Integral and series representations of the dirac delta function

@article{Li2007IntegralAS,
  title={Integral and series representations of the dirac delta function},
  author={You Tang Li and Roderick S. C. Wong},
  journal={Communications on Pure and Applied Analysis},
  year={2007},
  volume={7},
  pages={229-247}
}
  • Y. Li, R. Wong
  • Published 2007
  • Mathematics
  • Communications on Pure and Applied Analysis
Mathematical justifications are given for several integral and series representations of the Dirac delta function which appear in the physics literature. These include integrals of products of Airy functions, and of Coulomb wave functions; they also include series of products of Laguerre polynomials and of spherical harmonics. The methods used are essentially based on the asymptotic behavior of these special functions. 

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References

SHOWING 1-10 OF 14 REFERENCES
Generalized Functions: Theory and Technique
The Dirac delta function and delta sequences the Heaviside function the Dirac delta function the delta sequences a unit dipole the Heaviside sequences exercises. (Part contents)
Coulomb functions for attractive and repulsive potentials and for positive and negative energies
The paper gives a review of the mathematical properties of the Coulomb radial wave functions, for attractive and repulsive potentials and for positive and negative energies, together withExpand
Airy Functions And Applications To Physics
The use of special functions, and in particular Airy functions, is rather common in physics. The reason may be found in the need, and even in the necessity, to express a physical phenomenon in termsExpand
The Confluent Hypergeometric Function
The routines S22BA and S22BB, new at Mark 24, provide the functionality to calculate the confluent hypergeometric function 1F1(a; b; x), also known as Kummer’s function M(a, b, x). This has a wideExpand
DEFINITION AND SIMPLEST PROPERTIES OF GENERALIZED FUNCTIONS
A generalized function is defined as any linear continuous functional defined on K . The functionals [RK1]
Asymptotics and Special Functions
A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.
Methods of theoretical physics
Allis and Herlin Thermodynamics and Statistical Mechanics Becker Introduction to Theoretical Mechanics Clark Applied X-rays Collin Field Theory of Guided Waves Evans The Atomic Nucleus FinkelnburgExpand
...
1
2
...