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Dirac delta distribution

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1: 1.16 Distributions
§1.16(iii) Dirac Delta Distribution
The Dirac delta distribution is singular. …
1.16.15 D H = δ ,
1.16.16 D H ( x - x 0 ) = δ x 0 .
2: 2.6 Distributional Methods
2.6.17 f , ϕ = s = 0 n - 1 a s t - s - α , ϕ - s = 1 n c s δ ( s - 1 ) , ϕ + f n , ϕ
The Dirac delta distribution in (2.6.17) is given by
2.6.19 δ ( s ) , ϕ = ( - 1 ) s ϕ ( s ) ( 0 ) , s = 0 , 1 , 2 , ;
2.6.20 f , ϕ = s = 0 n - 1 a s t - s - 1 , ϕ - s = 1 n d s δ ( s - 1 ) , ϕ + f n , ϕ
Furthermore, K + contains the distributions H , δ , and t λ , t > 0 , for any real (or complex) number λ , where H is the distribution associated with the Heaviside function H ( t ) 1.16(iv)), and t λ is the distribution defined by (2.6.12)–(2.6.14), depending on the value of λ . …
3: 1.17 Integral and Series Representations of the Dirac Delta
In applications in physics and engineering, the Dirac delta distribution1.16(iii)) is historically and customarily replaced by the Dirac delta (or Dirac delta function) δ ( x ) . …
4: 14.30 Spherical and Spheroidal Harmonics
14.30.8 0 2 π 0 π Y l 1 , m 1 ( θ , ϕ ) ¯ Y l 2 , m 2 ( θ , ϕ ) sin θ d θ d ϕ = δ l 1 , l 2 δ m 1 , m 2 .
Distributional Completeness
For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii). …
5: 20.13 Physical Applications
is also a solution of (20.13.2), and it approaches a Dirac delta1.17) at t = 0 . … In the singular limit τ 0 + , the functions θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …
6: Bibliography L
  • D. A. Levine (1969) Algorithm 344: Student’s t-distribution [S14]. Comm. ACM 12 (1), pp. 37–38.
  • Y. T. Li and R. Wong (2008) Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.
  • J. E. Littlewood (1914) Sur la distribution des nombres premiers. Comptes Rendus de l’Academie des Sciences, Paris 158, pp. 1869–1872 (French).