integral representations for Dirac delta
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1: 1.17 Integral and Series Representations of the Dirac Delta
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§1.17(ii) Integral Representations
… ►Sine and Cosine Functions
… ►Bessel Functions and Spherical Bessel Functions (§§10.2(ii), 10.47(ii))
… ►Coulomb Functions (§33.14(iv))
… ►Airy Functions (§9.2)
…2: 10.59 Integrals
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►For an integral representation of the Dirac delta in terms of a product of spherical Bessel functions of the first kind see §1.17(ii), and for a generalization see Maximon (1991).
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3: 33.14 Definitions and Basic Properties
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►The function has the following properties:
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4: 9.11 Products
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►For an integral representation of the Dirac delta involving a product of two functions see §1.17(ii).
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5: Bibliography L
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Integral and series representations of the Dirac delta function.
Commun. Pure Appl. Anal. 7 (2), pp. 229–247.
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6: 10.22 Integrals
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►See also §1.17(ii) for an integral representation of the Dirac delta in terms of a product of Bessel functions.
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7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►of the Dirac delta distribution.
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►Applying the representation (1.17.13), now symmetrized as in (1.17.14), as ,
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►These latter results also correspond to use of the as defined in (1.17.12_1) and (1.17.12_2).
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►Thus, 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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►For 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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