze2πmi) =e2πmiaΓ(a,z)+(1-e2πmia)Γ(a).

(8.2.9) also holds when a is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. For example, in the case m=-1 we have

8.2.10 e-πiaΓ(a,zeπi)-eπiaΓ(a,ze-πi)=-2πiΓ(1-a),

without restriction on a.

Lastly,

8.2.11 Γ(a,ze±πi)=Γ(a)(1-zae±πiaγ*(a,-z)).

§8.2(iii) Differential Equations

If w=γ(a,z) or Γ(a,z), then

8.2.12 d2wdz2+(1+1-az)dwdz=0.

If w=ezz1-aΓ(a,z), then

8.2.13 d2wdz2-(1+1-az)dwdz+1-az2w=0.

Also,

8.2.14 zd2γ*dz2+(a+1+z)dγ*dz+aγ*=0.