Question

The Heaviside step function is defined by H(x) = { 1, x > 0. A representation of the Dirac delta function is given by the derivative of the function H(x) in the form Î´(x) = dH(x)/dx. To justify this important relationship between the Dirac delta function and the derivative of the Heaviside function, carry out the following series of steps: The Heaviside function is evidently a discontinuous function at x = 0. An approximation to the function Î´(x) by means of a continuous function is to use (for example) the function T(x) = 0.5 + 0.5tanh((x - Îµ)/Îµ), where tanh is the hyperbolic tangent function and Îµ is a small positive number. This function T(x) approaches the Heaviside function H(x) as Îµ approaches zero. Draw T(x) for four different values of Îµ to visually indicate how Î´(x - 0) tends to be small if Îµ is small. Calculate the derivative dT(x)/dx and draw it for three different values of Îµ.

          The Heaviside step function is defined by
H(x) = { 1, x > 0.
A representation of the Dirac delta function is given by the derivative of the function H(x) in the form Î´(x) = dH(x)/dx. To justify this important relationship between the Dirac delta function and the derivative of the Heaviside function, carry out the following series of steps:

The Heaviside function is evidently a discontinuous function at x = 0. An approximation to the function Î´(x) by means of a continuous function is to use (for example) the function
T(x) = 0.5 + 0.5tanh((x - Îµ)/Îµ),

where tanh is the hyperbolic tangent function and Îµ is a small positive number. This function T(x) approaches the Heaviside function H(x) as Îµ approaches zero.

Draw T(x) for four different values of Îµ to visually indicate how Î´(x - 0) tends to be small if Îµ is small.

Calculate the derivative dT(x)/dx and draw it for three different values of Îµ.

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Added by Alejandro K.

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