# Dirac delta; fourier representation

1. Dec 11, 2017

### Physgeek64

1. The problem statement, all variables and given/known data
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?

2. Relevant equations

3. 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

2. Dec 11, 2017

### 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 $[0,L]$

3. Dec 11, 2017

### BvU

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

4. Dec 11, 2017

### Physgeek64

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

5. Dec 11, 2017

### George Jones

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