distributional derivative
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1: 1.16 Distributions
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§1.16(ii) Derivatives of a Distribution
… βΊIf is a locally integrable function then its distributional derivative is . In the situation of (1.16.3_5) we have … βΊThe derivatives of tempered distributions are defined in the same way as derivatives of distributions. … βΊThe distributional derivative of is defined by …2: 2.6 Distributional Methods
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βΊMotivated by the definition of distributional derivatives, we can assign them the distributions defined by
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βΊTo define convolutions of distributions, we first introduce the space of all distributions of the form , where is a nonnegative integer, is a locally integrable function on which vanishes on , and denotes the th derivative of the distribution associated with .
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2.6.37
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2.6.40
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βΊSince the function and all its derivatives are locally absolutely continuous in , the distributional derivatives in the first sum in (2.6.44) can be replaced by the corresponding ordinary derivatives.
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3: 10.21 Zeros
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§10.21(i) Distribution
… βΊ§10.21(ix) Complex Zeros
βΊThis subsection describes the distribution in of the zeros of the principal branches of the Bessel functions of the second and third kinds, and their derivatives, in the case when the order is a positive integer . … βΊ …4: 1.1 Special Notation
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real variables. | |
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action of distribution on test function . | |
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primes | derivatives with respect to the variable, except where indicated otherwise. |
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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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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 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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The Riemann zeta-function and its derivatives.
Proc. Roy. Soc. London Ser. A 450, pp. 477–499.
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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: 1.17 Integral and Series Representations of the Dirac Delta
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βΊIn applications in physics, engineering, and applied mathematics, (see Friedman (1990)), the Dirac delta distribution (§1.16(iii)) is historically and customarily replaced by the Dirac delta (or Dirac delta function) .
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βΊIn the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non- improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly.
Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv).
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9: 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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10.73.2
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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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10: 7.20 Mathematical Applications
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βΊFurthermore, because , the angle between the -axis and the tangent to the spiral at is given by .
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βΊThe normal distribution function with mean and standard deviation is given by
…For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).