Dirac delta; fourier representation

[Physgeek64]

Homework Statement

I know that we can write ## \int_{-\infinity}^{\infinity}{e^{ikx}dx}= 2\pi \delta (k) ##

But is there an equivalent if the interval which we are considering is finite? i.e. is there any meaning in ##\int_{-0}^{-L}{e^{i(k-a)x}dx} ## is a lies within 0 and L?

Homework Equations

The Attempt at a Solution

I get the feeling the solution, if one exists, will be in the form ##\frac{2\pi}{L}## but I'm not sure if this is right,

Many thanks

 

[BvU]

Yes.

The outcome is still ##2\pi\delta(k-a)##, though. the integral over the remainder of the domain gives zero. Check out distributions

Edit, sorry, too quick. a can not lie in ##[0,L]##

 

[BvU]

So you get a transform of the (sin x)/x kind (#6 here). Something that in the limit goes towards a delta function.

 

[Physgeek64]

Yes.

The outcome is still ##2\pi\delta(k-a)##, though. the integral over the remainder of the domain gives zero. Check out distributions

Edit, sorry, too quick. a can not lie in ##[0,L]##

Why can a not lie in [0,L]?

 

[George Jones]

But is there an equivalent if the interval which we are considering is finite? i.e. is there any meaning in ##\int_{-0}^{-L}{e^{i(k-a)x}dx} ## is a lies within 0 and L?

To see what happens, why don't you just do the integral? Hint: if you feel uneasy integrating an imaginary exponential, use ##e^{i\theta}= \cos\theta + i \sin\theta##.