in terms of parabolic cylinder functions

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11: 12.9 Asymptotic Expansions for Large Variable

§12.9 Asymptotic Expansions for Large Variable

… ►

12.9.1

U\left(a,z\right)\sim e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum_{s=0}^{% \infty}(-1)^{s}\frac{{\left(\frac{1}{2}+a\right)_{2s}}}{s!(2z^{2})^{s}},

|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $z$ : complex variable, $s$ : nonnegative integer, $a$ : real or complex parameter and $\delta$ : arbitrary small positive constant
A&S Ref:: 19.8.1 (modification of)
Referenced by:: §12.9(i), §12.9(ii)
Permalink:: http://dlmf.nist.gov/12.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §12.9(i), §12.9 and Ch.12

… ►To obtain approximations for

U\left(a,-z\right)

and

V\left(a,-z\right)

z\to\infty

combine the results above with (12.2.15) and (12.2.16). … ►

§12.9(ii) Bounds and Re-Expansions for the Remainder Terms

►Bounds and re-expansions for the error term in (12.9.1) can be obtained by use of (12.7.14) and §§13.7(ii), 13.7(iii). …

12: 14.15 Uniform Asymptotic Approximations

… ►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials

P_{n}\left(\cos\theta\right)

n\to\infty

with

\theta

fixed. … ►In (14.15.15)–(14.15.18) … ►Here we introduce the envelopes of the parabolic cylinder functions

U\left(-c,x\right)

\overline{U}\left(-c,x\right)

, which are defined in §12.2. For

U\left(-c,x\right)

\overline{U}\left(-c,x\right)

, with

c

and

x

nonnegative, …where

x=X_{c}

denotes the largest positive root of the equation

U\left(-c,x\right)=\overline{U}\left(-c,x\right)

. …

13: Bibliography R

… ►

H. A. Ragheb, L. Shafai, and M. Hamid (1991) Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric. IEEE Trans. Antennas and Propagation 39 (2), pp. 218–223.

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External Links:: Document
Referenced by:: 6th item.
Encodings:: BibTeX

… ►

R. Reynolds and A. Stauffer (2021) Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

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External Links:: Link, ISSN 2227-7390, Document
Referenced by:: §8.15.
Encodings:: BibTeX

… ►

S. O. Rice (1954) Diffraction of plane radio waves by a parabolic cylinder. Calculation of shadows behind hills. Bell System Tech. J. 33, pp. 417–504.

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External Links:: MathReview (J. Shmoys)
Referenced by:: §12.17.
Encodings:: BibTeX

… ►

W. Rudin (1973) Functional Analysis. McGraw-Hill Book Co., New York.

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Notes:: McGraw-Hill Series in Higher Mathematics
External Links:: MathReview (F. Smithies), ZentralBlatt
Referenced by:: §1.18(x), §2.6(i).
Encodings:: BibTeX

… ►

J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

…

14: 18.15 Asymptotic Approximations

… ►These expansions are in terms of Whittaker functions (§13.14). … ►

In Terms of Elementary Functions

… ►

In Terms of Bessel Functions

… ►

In Terms of Airy Functions

… ►With

\mu=\sqrt{2n+1}

the expansions in Chapter 12 are for the parabolic cylinder function

U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)

, which is related to the Hermite polynomials via …

15: 12.11 Zeros

… ►

§12.11(i) Distribution of Real Zeros

… ►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

x=0

. ►

§12.11(ii) Asymptotic Expansions of Large Zeros

… ►

§12.11(iii) Asymptotic Expansions for Large Parameter

… ►For example, let the

s

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. …

16: 13.20 Uniform Asymptotic Approximations for Large $\mu$

… ►For the parabolic cylinder function

U

see §12.2. … ►For the parabolic cylinder functions

U

and

\overline{U}

see §12.2, and for the

\mathrm{env}

functions associated with

U

and

\overline{U}

see §14.15(v). … ►11) in this reference. … ►It should be noted that (13.20.11), (13.20.16), and (13.20.18) differ only in the common error terms. … ►These approximations are in terms of Airy functions. …

