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1: 27.2 Functions
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27.2.1
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
Tables of primes (§27.21) reveal great irregularity in their distribution.
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27.2.3
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2: Bibliography F
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3: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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On the distribution of the length of the longest increasing subsequence of random permutations.
J. Amer. Math. Soc. 12 (4), pp. 1119–1178.
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Moment equations for probability distributions in classical and quantum mechanics.
Phys. Rev. A 58 (3), pp. 1799–1809.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and Procrustes analysis.
J. Multivariate Anal. 41 (2), pp. 314–337.
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4: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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The computation of lattice frequency distribution functions by means of continued fractions.
Physica 39 (3), pp. 413–423.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Algorithm 322. F-distribution.
Comm. ACM 11 (2), pp. 116–117.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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5: Bibliography P
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Handbook of the Normal Distribution.
Statistics: Textbooks and Monographs, Vol. 40, Marcel Dekker Inc., New York.
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The exact distribution of the Wald statistic.
Econometrica 54 (4), pp. 881–895.
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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6: Foreword
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►That 1046-page tome proved to be an invaluable reference for the many scientists and engineers who use the special functions of applied mathematics in their day-to-day work, so much so that it became the most widely distributed and most highly cited NIST publication in the first 100 years of the institution’s existence.
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►November 20, 2009
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7: Bibliography C
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Ellipsoidal distributions of charge or mass.
J. Mathematical Phys. 2, pp. 441–450.
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Algorithm AS 310. Computing the non-central beta distribution function.
Appl. Statist. 46 (1), pp. 146–156.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Calculation of angular distributions in complex angular momentum theories of elastic scattering.
Molecular Physics 37 (6), pp. 1703–1712.
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Some non-central distribution problems in multivariate analysis.
Ann. Math. Statist. 34 (4), pp. 1270–1285.
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8: 3.8 Nonlinear Equations
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►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013).
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3.8.15
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►Consider and .
We have and .
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3.8.16
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9: 10.73 Physical Applications
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►Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid.
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►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).
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►The analysis of the current distribution in circular conductors leads to the Kelvin functions , , , and .
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10: Bibliography K
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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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Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution.
Justus von Liebig Verlag, Darmstadt (German, English).
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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The Askey scheme as a four-manifold with corners.
Ramanujan J. 20 (3), pp. 409–439.
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