distributional%20derivative
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1: 10.73 Physical Applications
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βΊLaplace’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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10.73.1
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βΊSee 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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10.73.4
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βΊThe analysis of the current distribution in circular conductors leads to the Kelvin functions , , , and .
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2: 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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3: 3.8 Nonlinear Equations
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βΊFor other efficient derivative-free methods, see Le (1985).
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βΊFor 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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βΊConsider and .
We have and .
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3.8.16
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4: 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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5: Bibliography W
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The zeros of Euler’s psi function and its derivatives.
J. Math. Anal. Appl. 332 (1), pp. 607–616.
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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter
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Comput. Methods Funct. Theory 9 (2), pp. 579–591.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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Rapid approximation to the Voigt/Faddeeva function and its derivatives.
J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.
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The generalised product moment distribution in samples from a normal multivariate population.
Biometrika 20A, pp. 32–52.
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6: 9.9 Zeros
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§9.9(i) Distribution and Notation
… βΊThey are denoted by , , , , respectively, arranged in ascending order of absolute value for … βΊFor the distribution in of the zeros of , where is an arbitrary complex constant, see MuraveΔ (1976) and Gil and Segura (2014). … βΊ§9.9(iii) Derivatives With Respect to
… βΊFor error bounds for the asymptotic expansions of , , , and see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …7: 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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