17: Bibliography M

… ►

J. C. P. Miller (Ed.) (1955) Tables of Weber Parabolic Cylinder Functions. Her Majesty’s Stationery Office, London.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §10.16, §10.39, §12.14(i), §12.14(ii), §12.14(iv), §12.14(v), §12.14(v), §12.14(vi), §12.14(vii), §12.14(viii), §12.14(x), §12.14(x), 2nd item, §12.2(i), §12.2(ii), §12.2(iii), §12.2(iv), §12.2(v), §12.2(vi), §12.2(vi), §12.4, §12.5(i), §12.5(ii), §12.5(iii), §12.7(i), §12.7(ii), §12.7(iii), §12.7(iii), §12.7(iv), §12.8(i), Ch.12.
Encodings:: BibTeX

… ►

W. Miller (1974) Lie theory and separation of variables. I: Parabolic cylinder coordinates. SIAM J. Math. Anal. 5 (4), pp. 626–643.

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External Links:: Document, MathReview (G. R. Allcock), ZentralBlatt
Referenced by:: §12.13(ii), §12.17.
Encodings:: BibTeX

… ►

H. J. W. Müller (1966b) Asymptotic expansions of ellipsoidal wave functions in terms of Hermite functions. Math. Nachr. 32, pp. 49–62.

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External Links:: Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §29.11, §29.7(ii).
Encodings:: BibTeX

… ►

K. H. Müller (1988) Elastodynamics in parabolic cylinders. Z. Angew. Math. Phys. 39 (5), pp. 748–752.

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External Links:: ISSN 0044-2275, Document, MathReview (Sergej M. Zverev), ZentralBlatt
Referenced by:: §12.17.
Encodings:: BibTeX

… ►

J. Murzewski and A. Sowa (1972) Tables of the functions of the parabolic cylinder for negative integer parameters. Zastos. Mat. 13, pp. 261–273.

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External Links:: MathReview, ZentralBlatt
Referenced by:: 7th item.
Encodings:: BibTeX

…

18: 28.8 Asymptotic Expansions for Large $q$

… ►These results are derived formally in Sips (1949, 1959, 1965). … ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►The approximations are expressed in terms of Whittaker functions

W_{\kappa,\mu}\left(z\right)

and

M_{\kappa,\mu}\left(z\right)

with

\mu=\tfrac{1}{4}

; compare §2.8(vi). …With additional restrictions on

z

, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions

\operatorname{me}_{\nu}\left(z,q\right)

(§28.12(ii)) and modified Mathieu functions

{\operatorname{M}^{(j)}_{\nu}}\left(z,h\right)

(§28.20(iii)). …

19: 13.6 Relations to Other Functions

… ►

§13.6(i) Elementary Functions

… ►

§13.6(iv) Parabolic Cylinder Functions

… ►

§13.6(vi) Generalized Hypergeometric Functions

… ►

§13.6(vii) Coulomb Functions

►For representations of Coulomb functions in terms of Kummer functions see (33.2.4), (33.2.8) and (33.14.5).

20: 18.30 Associated OP’s

… ►Assuming equation (18.2.8) with its initialization defines a set of OP’s,

p_{n}(x)

, the corresponding associated orthogonal polynomials of order

c

are the

p_{n}(x;c)

as defined by shifting the index

n

in the recurrence coefficients by adding a constant

c

, functions of

n

, say

f(n)

, being replaced by

f(n+c)

. …However, if the recurrence coefficients are polynomial, or rational, functions of

n

, polynomials of degree

n

may be well defined for

c\in\mathbb{R}

provided that

A_{n+c}B_{n+c}\neq 0,n=0,1,\dots

Askey and Wimp (1984). … ►For the parabolic cylinder function

U

see §12.2(i). … ►They can be expressed in terms of type 3 Pollaczek polynomials (which are also associated type 2 Pollaczek polynomials) by (18.35.10). … ►The ratio

p_{n}^{(0)}(z)/p_{n}(z)

, as defined here, thus provides the same statement of Markov’s Theorem, as in (18.2.9_5), but now in terms of differently obtained numerator and denominator polynomials. …