Dirac delta function | Wikipedia audio article
Channel: wikipedia tts
Date: 04/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Dirac_delta_function


00:01:41 1 Motivation and overview
00:07:39 2 History
00:07:53 3 Definitions
00:08:30 3.1 As a measure
00:16:37 3.2 As a distribution
00:17:36 3.3 Generalizations
00:17:56 4 Properties
00:21:58 4.1 Scaling and symmetry
00:24:21 4.2 Algebraic properties
00:27:52 4.3 Translation
00:31:15 4.4 Composition with a function
00:31:24 4.5 Properties in in/i dimensions
00:32:53 5 Fourier transform
00:33:49 6 Distributional derivatives
00:35:47 6.1 Higher dimensions
00:37:20 7 Representations of the delta function
00:40:26 7.1 Approximations to the identity
00:46:10 7.2 Probabilistic considerations
00:54:07 7.3 Semigroups
00:59:07 7.4 Oscillatory integrals
01:02:30 7.5 Plane wave decomposition
01:03:42 7.6 Fourier kernels
01:07:50 7.7 Hilbert space theory
01:09:52 7.7.1 Spaces of holomorphic functions
01:17:59 7.7.2 Resolutions of the identity
01:21:52 7.8 Infinitesimal delta functions
01:31:22 8 Dirac comb
01:37:51 9 Sokhotski–Plemelj theorem
01:39:27 10 Relationship to the Kronecker delta
01:42:03 11 Applications
01:47:39 11.1 Probability theory
01:49:26 11.2 Quantum mechanics
01:51:52 11.3 Structural mechanics
01:55:11 12 See also
01:56:57 13 Notes
01:57:06 14 References
02:01:20 15 External links
02:08:22 Structural mechanics
02:11:01 Fd acting on the beam. Now, let the distance d approach the limit zero, while M is kept constant. The load distribution, assuming a clockwise moment acting at x
02:12:59 See also



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SUMMARY
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In mathematics, the Dirac delta function (δ function) is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.
In engineering and signal processing, the delta function, also known as the unit impulse symbol, may be regarded through its Laplace transform, as coming from the boundary values of a complex analytic function of a complex variable.  The formal rules obeyed by this function are part of the operational calculus, a standard tool kit of physics and engineering. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin (in theory of distributions, this is a true limit). The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.