Dirac%20delta
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1: 1.17 Integral and Series Representations of the Dirac Delta
§1.17 Integral and Series Representations of the Dirac Delta
βΊ§1.17(i) Delta Sequences
… βΊSine and Cosine Functions
… βΊCoulomb Functions (§33.14(iv))
… βΊAiry Functions (§9.2)
…2: 25.20 Approximations
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βΊ
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Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.
3: 25.12 Polylogarithms
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βΊ
§25.12(iii) Fermi–Dirac and Bose–Einstein Integrals
βΊThe Fermi–Dirac and Bose–Einstein integrals are defined by βΊ
25.12.14
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βΊIn terms of polylogarithms
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βΊFor a uniform asymptotic approximation for see Temme and Olde Daalhuis (1990).
4: 1.16 Distributions
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βΊ
§1.16(iii) Dirac Delta Distribution
… βΊThe Dirac delta distribution is singular. … βΊWe use the notation of the previous subsection and take and in (1.16.35). … βΊSince , we have …Since the quantity on the extreme right of (1.16.41) is equal to , as distributions, the result in this equation can be stated as …5: 33.1 Special Notation
6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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βΊof the Dirac delta distribution.
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βΊApplying the representation (1.17.13), now symmetrized as in (1.17.14), as ,
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βΊThese latter results also correspond to use of the as defined in (1.17.12_1) and (1.17.12_2).
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βΊThus, and this is a case where is not continuous, if , , there will be an eigenfunction localized in the vicinity of , with a negative eigenvalue, thus disjoint from the continuous spectrum on .
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βΊFor fixed angular momentum the appropriate self-adjoint extension of the above operator may have both a discrete spectrum of negative eigenvalues , with corresponding eigenfunctions , and also a continuous spectrum , with Dirac-delta normalized eigenfunctions , also with measure .
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7: 33.14 Definitions and Basic Properties
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βΊThe function has the following properties:
βΊ
33.14.13
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βΊwhere the right-hand side is the Dirac delta (§1.17).
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33.14.15
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8: 25.19 Tables
9: Bibliography P
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Evaluation of Fermi-Dirac Integral.
In Nonlinear Numerical Methods and Rational Approximation
(Wilrijk, 1987), A. Cuyt (Ed.),
Mathematics and Its Applications, Vol. 43, pp. 435–444.
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Evaluation of the Fermi-Dirac integral of half-integer order.
Zastos. Mat. 21 (2), pp. 289–301.
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Numerical calculation of the generalized Fermi-Dirac integrals.
Comput. Phys. Comm. 55 (2), pp. 127–136.
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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