Re: delta functions and Green functions

["John S. Denker" <jsd@MONMOUTH.COM>]

At 07:50 PM 3/5/01 -0500, Robert B Zannelli wrote:
> I must confess I don't have a clue what a delta function is.

A delta function is an infinitely tall, infinitely skinny spike with unit
area under it.  No, I'm not going to draw a picture of it.  The integral of
a delta function is a step function.

> I went to the GIF and I saw what looks like a quarter wave damped sine
> wave which started some time after time zero.

That's not the delta function.  That's the Green function.  (Yes, that
Green.)  This particular Green function is the _response_ of a
mass-on-a-spring to a delta-function shaped force, i.e. an impulse.

And it doesn't turn on "after" time zero... It turns on exactly at time
zero.  Take another look at the labels on the axis.

Other linear systems have their own Green functions.  For instance, for an
RC circuit its Green function is zero for t<0, steps up at t=0, and then
exponentially decays.

> Perhaps I know what it is under a different name.

It is also know as the Dirac delta function.  (Yes, that Dirac.)

It is AKA the delta distribution (since it technically doesn't meet all the
criteria for being called a function).

It plays the same role for integrals that the Kronecker delta (delta_i_j)
plays for sums.