relation to parabolic cylinder functions
(0.013 seconds)
11—20 of 31 matching pages
11: 16.18 Special Cases
§16.18 Special Cases
… ►This is a consequence of the following relations: …As a corollary, special cases of the and functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer -function. Representations of special functions in terms of the Meijer -function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).12: 18.11 Relations to Other Functions
§18.11 Relations to Other Functions
… ►Ultraspherical
… ►Laguerre
… ►Hermite
… ►For the parabolic cylinder function , see §12.2. …13: Bibliography S
…
►
Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments.
Comput. Phys. Comm. 115 (1), pp. 69–86.
►
Evaluation of associated Legendre functions off the cut and parabolic cylinder functions.
Electron. Trans. Numer. Anal. 9, pp. 137–146.
…
►
On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
►
On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series.
J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
…
►
On parabolic cylinder functions.
J. Inst. Math. Appl. 4 (1), pp. 106–112.
…
14: Bibliography B
…
►
A note on integrals involving parabolic cylinder functions.
SIAM J. Appl. Math. 16 (1), pp. 71–74.
…
►
Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind.
U.S. Government Printing Office, Washington, D.C..
►
Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind.
U.S. Government Printing Office, Washington, D.C..
…
►
The asymptotic analysis of canonical problems in high-frequency scattering theory. II. The circular and parabolic cylinders.
Proc. Cambridge Philos. Soc. 74, pp. 313–332.
…
►
Exponential asymptotics for an eigenvalue of a problem involving parabolic cylinder functions.
Proc. Amer. Math. Soc. 114 (4), pp. 1025–1032.
…
15: 28.8 Asymptotic Expansions for Large
…
►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3).
…
►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5).
…
►Then as
…
►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8.
…
►For related results see Langer (1934) and Sharples (1967, 1971).
…
16: 18.30 Associated OP’s
…
►Associated polynomials and the related corecursive polynomials appear in Ismail (2009, §§2.3, 2.6, and 2.10), where the relationship of OP’s to continued fractions is made evident.
…
►
§18.30(i) Associated Jacobi Polynomials
… ►For the parabolic cylinder function see §12.2(i). … ►Defining associated orthogonal polynomials and their relationship to their corecursive counterparts is particularly simple via use of the recursion relations for the monic, rather than via those for the traditional polynomials. … ►See Ismail (2009, p. 46 ), where the th corecursive polynomial is also related to an appropriate continued fraction, given here as its th convergent, …17: 12.8 Recurrence Relations and Derivatives
§12.8 Recurrence Relations and Derivatives
►§12.8(i) Recurrence Relations
… ►(12.8.1)–(12.8.4) are also satisfied by . … ►§12.8(ii) Derivatives
… ►
12.8.10
…
18: Bibliography L
…
►
Functions on a parabolic cylinder with a negative integer index.
Differ. Uravn. 21 (11), pp. 2001–2003, 2024 (Russian).
…
►
Recurrence relations for hypergeometric functions of unit argument.
Math. Comp. 45 (172), pp. 521–535.
►
Corrigenda: “Recurrence relations for hypergeometric functions of unit argument” [Math. Comp. 45 (1985), no. 172, 521–535; MR 86m:33004].
Math. Comp. 48 (178), pp. 853.
…
►
Monotonic sequences related to zeros of Bessel functions.
Numer. Algorithms 49 (1-4), pp. 221–233.
…
►
Integral representation of the Hankel function in terms of parabolic cylinder functions.
Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.
…
19: Bibliography M
…
►
Evaluation of complex logarithms and related functions.
SIAM J. Numer. Anal. 18 (4), pp. 744–750.
…
►
Tables of Weber Parabolic Cylinder Functions.
Her Majesty’s Stationery Office, London.
…
►
Lie theory and separation of variables. I: Parabolic cylinder coordinates.
SIAM J. Math. Anal. 5 (4), pp. 626–643.
…
►
Elastodynamics in parabolic cylinders.
Z. Angew. Math. Phys. 39 (5), pp. 748–752.
…
►
Tables of the functions of the parabolic cylinder for negative integer parameters.
Zastos. Mat. 13, pp. 261–273.
…
20: 32.11 Asymptotic Approximations for Real Variables
…
►Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to
, for some nonzero real , where denotes the Airy function (§9.2).
…
►The connection formulas relating (32.11.25) and (32.11.26) are
…
►Any nontrivial solution of (32.11.29) that satisfies (32.11.30) is asymptotic to
as , where
is a constant.
Conversely, for any
there is a unique solution of (32.11.29) that is asymptotic to
as .
Here denotes the parabolic cylinder function (§12.2).
…