# Special Functions

The Dirac Delta Function

The function d(t) is the Dirac delta function.  The function u(t) is the Heaviside step function.

d(t) looks like the plot below,  i.e. it has a value of ¥ at time t=0, and a value of zero everywhere else.

It follows that for any arbitrary constant, t,

The delta function is also defined to have the following property:

That is, the area under the delta function is unity.

One way to view the delta function is as the limit of the triangle function below.

The area is ½ bh, according to simple geometry.  Thus, regardless of the value of H, the area is:

Now let H go to ¥.  The triangle gets taller and narrower until it looks like the delta function (infinitely tall and infinitely narrow).

An important property of the delta function is that whenever any function is integrated against it, the result is the value of the function at time t=0.

This follows directly from the statement above that the integral of the delta function is 1.

because is zero at all values except 0.

(since f(0) is continuous, its value at 0+  is the same as its value at 0-).

Similarly,

The Laplace Transform of the function f(t) is defined as:

Thus, the Laplace Transform of the delta function is:

## The Heaviside Step Function

The step function is:

 t

It is used to specify functions which are “turned on” at time t=0.

The Laplace Transform of the step function is:

 d’(t)

 t

Note that the derivative of the step function is the delta function since the derivative is 0 everywhere and infinite at t=0.  We can also talk about the derivative of the delta function, which is called the unit doublet.  This is infinitely large and positive at 0- and infinitely negative at 0+.  It looks like the figure to the right.  When any function is integrated against this function, the derivative is obtained.  To see this, do a u substitution with u=f(t), du=f’(t)dt, v=d(t), dv=d’(t)dt

The corresponding Laplace transform is s.

Steven A. Jones

Biomedical Systems (BIEN 225), Louisiana Tech University