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§12.10 Uniform Asymptotic Expansions for Large Parameter

Contents

§12.10(i) Introduction

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Referenced by:
§18.15(v)
Permalink:
http://dlmf.nist.gov/12.10.i

In this section we give asymptotic expansions of PCFs for large values of the parameter a that are uniform with respect to the variable z, when both a and z (=x) are real. These expansions follow from Olver (1959), where detailed information is also given for complex variables.

With the transformations

(12.2.2) becomes

12.10.2\frac{{d}^{2}w}{{dt}^{2}}=\mu^{4}(t^{2}\pm 1)w.
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Symbols:
\frac{df}{dx}: derivative of f with respect to x
Referenced by:
§12.10(i)
Permalink:
http://dlmf.nist.gov/12.10.E2
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With the upper sign in (12.10.2), expansions can be constructed for large \mu in terms of elementary functions that are uniform for t\in(-\infty,\infty) (§2.8(ii)). With the lower sign there are turning points at t=\pm 1, which need to be excluded from the regions of validity. These cases are treated in §§12.10(ii)–12.10(vi).

The turning points can be included if expansions in terms of Airy functions are used instead of elementary functions (§2.8(iii)). These cases are treated in §§12.10(vii)–12.10(viii).

Throughout this section the symbol \delta again denotes an arbitrary small positive constant.

§12.10(ii) Negative a, 2\sqrt{-a}<x<\infty

As a\to-\infty

uniformly for t\in[1+\delta,\infty), where

The coefficients are given by

12.10.8{\cal A}_{s}(t)=\frac{u_{s}(t)}{(t^{2}-1)^{{\frac{3}{2}s}}},~{}{\cal B}_{s}(t)% =\frac{v_{s}(t)}{(t^{2}-1)^{{\frac{3}{2}s}}},
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Symbols:
s: nonnegative integer, \mathcal{A}_{s}(t): coefficients, \mathcal{B}_{s}(t): coefficients, u_{s}(t): solution and v_{s}(t): solution
Referenced by:
§12.10(iv)
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http://dlmf.nist.gov/12.10.E8
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where u_{s}(t) and v_{s}(t) are polynomials in t of degree 3s, (s odd), 3s-2 (s even, s\geq 2). For s=0,1,2,

12.10.9
u_{0}(t)=1,
u_{1}(t)=\frac{t(t^{2}-6)}{24},
u_{2}(t)=\frac{-9t^{4}+249t^{2}+145}{1152},
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Symbols:
s: nonnegative integer and u_{s}(t): solution
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12.10.10
v_{0}(t)=1,
v_{1}(t)=\frac{t(t^{2}+6)}{24},
v_{2}(t)=\frac{15t^{4}-327t^{2}-143}{1152}.
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Symbols:
s: nonnegative integer and v_{s}(t): solution
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Higher polynomials u_{s}(t) can be calculated from the recurrence relation

where

12.10.128r_{s}(t)=(3t^{2}+2)u_{s}(t)-12(s+1)tr_{{s-1}}(t)+4(t^{2}-1)r^{{\prime}}_{{s-1% }}(t),
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Symbols:
s: nonnegative integer, u_{s}(t): solution and r_{s}(t): polynomial
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http://dlmf.nist.gov/12.10.E12
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and the v_{s}(t) then follow from

Lastly, the function g(\mu) in (12.10.3) and (12.10.4) has the asymptotic expansion:

12.10.14g(\mu)\sim h(\mu)\left(1+\frac{1}{2}\sum_{{s=1}}^{\infty}\frac{\gamma_{s}}{(% \frac{1}{2}\mu^{2})^{s}}\right),

where

12.10.15h(\mu)=2^{{-\frac{1}{4}\mu^{2}-\frac{1}{4}}}e^{{-\frac{1}{4}\mu^{2}}}\mu^{{% \frac{1}{2}\mu^{2}-\frac{1}{2}}},
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Symbols:
e: base of exponential function and h(\mu): expansion
Referenced by:
§12.10(vi)
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and the coefficients \gamma_{s} are defined by

compare (5.11.8). For s\leq 4

12.10.17
\gamma_{0}=1,
\gamma_{1}=-\tfrac{1}{24},
\gamma_{2}=\tfrac{1}{1152},
\gamma_{3}=\tfrac{1003}{4\;14720},
\gamma_{4}=-\tfrac{4027}{398\;13120}.
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Symbols:
\gamma_{s}: coefficients
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http://dlmf.nist.gov/12.10.E17
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§12.10(iii) Negative a, -\infty<x<-2\sqrt{-a}

