§8.13 Zeros

The function ${\gamma }^{\ast }(a,x)$ has no real zeros for $a\ge 0$. For and $n=1,2,3,\mathrm{\dots }$, there exist:

1. (a)

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

2. (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

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

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

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

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

1. (a)

   two zeros in each of the intervals when ;

2. (b)

   two zeros in each of the intervals when ;

3. (c)

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

As $x$ increases the positive zeros coalesce to form a double zero at ($a_n^{\ast },x_n^{\ast }$). 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 $a_n^{\ast }$, $x_n^{\ast }$ for large $n$ can be found in Kölbig (1970). When $x>x_n^{\ast }$ a pair of conjugate trajectories emanate from the point $a=a_n^{\ast }$ in the complex $a$-plane. See Kölbig (1970, 1972b) for further information.