# Sinusoidal and exponential series

1. May 2, 2014

### Jhenrique

If is possible to expess periodic functions as a serie of sinusoids, so is possible to express periodic functions with exponential variation through of a serie of sinusoids multiplied by a serie of exponentials? Also, somebody already thought in the ideia of express any function how a serie of exponential?

2. May 2, 2014

### HallsofIvy

What do you mean by "periodic functions wit exponential variation"?

3. May 2, 2014

### Jhenrique

An exemple of a periodic function that can be approximate by fourier series is:

And another exemple of a "periodic function with exponential variation" is a function like this:

So, if exist a exponential factor in the fourier series, this serie would be perfect for represent this second graph. Yeah!?

4. May 2, 2014

### Staff: Mentor

The function in the graph is not periodic. For a periodic function whose period is p, f(x) = f(x + p), for any x.

5. May 2, 2014

### Jhenrique

True! But, what say about a fourier serie with factor exponential?

6. May 2, 2014

### Staff: Mentor

The Fourier series for a function is periodic, but if you multiply that series by an exponential function, the product is no longer periodic. I'm not sure I understand what you're asking, though.

7. May 2, 2014

### Jhenrique

The fourier series, roughly speaking, is $f(t) = \sum_{-\infty }^{+\infty } A_\omega \cos(\omega t - \varphi_\omega ) \Delta \omega$, I was thinking in a serie like this: $f(t) = \sum_{-\infty }^{+\infty } \sum_{-\infty }^{+\infty } A_{\omega \sigma} \exp(\sigma t) \cos(\omega t - \varphi_{\omega \sigma}) \Delta \omega \Delta \sigma$ with the intention of express any function through this serie.