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Notes:
See Olver (1959).
Permalink:
http://dlmf.nist.gov/12.10.iii

When \mu\to\infty, asymptotic expansions for the functions \mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right) and \mathop{V\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right) that are uniform for t\in[1+\delta,\infty) are obtainable by substitution into (12.2.15) and (12.2.16) by means of (12.10.3) and (12.10.5). Similarly for {\mathop{U\/}\nolimits^{{\prime}}}\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right) and {\mathop{V\/}\nolimits^{{\prime}}}\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right).

§12.10(iv) Negative a, -2\sqrt{-a}<x<2\sqrt{-a}

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Notes:
See Olver (1959).
Referenced by:
§18.15(v)
Permalink:
http://dlmf.nist.gov/12.10.iv

As a\to-\infty

uniformly for t\in[-1+\delta,1-\delta]. The quantities \kappa and \chi are defined by

12.10.22
\kappa=\mu^{2}\eta-\tfrac{1}{4}\pi,
\chi=\mu^{2}\eta+\tfrac{1}{4}\pi,
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Symbols:
\kappa, \chi and \eta
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where

12.10.23\eta=\tfrac{1}{2}\mathop{\mathrm{arccos}\/}\nolimits t-\tfrac{1}{2}t\sqrt{1-t^% {2}},
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Symbols:
\mathop{\mathrm{arccos}\/}\nolimits z: arccosine function and \eta
Referenced by:
§12.10(vii), ‣ §12.14(ix)
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and the coefficients \widetilde{\cal{A}}_{s}(t) and \widetilde{\cal{B}}_{s}(t) are given by

12.10.24
\widetilde{\cal{A}}_{s}(t)=\frac{u_{s}(t)}{(1-t^{2})^{{\frac{3}{2}s}}},
\widetilde{\cal{B}}_{s}(t)=\frac{v_{s}(t)}{(1-t^{2})^{{\frac{3}{2}s}}};

compare (12.10.8).

§12.10(v) Positive a, -\infty<x<\infty

As a\to\infty

uniformly for t\in\Real. Here bars do not denote complex conjugates; instead

12.10.27\overline{u}_{s}(t)=i^{s}u_{s}(-it),

and the function \overline{g}(\mu) has the asymptotic expansion

where h(\mu) and \gamma_{s} are as in §12.10(ii).

§12.10(vi) Modifications of Expansions in Elementary Functions

In Temme (2000) modifications are given of Olver’s expansions. An example is the following modification of (12.10.3)

where \xi and h(\mu) are as in (12.10.7) and (12.10.15) ,

12.10.32\tau=\frac{1}{2}\left(\frac{t}{\sqrt{t^{2}-1}}-1\right),
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Symbols:
\tau
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and the coefficients \mathsf{A}_{s}(\tau) are the product of \tau^{s} and a polynomial in \tau of degree 2s. They satisfy the recursion

starting with \mathsf{A}_{o}(\tau)=1. Explicitly,

12.10.34
\mathsf{A}_{1}(\tau)=-\tfrac{1}{12}\tau(20\tau^{2}+30\tau+9),
\mathsf{A}_{2}(\tau)=\tfrac{1}{288}\tau^{2}(6160\tau^{4}+18480\tau^{3}+19404% \tau^{2}+8028\tau+945).
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Symbols:
\tau and \mathsf{A}_{s}(\tau): coefficients
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http://dlmf.nist.gov/12.10.E34
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The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when \mu\to\infty uniformly with respect to t\in[1+\delta,\infty). In addition, it enjoys a double asymptotic property: it holds if either or both \mu and t tend to infinity. Observe that if t\to\infty, then \mathsf{A}_{s}(\tau)=\mathop{O\/}\nolimits\!\left(t^{{-2s}}\right), whereas {\cal A}_{s}(t)=\mathop{O\/}\nolimits(1) or \mathop{O\/}\nolimits\!\left(t^{{-2}}\right) according as s is even or odd. The proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv).

