In mathematics, the Dirac delta function (δ function), also known as the unit impulse symbol,[1] is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.[2][3][4] It can also be interpreted as a linear functionalthat maps every function to its value at zero,[5][6] or as the weak limit of a sequence of bump functions, which are zero over most of the real line, with a tall spike at the origin. Bump functions are thus sometimes called "approximate" or "nascent" delta functions.
The delta function was introduced by physicist Paul Dirac as a tool for the normalization of state vectors. It also has uses in probability theory and signal processing. As it is not a true mathematical function, some mathematicians[who?] objected to it as nonsense until Laurent Schwartz developed the theory of distributions.
The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is the discrete analog of the Dirac delta function.
https://en.wikipedia.org/wiki/Dirac_delta_function
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