Fourier Series

Last revised: August 15, 2003
Euler's discovery that any periodic function can be expanded in a series whose terms are 
sine and cosine functions leads to many different series whose terms are rational numbers 
and whose sum involves pi. 

Requires:
Derivatives and Integrals
Power Series
Reference:
Knopp, pages 350-356
Knopp, pages 372-378
Minilecture 1: Euler's formula for Fourier coefficients
Assume that the series 

converges to a function f(x).  Show that the coefficients an and bn are given by Euler's formulas

an = (1/p) 
 = (1/p)
This leaves out the hard part of Fourier series, which is the tricky issue of convergence. You might want to look through pages 356-372 of Knopp to see what you're missing.
Prove the orthogonality relations in the middle of Knopp page 353.  Once you have these, the rest is easy.

Minilecture 2: Fourier series for p
Use Euler's formulas to derive three different series, two involving only sine functions and the other involving only cosine functions, for a periodic function that equals x on 0 < x < p.
Show how substitution of x = p/2 gives a new derivation of known series involving p.

One the course Web site is a Windows application program fourier.exe that you can use to
illustrate your presentation. It shows very nicely that the convergence of Fourier series is
strange near points where the function being approximated (or its derivative) has a discontinuity.

The three series that you want are based on examples 2a and 2b at the bottom of p. 375 of Knopp and example 2c on the next page, but you have to be careful with coefficients in order to make the functions precisely the same as f(x) = x on 0 < x < p.  The series from example 2a is

x = p - 2(Knopp, page 376, 211)
The series from example 2b is

x =  2(Knopp, bottom of p. 375, multiplied by 2)
The series from example 2c is

x = p/2 - 
In every case, you can exploit symmetry properties of the functions that you are expanding and
the sine and cosine functions to simplify the integrals that you need to evaluate. Then you need

  Be sure to sketch graphs of the functions, the first two of which have discontinuities.
Show that substituting x= p/2 into the first two of these series gives Gregory's formula for p.
Substituting x= p/2 into the third series gives something unfamiliar:

Another way to obtain this formula is to start with Euler's

Multiply both sides of this equation by 1/4 and subtract from the original equation.

Minilecture 3: A Fourier proof of an Eulerian formula
Assuming Euler's formulas as given above, show that expansion in a Fourier series, for non-integral values of a, of the continuous function
 for 0 <= x <= p
 for p <= x <= 2p
leads to a Fourier series that can be converted into 

This is example 3 on page 377 of Knopp, but be sure to look back to page 373 to find the three methods for "preparing a function for the generation of a Fourier series:" this explains the name f2(x) that Knopp uses.  Sketch the function f2(x).

The program fourier.exe on the course Web site lets you show how the series is built up term by term.
The value a=3/2 is built in, but you can change it to anything else.  When you demonstrate
the program, point out how the convergence is much worse near the points where the derivative
has a discontinuity.
Minilecture 4 (computational):
By substituting a numerical value into a Fourier series, you express x (some quantity proportional to p, like p/4, in terms of trigonometric functions of p whose values you know.  Series expansions of different functions, along with different values of x, provide an unlimited variety of series whose sum is proportional to p or to a power of p.  For the most part, alas, these don't converge very fast.  Still, see if you can invent a new series and sum it to get at least 15 digits of p correct. Use Mathematica or C++ as you choose.  Ordinary double-precision floating point is fine.