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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.13 Zeros

  1. §8.13(i) x-Zeros of γ(a,x)
  2. §8.13(ii) λ-Zeros of γ(a,λa) and Γ(a,λa)
  3. §8.13(iii) a-Zeros of γ(a,x)

§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 12n<a<22n;

  2. (b)

    one negative zero x(a) and one positive zero x+(a) when 2n<a<12n.

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

8.13.1 1+a1<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<22n when x<0;

  2. (b)

    two zeros in each of the intervals 2n<a<12n 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