# Solving an equation with Dirac delta functions

1. May 23, 2012

### eliotsbowe

Hello, I'm dealing with the following equation:

$$A e^{jat} + B e^{jbt} = C e^{jct}$$ $$\forall t \in \mathbb{R}$$

My book says: given nonzero constants A,B,C, if the above equation yelds for any real t, then the a,b,c constants must be equal.

The above statement is prooved by taking the Fourier transform of the complex exponentials:
$$A2\pi \delta(t+a) + B2\pi \delta(t+b) = C2\pi \delta(t+c)$$$$A \delta(t+a) + B \delta(t+b) = C \delta(t+c)$$

But I don't really get how such an equation as the last one can force a,b,c to be equal.

Any help would be appreciated, thanks in advance.

2. May 23, 2012

### Vargo

The left side of the equation has a spike at -a and at -b and is 0 everywhere else. The right side has a spike at -c and is 0 everywhere else.

3. May 23, 2012

### eliotsbowe

I think I got it, thanks!

4. May 23, 2012

### lugita15

ellotsbowe, that is a truly marvelous proof, using distributions to get more or less a result of elementary algebra. What book did you get it from?

5. May 23, 2012

### theorem4.5.9

I thought the same thing! My first thought was successive derivatives and set $t=0$. This is very slick.