The purpose of this problem is to explore the Gibbs phenomenon. This phenomenon occurs as a result of truncating the Fourier series of a discontinuous function. Examine, for example, this henomenon in detail for the function y3(x) given in P8. The function under consideration is given analytically by:
a. Find the value where the truncated Fourier series overshoots the value of 0.5. (Answer: The limiting value of this first maximum is 0.58949).
b. Find the limiting value of the first local minimum. (Answer: The limiting value of this first minimum is 0.45142).
c. Derive, from first principles, the answers to parts (a) and (b). (Hint: Look up in a standard integral table the sine integral function.)
NOTE An important goal of filter theory is to find methods to smooth these kinds of oscillations.