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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.13 Zeros


§8.13(i) x-Zeros of γ*(a,x)

The function γ*(a,x) has no real zeros for a0. For a<0 and n=1,2,3,, there exist:

  1. (a)

    one negative zero x-(a) and no positive zeros when 1-2n<a<2-2n;

  2. (b)

    one negative zero x-(a) and one positive zero x+(a) when -2n<a<1-2n.

The negative zero x-(a) decreases monotonically in the interval -1<a<0, and satisfies

8.13.1 1+a-1<x-(a)<ln|a|,

When -5a4 the behavior of the x-zeros as functions of a can be seen by taking the slice γ*(a,x)=0 of the surface depicted in Figure 8.3.6. Note that from (8.4.12) γ*(-n,0)=0, n=1,2,3,.

For asymptotic approximations for x+(a) and x-(a) as a- see Tricomi (1950b), with corrections by Kölbig (1972b). For more accurate asymptotic approximations see Thompson (2012).

§8.13(ii) λ-Zeros of γ(a,λa) and Γ(a,λa)

For information on the distribution and computation of zeros of γ(a,λa) and Γ(a,λa) in the complex λ-plane for large values of the positive real parameter a see Temme (1995a).

§8.13(iii) a-Zeros of γ*(a,x)

For fixed x and n=1,2,3,, γ*(a,x) has:

  1. (a)

    two zeros in each of the intervals -2n<a<2-2n when x<0;

  2. (b)

    two zeros in each of the intervals -2n<a<1-2n when 0<xxn*;

  3. (c)

    zeros at a=-n when x=0.

As x increases the positive zeros coalesce to form a double zero at (an*,xn*). The values of the first six double zeros are given to 5D in Table 8.13.1. For values up to n=10 see Kölbig (1972b). Approximations to an*, xn* for large n can be found in Kölbig (1970). When x>xn* a pair of conjugate trajectories emanate from the point a=an* in the complex a-plane. See Kölbig (1970, 1972b) for further information.

Table 8.13.1: Double zeros (an*,xn*) of γ*(a,x).
n an* xn*
1 -1.64425 0.30809
2 -3.63887 0.77997
3 -5.63573 1.28634
4 -7.63372 1.80754
5 -9.63230 2.33692
6 -11.63126 2.87150