§12.11 Zeros

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§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

Notes:: See Whittaker and Watson (1927, p. 354), Dean (1966).
Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.11.i

If $a\geq-\tfrac{1}{2},$ then $\mathop{U\/}\nolimits\!\left(a,x\right)$ has no real zeros. If $-\tfrac{3}{2}<a<-\tfrac{1}{2}$ , then $\mathop{U\/}\nolimits\!\left(a,x\right)$ has no positive real zeros. If $-2n-\tfrac{3}{2}<a<-2n+\tfrac{1}{2}$ , $n=1,2,\dots$ , then $\mathop{U\/}\nolimits\!\left(a,x\right)$ has $n$ positive real zeros. Lastly, when $a=-n-\tfrac{1}{2}$ , $n=1,2,\dots$ (Hermite polynomial case) $\mathop{U\/}\nolimits\!\left(a,x\right)$ 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 $a>-\tfrac{1}{2},$ then $\mathop{V\/}\nolimits\!\left(a,x\right)$ has no positive real zeros, and if $a=\tfrac{3}{2}-2n$ , $n\in\Integer$ , then $\mathop{V\/}\nolimits\!\left(a,x\right)$ has a zero at $x=0$ .

§12.11(ii) Asymptotic Expansions of Large Zeros

Notes:: See Riekstynš (1991, p. 195).
Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.11.ii

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $e$ : base of exponential function, $z$ : complex variable, $s$ : nonnegative integer, $a$ : real or complex parameter, $\tau _{s}$ and $\lambda _{s}$
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with

12.11.2 $\tau _{s}=\left(2s+\tfrac{1}{2}-a\right)\pi+i\mathop{\ln\/}\nolimits\!\left(\pi^{{-\frac{1}{2}}}2^{{-a-\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+a\right)\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $s$ : nonnegative integer, $a$ : real or complex parameter and $\tau _{s}$
Permalink:: http://dlmf.nist.gov/12.11.E2
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and

12.11.3 $\lambda _{s}=\mathop{\ln\/}\nolimits\tau _{s}-\tfrac{1}{2}\pi i.$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $s$ : nonnegative integer, $\tau _{s}$ and $\lambda _{s}$
Permalink:: http://dlmf.nist.gov/12.11.E3
Encodings:: TeX, pMML, png

When $a=\tfrac{1}{2}$ these zeros are the same as the zeros of the complementary error function $\mathop{\mathrm{erfc}\/}\nolimits(z/\sqrt{2})$ ; 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).

§12.11(iii) Asymptotic Expansions for Large Parameter

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 $w^{5}$ .
Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.11.iii

For large negative values of $a$ the real zeros of $\mathop{U\/}\nolimits\!\left(a,x\right)$ , ${\mathop{U\/}\nolimits^{{\prime}}}\!\left(a,x\right)$ , $\mathop{V\/}\nolimits\!\left(a,x\right)$ , and ${\mathop{V\/}\nolimits^{{\prime}}}\!\left(a,x\right)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). For example, let the $s$ th real zeros of $\mathop{U\/}\nolimits\!\left(a,x\right)$ and ${\mathop{U\/}\nolimits^{{\prime}}}\!\left(a,x\right)$ , 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 $u_{{a,s}}\sim 2^{{\frac{1}{2}}}\mu\left(p_{0}(\alpha)+\frac{p_{1}(\alpha)}{\mu^{4}}+\frac{p_{2}(\alpha)}{\mu^{8}}+\cdots\right),$

Symbols:: $\sim$ : asymptotic equality, $s$ : nonnegative integer, $a$ : real or complex parameter, $u_{{a,s}}$ : zeros, $\alpha$ and $p_{n}(\zeta)$ : coefficients
Referenced by:: §12.11(iii)
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as $\mu$ ( $=\sqrt{-2a}$ ) $\to\infty$ , $s$ fixed. Here $\alpha=\mu^{{-\frac{4}{3}}}a_{s}$ , $a_{s}$ denoting the $s$ th negative zero of the function $\mathop{\mathrm{Ai}\/}\nolimits$ (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
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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
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Similarly, for the zeros of ${\mathop{U\/}\nolimits^{{\prime}}}\!\left(a,x\right)$ we have

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

Symbols:: $\sim$ : asymptotic equality, $s$ : nonnegative integer, $a$ : real or complex parameter, $u^{{\prime}}_{{a,s}}$ : zeros and $\beta$
Permalink:: http://dlmf.nist.gov/12.11.E7
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where $\beta=\mu^{{-\frac{4}{3}}}a^{{\prime}}_{s}$ , $a^{{\prime}}_{s}$ denoting the $s$ th negative zero of the function ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}$ and

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

Symbols:: $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/12.11.E8
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For the first zero of $\mathop{U\/}\nolimits\!\left(a,x\right)$ we also have

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

Symbols:: $\sim$ : asymptotic equality, $a$ : 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

where the numerical coefficients have been rounded off.

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

§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