Examples

 (a) PROBLEMS UNDER THE INTERVAL (-л, л)

Example 2.2.a(1): Find the Fourier expansion of f(x) = x in

Example 2.2.a(2): Find the Fourier series for f(x) eries for f(x) = x² ppppppppppp and deduce that

Example 2.2.a(4): What is the Fourier expansion of the periodic function whose definition of one period is

Example 2.2.a(5): In -л <x<л, express sinh ax and cosh ax in Fourier series of periodicity.

Example 2.2.a(6) Obtain the Fourier series of periodicity 2л for

f(x) = -x, when -л < x < 0 and f (x) = x, when 0 < x < л

[OR]

Example 2.2.a(7): Find the Fourier series representation for f(x) = | sin x | inл<x<л.

Example 2.2.a(8) Express f (x) = x sinx as a Fourier series in the interval (-л, л).

Solution: Given

Example 2.2.a(9) : Expand f(x) = | cos x in a Fourier series in the interval (-, л)

Example 2.2.a(10): Determine the Fourier series for the function

Example 2.2.a(11) : Obtain the Fourier series for f(x)=1+x+x² in (-л, л). Deduce that

Example 2.2.a(12): Prove that