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21: 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)).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: 6.16 Mathematical Applications
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►Hence, if is fixed and , then , , or according as , , or ; compare (6.2.14).
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►Hence if and , then the limiting value of overshoots by approximately 18%.
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►If we assume Riemann’s hypothesis that all nonreal zeros of have real part of (§25.10(i)), then
…where is the number of primes less than or equal to
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24: 9 Airy and Related Functions
Chapter 9 Airy and Related Functions
…25: 11 Struve and Related Functions
Chapter 11 Struve and Related Functions
…26: Bibliography L
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Algorithm 917: complex double-precision evaluation of the Wright function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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Recurrence relations for hypergeometric functions of unit argument.
Math. Comp. 45 (172), pp. 521–535.
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Monotonic sequences related to zeros of Bessel functions.
Numer. Algorithms 49 (1-4), pp. 221–233.
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New recursion relation for the Clebsch-Gordan coefficients.
Phys. Rev. (2) 110 (4), pp. 815–816.
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27: Bibliography I
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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Special Functions, -Series and Related Topics.
Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.
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Two families of orthogonal polynomials related to Jacobi polynomials.
Rocky Mountain J. Math. 21 (1), pp. 359–375.
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Bounds for the small real and purely imaginary zeros of Bessel and related functions.
Methods Appl. Anal. 2 (1), pp. 1–21.
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The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent.
Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).
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28: 16.24 Physical Applications
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►For an extension to two-loop integrals see Moch et al. (2002).
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