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The Dirac Delta Function

The Dirac Delta Function. The Dirac delta or Dirac's delta is a mathematical construct introduced by theoretical physicist Paul Dirac. Informally, it is a generalized function representing an infinitely sharp peak bounding unit area: a

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The Dirac Delta Function

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  1. The Dirac Delta Function The Dirac delta or Dirac's delta is a mathematical construct introduced by theoretical physicist Paul Dirac. Informally, it is a generalized function representing an infinitely sharp peak bounding unit area: a 'function' δ(x) that has the value zero everywhere except at x = 0 where its value is infinitely large in such a way that its total integral is 1. The Dirac delta is not strictly a function. While for many purposes it can be manipulated as such, formally it can be defined as a distribution. In many applications, the Dirac delta is regarded as limit of a sequence of functions having a tall spike at the origin. The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.

  2. Representations of the delta function

  3. Demonstrate that Another representation: Useful properties:

  4. The Dirac Delta Function in 3D Useful representation: Useful application: scalar function of r

  5. Let us now take f(r)=rn But what about n=-1? if the zero vector is not in V If the zero vector is in V: For S we take a sphere of radius r centered around (0,0,0) Hence

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