The Deltafunction as the Limit of Some Special Functions

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Another Demonstration gives some representations for the Dirac deltafunction as the limit of elementary functions (see the related links below).This Demonstration illustrates several additional limiting relations involving special functions and the deltafunction:

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Bessel: , (

Airy: .

Hermite: The derivative of the deltafunction is given by . Shown is with .

Dirichlet kernel: , where the delta-comb or shah function is defined by . This is a periodic extension of the deltafunction.

Fejér kernel: , with the same periodicity as the Dirichlet kernel.

Sigmoid: The derivative of the sigmoid function: .

Closure: An orthonormal set of functions {} obeys the closure relation . This is shown for orthonormalized Hermite functions with .

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Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Snapshot 1: Bessel function representation with All but the leftmost peak will produce canceling contributions as

Snapshot 2: approach to using Hermite polynomial

Snapshot 3: closure relation approaching



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