Chester–Friedman–Ursell method
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21: 16.25 Methods of Computation
§16.25 Methods of Computation
►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …22: 32.17 Methods of Computation
§32.17 Methods of Computation
►The Painlevé equations can be integrated by Runge–Kutta methods for ordinary differential equations; see §3.7(v), Hairer et al. (2000), and Butcher (2003). …23: 29.20 Methods of Computation
§29.20 Methods of Computation
… ►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …The numerical computations described in Jansen (1977) are based in part upon this method. ►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. …24: 1.17 Integral and Series Representations of the Dirac Delta
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►In applications in physics, engineering, and applied mathematics, (see Friedman (1990)), the Dirac delta distribution (§1.16(iii)) is historically and customarily replaced by the Dirac delta (or Dirac delta function) .
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►See Friedman (1990, p. 250).
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►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non- improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly.
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►A comprehensive and detailed applied mathematics approach is that of Friedman (1990, Ch. 3 and 4 ).
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25: 3.8 Nonlinear Equations
26: 19.29 Reduction of General Elliptic Integrals
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►For an implementation by James FitzSimons of the method for reducing to basic integrals and extensive tables of such reductions, see Carlson (1999) and Carlson and FitzSimons (2000).
►Another method of reduction is given in Gray (2002).
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