Solving Dirac Delta Cosx: Find Range of n and a_n, x_n

[apw235]

Homework Statement

The function $$\delta(cosx)$$ can be written as a sum of Dirac delta functions:
$$\delta(cosx)=\sum_{n} a_{n}\delta(x-x_{n})$$
Find the range for n and the values for $$a_{n}$$ and $$x_{n}$$

The Attempt at a Solution

Well, taking the integral of $$\delta(cosx)$$, we only get spikes when x is an even multiple of $$\frac{\pi}{2}$$. So shouldn't n run to infinity? Thats all i have so far, any help would be appreciated. thanks.

-Adrian

 

[Dick]

You get 'spikes' where cos(x)=0. I wouldn't describe those as 'even multiples of pi/2'. In general if x_i are the roots of f(x)=0, then delta(f(x)) is the sum of delta(x_i)/|f'(x_i)|.