12 Parabolic Cylinder Functions Properties 12.10 Uniform Asymptotic Expansions for Large Parameter 12.12 Integrals

§12.11 Zeros

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Permalink:: http://dlmf.nist.gov/12.11
See also:: Annotations for Ch.12

§12.11(i) Distribution of Real Zeros
§12.11(ii) Asymptotic Expansions of Large Zeros
§12.11(iii) Asymptotic Expansions for Large Parameter

§12.11(i) Distribution of Real Zeros

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Keywords:: distribution, parabolic cylinder functions, zeros
Notes:: See Whittaker and Watson (1927, p. 354), Dean (1966).
Permalink:: http://dlmf.nist.gov/12.11.i
See also:: Annotations for §12.11 and Ch.12

If $a\geq-\tfrac{1}{2}$ , then $U\left(a,x\right)$ has no real zeros. If $-\tfrac{3}{2}<a<-\tfrac{1}{2}$ , then $U\left(a,x\right)$ has no positive real zeros. If $-2n-\tfrac{3}{2}<a<-2n+\tfrac{1}{2}$ , $n=1,2,\dots$ , then $U\left(a,x\right)$ has positive real zeros. Lastly, when $a=-n-\tfrac{1}{2}$ , $n=1,2,\dots$ (Hermite polynomial case) $U\left(a,x\right)$ has zeros and they lie in the interval $[-2\sqrt{-a},2\sqrt{-a}\,]$ . For further information on these cases see Dean (1966).

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

§12.11(ii) Asymptotic Expansions of Large Zeros

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Keywords:: asymptotic expansions for large variable, parabolic cylinder functions, zeros
Notes:: See Riekstynš (1991, p. 195).
Referenced by:: Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/12.11.ii
See also:: Annotations for §12.11 and Ch.12

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

12.11.1 $z_{a,s}=e^{\frac{3}{4}\pi i}\sqrt{2\tau_{s}}\left(1-\frac{ia\lambda_{s}}{2\tau% _{s}}+\frac{2a^{2}\lambda_{s}^{2}-8a^{2}\lambda_{s}+4a^{2}+3}{16\tau_{s}^{2}}+% O\left(\lambda_{s}^{3}\tau_{s}^{-3}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : 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, : complex variable, : nonnegative integer, : 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 $\tau_{s}=\left(2s+\tfrac{1}{2}-a\right)\pi+i\ln\left(\pi^{-\frac{1}{2}}2^{-a-% \frac{1}{2}}\Gamma\left(\tfrac{1}{2}+a\right)\right),$

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Defines:: $\tau_{s}$ (locally)
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer and : 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 $\lambda_{s}=\ln\tau_{s}-\tfrac{1}{2}\pi i.$

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Defines:: $\lambda_{s}$ (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, : 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 $a=\tfrac{1}{2}$ these zeros are the same as the zeros of the complementary error function $\operatorname{erfc}(z/\sqrt{2})$ ; compare (12.7.5). Numerical calculations in this case show that $z_{\frac{1}{2},s}$ corresponds to the th zero on the string; compare §7.13(ii).

§12.11(iii) Asymptotic Expansions for Large Parameter

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Keywords:: asymptotic expansions for large parameter, parabolic cylinder functions, zeros
Notes:: See Olver (1959). (12.11.9) is obtained by truncating (12.11.4) at its second term, and applying (12.10.41) with terms up to and including $w^{5}$ .
Permalink:: http://dlmf.nist.gov/12.11.iii
See also:: Annotations for §12.11 and Ch.12

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

12.11.4 $u_{a,s}\sim 2^{\frac{1}{2}}\mu\left(p_{0}(\alpha)+\frac{p_{1}(\alpha)}{\mu^{4}% }+\frac{p_{2}(\alpha)}{\mu^{8}}+\cdots\right),$

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Defines:: $u_{a,s}$ : zeros (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, : nonnegative integer, : 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 $\mu$ ( $=\sqrt{-2a}$ ) $\to\infty$ , fixed. Here $\alpha=\mu^{-\frac{4}{3}}a_{s}$ , $a_{s}$ denoting the th negative zero of the function $\operatorname{Ai}$ (see §9.9(i)). The first two coefficients are given by

12.11.5 $p_{0}(\zeta)=t(\zeta),$

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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 $p_{1}(\zeta)=\frac{t^{3}-6t}{24(t^{2}-1)^{2}}+\frac{5}{48((t^{2}-1)\zeta^{3})^% {\frac{1}{2}}}.$

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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 $U'\left(a,x\right)$ we have

12.11.7 $u^{\prime}_{a,s}\sim 2^{\frac{1}{2}}\mu\left(q_{0}(\beta)+\frac{q_{1}(\beta)}{% \mu^{4}}+\frac{q_{2}(\beta)}{\mu^{8}}+\cdots\right),$

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Defines:: $u^{\prime}_{a,s}$ : zeros (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, : nonnegative integer, : 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 $\beta=\mu^{-\frac{4}{3}}a^{\prime}_{s}$ , $a^{\prime}_{s}$ denoting the th negative zero of the function $\operatorname{Ai}'$ and

12.11.8 $q_{0}(\zeta)=t(\zeta).$

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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 $U\left(a,x\right)$ we also have

12.11.9 $u_{a,1}\sim 2^{\frac{1}{2}}\mu\left(1-1.85575\;708\mu^{-4/3}-0.34438\;34\mu^{-% 8/3}-0.16871\;5\mu^{-4}-0.11414\mu^{-16/3}-0.0808\mu^{-20/3}-\cdots\right),$

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Symbols:: $\sim$ : Poincaré asymptotic expansion, : 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).

12.10 Uniform Asymptotic Expansions for Large Parameter 12.12 Integrals

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§12.11 Zeros

Contents

§12.11(i) Distribution of Real Zeros

§12.11(ii) Asymptotic Expansions of Large Zeros

§12.11(iii) Asymptotic Expansions for Large Parameter