Evaluating integral of delta function

[elimenohpee]

Homework Statement

Evaluate the integral:

Homework Equations

To integrate this, should one use a dummy variable to get the delta function only of t, then integrate, then substitute back in after integration?

 

[quantumdude]

Homework Statement

Evaluate the integral:

Homework Equations

To integrate this, should one use a dummy variable to get the delta function only of t, then integrate,

The delta will be a function of the dummy variable, not t. But yes, that is what you want to do.

then substitute back in after integration?

It's a definite integral. Why would there be any need to substitute back after integration?

 

[Dick]

Substitute u=t+3 and use the definition of the delta function.

 

[elimenohpee]

Ok I just want to make sure I'm thinking correctly and logically. If I let t1 = t + 3, then I need to change all the 't' in the function to t1. I know that the Delta function is defined as

So if I can get it into this form with t1:

then the delta function should simplify as 1 correct?

The function would look like this:

then simplify to this:

then separate the exponential, pull the exponential not containing t1 out of the integral, integrate the exponential containing t1, and should be left with just this:

Does this look right? Or am I not thinking about this right.

 

[Dick]

The definition of the delta function is not that integral of delta(t)*dt=1. Many functions have that property. It's that the integral of f(t)*delta(t)*dt=f(0), if f is a continuous function.

 

[elimenohpee]

Oh ok I see what you mean. So if I have

it should still evaluate to e^3 right?

 

[Dick]

Yes. e^3. Sorry. I didn't read the post through to the end. But, once you have it in the form delta(t)*f(t)*dt you are done.

 

[elimenohpee]

Oh ok excellent! Thanks for the help, this has helped me 'de-mystify' the delta function :)