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21: 14 Legendre and Related Functions
Chapter 14 Legendre and Related Functions
…22: 16.25 Methods of Computation
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►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations.
They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19.
There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations).
…Instead a boundary-value problem needs to be formulated and solved.
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23: 26.9 Integer Partitions: Restricted Number and Part Size
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►Conjugation establishes a one-to-one correspondence between partitions of into at most parts and partitions of into parts with largest part less than or equal to
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It follows that also equals the number of partitions of into parts that are less than or equal to
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§26.9(iii) Recurrence Relations
… ►where the inner sum is taken over all positive divisors of that are less than or equal to . … ►As with fixed, …24: 19.10 Relations to Other Functions
§19.10 Relations to Other Functions
►§19.10(i) Theta and Elliptic Functions
►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. … ►§19.10(ii) Elementary Functions
… ►For relations to the Gudermannian function and its inverse (§4.23(viii)), see (19.6.8) and …25: Bibliography N
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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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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors.
2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
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Confluent hypergeometric equations and related solvable potentials in quantum mechanics.
J. Math. Phys. 41 (12), pp. 7964–7996.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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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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26: 9 Airy and Related Functions
Chapter 9 Airy and Related Functions
…27: 36.5 Stokes Sets
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►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set:
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►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4).
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►The first sheet corresponds to
and is generated as a solution of Equations (36.5.6)–(36.5.9).
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►This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the -axis by .
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►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets.
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28: 25.17 Physical Applications
§25.17 Physical Applications
… ►This relates to a suggestion of Hilbert and Pólya that the zeros are eigenvalues of some operator, and the Riemann hypothesis is true if that operator is Hermitian. … ►Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). It has been found possible to perform such regularizations by equating the divergent sums to zeta functions and associated functions (Elizalde (1995)).29: 36 Integrals with Coalescing Saddles
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