Sinusoidal and exponential series

[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?

 

[HallsofIvy]

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

 

[Jhenrique]

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

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!?

 

[Mark44]

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

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

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

 

[Jhenrique]

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

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

 

[Mark44]

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.

 

[Jhenrique]

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.

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.