# §8.13 Zeros

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

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

1. (a)

one negative zero $x_{-}(a)$ and no positive zeros when $1-2n;

2. (b)

one negative zero $x_{-}(a)$ and one positive zero $x_{+}(a)$ when $-2n.

The negative zero $x_{-}(a)$ decreases monotonically in the interval $-1, and satisfies

 8.13.1 $1+a^{-1} $-1. ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function and $a$: parameter Permalink: http://dlmf.nist.gov/8.13.E1 Encodings: TeX, pMML, png See also: Annotations for §8.13(i), §8.13 and Ch.8

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

For asymptotic approximations for $x_{+}(a)$ and $x_{-}(a)$ as $a\to-\infty$ see Tricomi (1950b), with corrections by Kölbig (1972b). For more accurate asymptotic approximations see Thompson (2012).

## §8.13(ii) $\lambda$-Zeros of $\gamma\left(a,\lambda a\right)$ and $\Gamma\left(a,\lambda a\right)$

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

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

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

1. (a)

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

2. (b)

two zeros in each of the intervals $-2n when $0;

3. (c)

zeros at $a=-n$ when $x=0$.

As $x$ 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 $n=10$ see Kölbig (1972b). Approximations to $a_{n}^{*}$, $x_{n}^{*}$ for large $n$ 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 $a$-plane. See Kölbig (1970, 1972b) for further information.