For additional information see Temme (2000). See also Olver (1997b, pp. 206–208) and Jones (2006).

§12.10(vii) Negative a, -2\sqrt{-a}<x<\infty. Expansions in Terms of Airy Functions

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Notes:
See Olver (1959). Equations (12.10.42)–(12.10.46) are rearrangements of Olver’s results and have the advantage of avoiding the many-valued functions in the explicit expressions for A_{s}(\zeta), B_{s}(\zeta), C_{s}(\zeta), and D_{s}(\zeta).
Keywords:
parabolic cylinder functions
Referenced by:
§12.10(i), §12.11(iii), ‣ §12.14(ix), §18.15(v), §2.8(iii)
Permalink:
http://dlmf.nist.gov/12.10.vii

The following expansions hold for large positive real values of \mu, uniformly for t\in[-1+\delta,\infty). (For complex values of \mu and t see Olver (1959).)

The variable \zeta is defined by

12.10.39
\tfrac{2}{3}\zeta^{{\frac{3}{2}}}=\xi,\quad\text{$1\leq t$},\text{($\zeta\geq 0% $)};
\tfrac{2}{3}(-\zeta)^{{\frac{3}{2}}}=\eta,\quad\text{$-1<t\leq 1$},\text{($% \zeta\leq 0$)},
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Symbols:
\xi, \eta and \zeta: change of variable
Referenced by:
§12.11(iii)
Permalink:
http://dlmf.nist.gov/12.10.E39
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where \xi,\eta are given by (12.10.7), (12.10.23), respectively, and

12.10.40\phi(\zeta)=\left(\frac{\zeta}{t^{2}-1}\right)^{{\frac{1}{4}}}.
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Symbols:
\zeta: change of variable and \phi(\zeta): function
Referenced by:
§12.10(vii)
Permalink:
http://dlmf.nist.gov/12.10.E40
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The function \zeta=\zeta(t) is real for t>-1 and analytic at t=1. Inversely, with w=2^{{-\frac{1}{3}}}\zeta,

12.10.41t=1+w-\tfrac{1}{10}w^{2}+\tfrac{11}{350}w^{3}-\tfrac{823}{63000}w^{4}+\tfrac{1% \;50653}{242\;55000}w^{5}+\cdots,|\zeta|<\left(\tfrac{3}{4}\pi\right)^{{\frac{2}{3}}}.
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Symbols:
\zeta: change of variable
Referenced by:
§12.11(iii), §12.11(iii)
Permalink:
http://dlmf.nist.gov/12.10.E41
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For g(\mu) see (12.10.14). The coefficients A_{s}(\zeta) and B_{s}(\zeta) are given by

where \phi(\zeta) is as in (12.10.40), u_{k}(t) is as in §12.10(ii), \alpha_{0}=1, and

12.10.43
\alpha_{m}=\frac{(2m+1)(2m+3)\cdots(6m-1)}{m!(144)^{m}},
\beta_{m}=-\frac{6m+1}{6m-1}\alpha_{m}.
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Symbols:
!: n!: factorial, \alpha_{m}: coefficients and \beta_{m}: coefficients
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The coefficients C_{s}(\zeta) and D_{s}(\zeta) in (12.10.36) and (12.10.38) are given by

where

12.10.45\chi(\zeta)=\frac{\phi^{{\prime}}(\zeta)}{\phi(\zeta)}=\frac{1-2t(\phi(\zeta))% ^{6}}{4\zeta}.

Explicitly,

where v_{k}(t) is as in §12.10(ii).

Modified Expansions

The expansions (12.10.35)–(12.10.38) can be modified, again see Temme (2000), and the new expansions hold if either or both \mu and t tend to infinity. This is provable by the methods used in §10.41(v).

§12.10(viii) Negative a, -\infty<x<2\sqrt{-a}. Expansions in Terms of Airy Functions

When \mu\to\infty, asymptotic expansions for \mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right) and \mathop{V\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right) that are uniform for t\in[-1+\delta,\infty) are obtained by substitution into (12.2.15) and (12.2.16) by means of (12.10.35) and (12.10.37). Similarly for {\mathop{U\/}\nolimits^{{\prime}}}\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right) and {\mathop{V\/}\nolimits^{{\prime}}}\!\left(-\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right).

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