hyperbolic%20cases
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1: 36.2 Catastrophes and Canonical Integrals
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Normal Forms for Umbilic Catastrophes with Codimension
… ►(hyperbolic umbilic). ►Canonical Integrals
… ►§36.2(ii) Special Cases
… ►§36.2(iv) Addendum to 36.2(ii) Special Cases
…2: 36.4 Bifurcation Sets
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►Hyperbolic umbilic bifurcation set (codimension three):
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►Hyperbolic umbilic cusp line (rib):
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Special Cases
… ►§36.4(ii) Visualizations
…3: 7.8 Inequalities
4: Bibliography M
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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case.
J. Multivariate Anal. 77 (1), pp. 21–53.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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5: 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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►The step from to is an ascending Landen transformation if (leading ultimately to a hyperbolic case of ) or a descending Gauss transformation if (leading to a circular case of ).
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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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