special%20cases
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1: 36.2 Catastrophes and Canonical Integrals
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►Special cases: , fold catastrophe; , cusp catastrophe; , swallowtail catastrophe.
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§36.2(ii) Special Cases
… ►Addendum: For further special cases see §36.2(iv) … ►§36.2(iv) Addendum to 36.2(ii) Special Cases
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36.2.29
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2: 19.36 Methods of Computation
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►The computation is slowest for complete cases.
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►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively.
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►Also, see Todd (1975) for a special case of .
For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
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►These special theorems are also useful for checking computer codes.
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3: Bibliography K
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Special Functions and Perturbations of Black Holes.
In Special Functions (Hong Kong, 1999),
pp. 140–151.
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Special functions and the Bieberbach conjecture.
Amer. Math. Monthly 95 (8), pp. 689–696.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Some special cases of the generalized hypergeometric function
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J. Comput. Appl. Math. 78 (1), pp. 79–95.
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4: 25.12 Polylogarithms
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►The special case
is the Riemann zeta function: .
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►(In the latter case (25.12.11) becomes (25.5.1)).
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5: 36.4 Bifurcation Sets
6: 7.8 Inequalities
7: Bibliography S
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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The zeros of special functions from a fixed point method.
SIAM J. Numer. Anal. 40 (1), pp. 114–133.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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On certain special sets of orthogonal polynomials.
Proc. Amer. Math. Soc. 1, pp. 731–737.
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8: 20.11 Generalizations and Analogs
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►In the case
identities for theta functions become identities in the complex variable , with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7).
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►However, in this case
is no longer regarded as an independent complex variable within the unit circle, because is related to the variable of the theta functions via (20.9.2).
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►For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii).
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9: Bibliography N
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Toda equation and its solutions in special functions.
J. Phys. Soc. Japan 65 (6), pp. 1589–1597.
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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Mathematical Approximation of Special Functions.
Nova Science Publishers Inc., Commack, NY.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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