Conical functions of purely imaginary order and argument

2013 ◽  
Vol 143 (5) ◽  
pp. 929-955
Author(s):  
T. M. Dunster
Keyword(s):  
Closed Geodesic ◽  
Asymptotic Approximations ◽  
Parabolic Cylinder ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Approximations And Expansions ◽  
Fourier Expansions ◽  
Parabolic Cylinder Functions ◽  
Conical Form

Associated Legendre functions are studied for the case where the degree is in conical form −½ + iτ (τ real), and the order iμ and argument ix are purely imaginary (μ and x real). Conical functions in this form have applications to Fourier expansions of the eigenfunctions on a closed geodesic. Real-valued numerically satisfactory solutions are introduced which are continuous for all real x. Uniform asymptotic approximations and expansions are then derived for the cases where one or both of μ and τ are large; these results (which involve elementary, Airy, Bessel and parabolic cylinder functions) are uniformly valid for unbounded x.

Legendre functions with both parameters large

1975 ◽  
Vol 278 (1279) ◽  
pp. 175-185 ◽  
Keyword(s):  
Differential Equations ◽  
Preceding Paper ◽  
Asymptotic Approximations ◽  
Parabolic Cylinder ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Error Terms ◽  
Parabolic Cylinder Functions ◽  
Explicit Bounds ◽  
Linear Differential

By application of the theory for second-order linear differential equations with two turning points developed in the preceding paper, some new asymptotic approximations are obtained for the associated Legendre functions when both the degree n and order m are large. The approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid with respect to x ∈ ( − 1 , 1 ) and m / ( n + 1 2 ) ∈ [ δ , 1 + Δ ] where δ and ∆ are arbitrary fixed numbers such that 0 < δ < 1 and ∆ > 0. The values of m and n + ½ are either both real, or both purely imaginary. In all cases explicit bounds are supplied for the error terms associated with the approximations.

Second-order linear differential equations with two turning points

1975 ◽  
Vol 278 (1279) ◽  
pp. 137-174 ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Real Variable ◽  
Real Parameter ◽  
Asymptotic Approximations ◽  
Parabolic Cylinder ◽  
Absolute Value ◽  
Liouville Transformation ◽  
Parabolic Cylinder Functions ◽  
Linear Differential

Differential equations of the form d 2 w / d x 2 = { u 2 f ( u , a , x ) + g ( u , a , x ) } w are considered for large values of the real parameter u . Here x is a real variable ranging over an open, possibly infinite, interval ( x 1 , x 2 ), and a is a bounded real parameter. It is assumed that f {u, a, x) and g{u,a, x) are free from singularity within ( x 1 , x 2 ), and f (u, a, x) has exactly two zeros, which depend continuously on a and coincide for a certain value of a . Except in the neighbourhoods of the zeros, g(u,a,x) is small in absolute value compared with u 2 f(u, a, x ). By application of the Liouville transformation, the differential equation is converted into one of four standard forms, with continuous coefficients. Asymptotic approximations for the solutions are then constructed in terms of parabolic cylinder functions. These approximations are valid for large u , uniformly with respect to x ε ( x 1 , x 2 ) and also uniformly with respect to a . Each approximation is accompanied by a strict and realistic error bound. The paper also includes some new properties of parabolic cylinder functions.

Whittaker functions with both parameters large: uniform approximations in terms of parabolic cylinder functions

1980 ◽  
Vol 86 (3-4) ◽  
pp. 213-234 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
General Theory ◽  
Error Bounds ◽  
Asymptotic Theory ◽  
Open Interval ◽  
Asymptotic Approximations ◽  
Parabolic Cylinder ◽  
Whittaker Functions ◽  
Second Order Differential Equations ◽  
Uniform Approximations ◽  
Parabolic Cylinder Functions

SynopsisAsymptotic approximations are derived for the Whittaker functions Wκ,μ (z), Mκ, μ (z), Wικ, ιμ (iz) and Mικ, ιμ(iZ) for large positive values of the parameter μ that are uniform with respect to unrestricted values of the argument z in the open interval (0, ∞), and bounded real values of the ratio κ/μ. The approximations are in terms of parabolic cylinder functions, and in most instances are accompanied by strict error bounds.The results are derived by application of a recently-developed asymptotic theory of second-order differential equations having coalescing turning points, and an extension of the general theory of equations of this kind is also included.

scholarly journals On certain infinite integrals involving Struve functions and parabolic cylinder functions

1946 ◽  
Vol 7 (4) ◽  
pp. 171-173 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Present Note ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Struve Functions ◽  
Image Position

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

scholarly journals Derivatives of addition theorems for Legendre functions

1995 ◽  
Vol 37 (2) ◽  
pp. 212-234 ◽  
Author(s):  
D.E. Winch ◽  
P.H. Roberts
Keyword(s):  
Spherical Harmonics ◽  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Addition Theorems ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Derivatives Of

AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

scholarly journals Spatially Restricted Integrals in Gradiometric Boundary Value Problems

2009 ◽  
Vol 44 (4) ◽  
pp. 131-148 ◽  
Author(s):  
M. Eshagh
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Legendre Functions ◽  
Gravity Gradiometry ◽  
Satellite Gravity Gradiometry ◽  
Satellite Gravity ◽  
Earth’S Gravity Field ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Slepian Functions

Spatially Restricted Integrals in Gradiometric Boundary Value ProblemsThe spherical Slepian functions can be used to localize the solutions of the gradiometric boundary value problems on a sphere. These functions involve spatially restricted integral products of scalar, vector and tensor spherical harmonics. This paper formulates these integrals in terms of combinations of the Gaunt coefficients and integrals of associated Legendre functions. The presented formulas for these integrals are useful in recovering the Earth's gravity field locally from the satellite gravity gradiometry data.

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