§12.11 Zeros

If $a\ge -\frac{1}{2}$, then $U(a,x)$ has no real zeros. If , then $U(a,x)$ has no positive real zeros. If , $n=1,2,\mathrm{\dots }$, then $U(a,x)$ has $n$ positive real zeros. Lastly, when $a=-n-\frac{1}{2}$, $n=1,2,\mathrm{\dots }$ (Hermite polynomial case) $U(a,x)$ has $n$ zeros and they lie in the interval $[-2\sqrt{-a},2\sqrt{-a}]$. For further information on these cases see Dean (1966).

If $a>-\frac{1}{2}$, then $V(a,x)$ has no positive real zeros, and if $a=\frac{3}{2}-2n$, $n\in \mathbb{Z}$, then $V(a,x)$ has a zero at $x=0$.

When $a>-\frac{1}{2}$, $U(a,z)$ has a string of complex zeros that approaches the ray $\mathrm{ph}z=\frac{3}{4}\pi$ as $z\to \mathrm{\infty }$, and a conjugate string. When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\overline{z_{a,s}}$, where $s$ is a large positive integer and

12.11.1		$$z_{a,s}={\mathrm{e}}^{\frac{3}{4}\pi \mathrm{i}}\sqrt{2{\tau }_s}\left(1-\frac{\mathrm{i}a{\lambda }_s}{2{\tau }_s}+\frac{2a^2{\lambda }_s^2-8a^2{\lambda }_s+4a^2+3}{16{\tau }_s^2}+O\left({\lambda }_s^3{\tau }_s^{-3}\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter, ${\tau }_s$ and ${\lambda }_s$ Permalink: http://dlmf.nist.gov/12.11.E1 Encodings: TeX, pMML, png See also: Annotations for §12.11(ii), §12.11 and Ch.12

with

12.11.2		$${\tau }_s=\left(2s+\frac{1}{2}-a\right)\pi +\mathrm{i}\ln \left({\pi }^{-\frac{1}{2}}{2}^{-a-\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)\right),$$
ⓘ Defines: ${\tau }_s$ (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $s$: nonnegative integer and $a$: real or complex parameter Permalink: http://dlmf.nist.gov/12.11.E2 Encodings: TeX, pMML, png See also: Annotations for §12.11(ii), §12.11 and Ch.12

and

12.11.3		$${\lambda }_s=\ln {\tau }_s-\frac{1}{2}\pi \mathrm{i}.$$
ⓘ Defines: ${\lambda }_s$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $s$: nonnegative integer and ${\tau }_s$ Permalink: http://dlmf.nist.gov/12.11.E3 Encodings: TeX, pMML, png See also: Annotations for §12.11(ii), §12.11 and Ch.12

When $a=\frac{1}{2}$ these zeros are the same as the zeros of the complementary error function $\mathrm{erfc}(z/\sqrt{2})$; compare (12.7.5). Numerical calculations in this case show that $z_{\frac{1}{2},s}$ corresponds to the $s$th zero on the string; compare §7.13(ii).

For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). For example, let the $s$th real zeros of $U(a,x)$ and $U^{\prime }(a,x)$, counted in descending order away from the point $z=2\sqrt{-a}$, be denoted by $u_{a,s}$ and $u_{a,s}^{\prime }$, respectively. Then

12.11.4		$$u_{a,s}\sim 2^{\frac{1}{2}}\mu \left(p_0(\alpha )+\frac{p_1(\alpha )}{{\mu }^4}+\frac{p_2(\alpha )}{{\mu }^8}+\mathrm{\cdots }\right),$$
ⓘ Defines: $u_{a,s}$: zeros (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $a$: real or complex parameter, $\alpha$ and $p_n(\zeta )$: coefficients Referenced by: §12.11(iii) Permalink: http://dlmf.nist.gov/12.11.E4 Encodings: TeX, pMML, png See also: Annotations for §12.11(iii), §12.11 and Ch.12

as $\mu$ ($=\sqrt{-2a}$) $\to \mathrm{\infty }$, $s$ fixed. Here $\alpha ={\mu }^{-\frac{4}{3}}{a}_s$, $a_s$ denoting the $s$th negative zero of the function $\mathrm{Ai}$ (see §9.9(i)). The first two coefficients are given by

12.11.5		$$p_0(\zeta )=t(\zeta ),$$
ⓘ Symbols: $\zeta$: change of variable and $p_n(\zeta )$: coefficients Permalink: http://dlmf.nist.gov/12.11.E5 Encodings: TeX, pMML, png See also: Annotations for §12.11(iii), §12.11 and Ch.12

where $t(\zeta )$ is the function inverse to $\zeta (t)$, defined by (12.10.39) (see also (12.10.41)), and

12.11.6		$$p_1(\zeta )=\frac{t^3-6t}{24{(t^2-1)}^2}+\frac{5}{48{((t^2-1){\zeta }^3)}^{\frac{1}{2}}}.$$
ⓘ Symbols: $\zeta$: change of variable and $p_n(\zeta )$: coefficients Permalink: http://dlmf.nist.gov/12.11.E6 Encodings: TeX, pMML, png See also: Annotations for §12.11(iii), §12.11 and Ch.12

Similarly, for the zeros of $U^{\prime }(a,x)$ we have

12.11.7		$$u_{a,s}^{\prime }\sim 2^{\frac{1}{2}}\mu \left(q_0(\beta )+\frac{q_1(\beta )}{{\mu }^4}+\frac{q_2(\beta )}{{\mu }^8}+\mathrm{\cdots }\right),$$
ⓘ Defines: $u_{a,s}^{\prime }$: zeros (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $a$: real or complex parameter and $\beta$ Permalink: http://dlmf.nist.gov/12.11.E7 Encodings: TeX, pMML, png See also: Annotations for §12.11(iii), §12.11 and Ch.12

where $\beta ={\mu }^{-\frac{4}{3}}{a}_s^{\prime }$, $a_s^{\prime }$ denoting the $s$th negative zero of the function ${\mathrm{Ai}}^{\prime }$ and

12.11.8		$$q_0(\zeta )=t(\zeta ).$$
ⓘ Symbols: $\zeta$: change of variable Permalink: http://dlmf.nist.gov/12.11.E8 Encodings: TeX, pMML, png See also: Annotations for §12.11(iii), §12.11 and Ch.12

For the first zero of $U(a,x)$ we also have

12.11.9		$$u_{a,1}\sim 2^{\frac{1}{2}}\mu (\begin{array}{l}1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}\\ -\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }),\end{array}$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros Referenced by: §12.11(iii) Permalink: http://dlmf.nist.gov/12.11.E9 Encodings: TeX, pMML, png See also: Annotations for §12.11(iii), §12.11 and Ch.12

where the numerical coefficients have been rounded off.

For further information, including associated functions, see Olver (1959).