Complex exponetial form of Fourier series

[Bipolarity]

I have some rather technical questions about the complex exponential form of the Fourier series:

1) What is the motivation behind the complex exponential form? Why not just use the real form (i.e. with sine and cosines)?

2) Surely the complex exponential form is an orthogonal set, i.e. $<e^{iπmx/p},e^{iπnx/p}>=0$ for all integers m,n not equal to one another.

3) Are the two forms equivalent, i.e. if you can express a function with the Fourier sine/cosine series such that the function converges to the Fourier sine/cosine series, then can you also express the same function with its complex exponential Fourier series such that the function converges to its complex exponential Fourier series? And what about the converse?

BiP

 

[micromass]

1) What is the motivation behind the complex exponential form? Why not just use the real form (i.e. with sine and cosines)?

I think that in first courses, the real form will always be most useful. But later one, the complex exponential form is important too. I really don't know very much about applications of Fourier series, but from math point-of-view, the complex exponential is important because it is a group homomorphism

$$\mathbb{R}\rightarrow S^1:x\rightarrow e^{ix}$$

This might seem like an insignificant fact to you, but it actually is the main reason that Fourier series work and it suggests a generalization.

2) Surely the complex exponential form is an orthogonal set, i.e. $<e^{iπmx/p},e^{iπnx/p}>=0$ for all integers m,n not equal to one another.

I'm not a fan of your notation. But yes.

3) Are the two forms equivalent, i.e. if you can express a function with the Fourier sine/cosine series such that the function converges to the Fourier sine/cosine series, then can you also express the same function with its complex exponential Fourier series such that the function converges to its complex exponential Fourier series? And what about the converse?

Yes, they are completely equivalent. It doesn't matter whether you use the sine/cosine version or the exponential version. It's all the same thing. Sometimes, it is simply more useful and natural to consider the exponential version. However, everything that can be done with the exponential version, can also be done with sines/cosines (and vice versa). So the difference is just an aesthetic one.