Fourier Series = Re(Power Series)

joeblow

Somebody posted a question about Fourier series yesterday that got me thinking about an argument I heard some time before.

If we have a (complex-valued) analytic function f, then any closed loop in the complex plane will be mapped by f to another closed loop. (If the loop doesn't enclose any singularities or branch points.) In fact, if start = finish of the original loop, then start = finish of the image curve. Thus, the function is periodic on this loop. In particular, its real- and imaginary parts (call them u and v) are also periodic on this loop. If we write [tex]f(r,\theta )=re^{i\theta}=u(re^{i\theta})+iv(re^{i\theta})[/tex], and set r= const., we obtain real and imaginary parts that have period 2π. (Since the loop is an origin-centered circle.)

Thus, since f is analytic, we have [Taylor series of f] = [Fourier series of u]+i [Fourier series of v]. Then, if we are given a harmonic function u of period 2π, then we can find an harmonic conjugate v, and find the Fourier coefficients by simply reading off the coefficients of the real-part of the Taylor series. If the period is different, the approach can be easily modified. The claimed benefit is that it is easier to find the coefficients. (Though in all likelihood, you'll be evaluating one integral while finding a conjugate.)

Question 1: do you like this idea?

Question 2: I do not work with Fourier series often, and I never gained an appreciation of when they'd ever be used. Every article I've read merely asserts that the decomposition of a function into its Fourier series is important. Thus, while I understand the existence of Fourier Series, I do not understand the need for them. In practice, do the coeffecients come first and we build the function? (If this is the case, the above method is probably useless.) Or is it the other way around?