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From: gearhead on 25 Apr 2010 17:19
I'm studying for a test tomorrow and got stuck on this.
How do you evaluate a proper integral of f(t) times the dirac delta
function of (t-a)
For example: the integral from -2 to 4 of (t^3+4) times delta(t-2)
with respect to t
I can't figure out how to make the sifting function work. In the
chapter on Laplace transforms, the book only shows the improper
integral (negative infinity to infinity), and it doesn't have any
examples of using the sifting property of the delta function to
evaluate a proper integral with limits. I picked out a problem at the
back of the chapter, the example I showed you, and couldn't do it.
I have the answer manual but it just gives the answer as 8+4, that's
it, no steps or explanation. I guess they thought it was so simple
and obvious it needed no explanation. But there isn't anything
anywhere in the book showing how to do it, and I can't find a method
to solve this problem using the sifting function anywhere on the web
either. Will the sifting property solve this problem?
From: Ray Vickson on 25 Apr 2010 17:45
On Apr 25, 2:19 pm, gearhead <nos...(a)billburg.com> wrote:
> I'm studying for a test tomorrow and got stuck on this.
> How do you evaluate a proper integral of f(t) times the dirac delta
> function of (t-a)
> For example:  the integral from -2 to 4 of (t^3+4) times delta(t-2)
> with respect to t
> I can't figure out how to make the sifting function work.  In the
> chapter on Laplace transforms, the book only shows the improper
> integral (negative infinity to infinity), and it doesn't have any
> examples of using the sifting property of the delta function to
> evaluate a proper integral with limits.  I picked out a problem at the
> back of the chapter, the example I showed you, and couldn't do it.
> I have the answer manual but it just gives the answer as 8+4, that's
> it, no steps or explanation.  I guess they thought it was so simple
> and obvious it needed no explanation.  But there isn't anything
> anywhere in the book showing how to do it, and I can't find a method
> to solve this problem using the sifting function anywhere on the web
> either.  Will the sifting property solve this problem?

The way that physicists and engineers do it is to say
integral(f(t)*delta(t-a) dt, t = t0..t1) = f(a) for an "arbitrary
continuous function" f and where 'a' lies in the interior of the
interval from t0 to t1. Mathematicians would argue with this, and
demand restrictions on f, etc. If you regard delta(t-a) as a limit of
actual functions D_n(t-a) as n --> infinity, and if you are sloppy
about interchanging limits and integrations, you get the physicist's
evaluation.

R.G. Vickson
From: gearhead on 25 Apr 2010 18:02
On Apr 25, 5:45 pm, Ray Vickson <RGVick...(a)shaw.ca> wrote:
> On Apr 25, 2:19 pm, gearhead <nos...(a)billburg.com> wrote:
>
>
>
> > I'm studying for a test tomorrow and got stuck on this.
> > How do you evaluate a proper integral of f(t) times the dirac delta
> > function of (t-a)
> > For example:  the integral from -2 to 4 of (t^3+4) times delta(t-2)
> > with respect to t
> > I can't figure out how to make the sifting function work.  In the
> > chapter on Laplace transforms, the book only shows the improper
> > integral (negative infinity to infinity), and it doesn't have any
> > examples of using the sifting property of the delta function to
> > evaluate a proper integral with limits.  I picked out a problem at the
> > back of the chapter, the example I showed you, and couldn't do it.
> > I have the answer manual but it just gives the answer as 8+4, that's
> > it, no steps or explanation.  I guess they thought it was so simple
> > and obvious it needed no explanation.  But there isn't anything
> > anywhere in the book showing how to do it, and I can't find a method
> > to solve this problem using the sifting function anywhere on the web
> > either.  Will the sifting property solve this problem?
>
> The way that physicists and engineers do it is to say
> integral(f(t)*delta(t-a) dt, t = t0..t1) = f(a) for an "arbitrary
> continuous function" f and where 'a' lies in the interior of the
> interval from t0 to t1. Mathematicians would argue with this, and
> demand restrictions on f, etc. If you regard delta(t-a) as a limit of
> actual functions D_n(t-a) as n --> infinity, and if you are sloppy
> about interchanging limits and integrations, you get the physicist's
> evaluation.
>
> R.G. Vickson

That's so simple it's criminal. Just evaluate f(t) at a, as long as a
lies within the limits of integration. Wow.
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