8 Incomplete Gamma and Related Functions Incomplete Gamma Functions 8.12 Uniform Asymptotic Expansions for Large Parameter 8.14 Integrals

§8.13 Zeros

Keywords:: incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.13

§8.13(i) -Zeros of $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,x\right)$
§8.13(ii) $\lambda$ -Zeros of $\mathop{\gamma\/}\nolimits\!\left(a,\lambda a\right)$ and $\mathop{\Gamma\/}\nolimits\!\left(a,\lambda a\right)$
§8.13(iii) -Zeros of $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,x\right)$

§8.13(i) -Zeros of $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,x\right)$

Notes:: See Erdélyi et al. (1953b, §9.6), Tricomi (1950b), and Lew (1994).
Permalink:: http://dlmf.nist.gov/8.13.i

The function $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,x\right)$ has no real zeros for $a\geq 0$ . For and $n=1,2,3,\dots$ , there exist:

(a)

one negative zero $x_{{-}}(a)$ and no positive zeros when ;
(b)

one negative zero $x_{{-}}(a)$ and one positive zero $x_{+}(a)$ when .

The negative zero $x_{{-}}(a)$ decreases monotonically in the interval , and satisfies

8.13.1 $1+a^{{-1}}<x_{{-}}(a)<\mathop{\ln\/}\nolimits|a|,$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : parameter
Permalink:: http://dlmf.nist.gov/8.13.E1
Encodings:: TeX, pMML, png

When $-5\leq a\leq 4$ the behavior of the -zeros as functions of can be seen by taking the slice $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,x\right)=0$ of the surface depicted in Figure 8.3.6. Note that from (8.4.12) $\mathop{\gamma^{{*}}\/}\nolimits\!\left(-n,0\right)=0$ , $n=1,2,3,\dots$ .

For asymptotic approximations for $x_{{+}}(a)$ and $x_{{-}}(a)$ as $a\to-\infty$ see Tricomi (1950b), with corrections by Kölbig (1972b).

§8.13(ii) $\lambda$ -Zeros of $\mathop{\gamma\/}\nolimits\!\left(a,\lambda a\right)$ and $\mathop{\Gamma\/}\nolimits\!\left(a,\lambda a\right)$

Permalink:: http://dlmf.nist.gov/8.13.ii

For information on the distribution and computation of zeros of $\mathop{\gamma\/}\nolimits\!\left(a,\lambda a\right)$ and $\mathop{\Gamma\/}\nolimits\!\left(a,\lambda a\right)$ in the complex $\lambda$ -plane for large values of the positive real parameter see Temme (1995a).

§8.13(iii) -Zeros of $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,x\right)$

Notes:: See Tricomi (1950b) and Kölbig (1972b).
Keywords:: incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.13.iii

For fixed and $n=1,2,3,\dots$ , $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,x\right)$ has:

(a)

two zeros in each of the intervals when ;
(b)

two zeros in each of the intervals when $0<x\leq x_{n}^{{*}}$ ;
(c)

zeros at when .

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