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11: Bibliography B
  • A. W. Babister (1967) Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations. The Macmillan Co., New York.
  • W. G. C. Boyd and T. M. Dunster (1986) Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions. SIAM J. Math. Anal. 17 (2), pp. 422–450.
  • W. G. C. Boyd (1990a) Asymptotic Expansions for the Coefficient Functions Associated with Linear Second-order Differential Equations: The Simple Pole Case. In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.), Lecture Notes in Pure and Applied Mathematics, Vol. 124, pp. 53–73.
  • 12: 10.25 Definitions
    Corresponding to the symbol 𝒞 ν introduced in §10.2(ii), we sometimes use 𝒵 ν ( z ) to denote I ν ( z ) , e ν π i K ν ( z ) , or any nontrivial linear combination of these functions, the coefficients in which are independent of z and ν . …
    13: Bibliography M
  • Meta.Numerics (website) David Wright’s software package for .NET programming language
  • mpmath (free python library)
  • 14: 10.2 Definitions
    The notation 𝒞 ν ( z ) denotes J ν ( z ) , Y ν ( z ) , H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) , or any nontrivial linear combination of these functions, the coefficients in which are independent of z and ν . …
    15: 2.9 Difference Equations
    Many special functions that depend on parameters satisfy a three-term linear recurrence relation …
    16: William P. Reinhardt
    He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. …
    17: 3.11 Approximation Techniques
    Also, in cases where f ( x ) satisfies a linear ordinary differential equation with polynomial coefficients, the expansion (3.11.11) can be substituted in the differential equation to yield a recurrence relation satisfied by the c n . … With b 0 = 1 , the last q equations give b 1 , , b q as the solution of a system of linear equations. … (3.11.29) is a system of n + 1 linear equations for the coefficients a 0 , a 1 , , a n . … More generally, let f ( x ) be approximated by a linear combination … We take n complex exponentials ϕ k ( x ) = e i k x , k = 0 , 1 , , n - 1 , and approximate f ( x ) by the linear combination (3.11.31). …
    18: Bibliography S
  • B. D. Sleeman (1969) Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.
  • 19: 16.14 Partial Differential Equations
    In addition to the four Appell functions there are 24 other sums of double series that cannot be expressed as a product of two F 1 2 functions, and which satisfy pairs of linear partial differential equations of the second order. …
    20: 9.15 Mathematical Applications
    Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …