Expressing the Function f(x) cos(x) for π ≤ x ≤ 2π in a Fourier Series

When dealing with functions that are defined over a specific interval, such as the function ( f(x) cos(x) ) for ( pi leq x leq 2pi ) and ( f(x) 0 ) for ( x 2pi ), the Fourier series provides a powerful tool for representing this function in a periodic manner. This article will walk you through the process of expressing this function in a Fourier series.

Introduction to Fourier Series

A Fourier series represents a periodic function as the sum of a series of sines and cosines. A function can be expressed in a Fourier series if it satisfies certain conditions, such as being periodic and piecewise continuous. In this case, our function ( f(x) ) is defined on the interval [ pi leq x leq 2pi ] and extended to be periodic. The general form of a Fourier series is given by:

General Form of Fourier Series

[ f(x) f_0 sum_{n1}^{infty} a_n cos left( frac{2pi n}{T} x right) sum_{n1}^{infty} b_n sin left( frac{2pi n}{T} x right) ]

Where ( f_0 ) is the average value of the function over one period, and ( a_n ) and ( b_n ) are the Fourier coefficients, which can be calculated using the following formulas:

Formulas for Fourier Coefficients

[ f_0 frac{1}{T} int_{0}^{T} f(x) , dx ]

[ a_n frac{2}{T} int_{0}^{T} f(x) cos left( frac{2pi n}{T} x right) , dx ]

[ b_n frac{2}{T} int_{0}^{T} f(x) sin left( frac{2pi n}{T} x right) , dx ]

In this specific example, the period ( T 2pi ).

Calculation of Fourier Coefficients

Let's calculate the Fourier coefficients for our function ( f(x) ) defined on the interval ( pi leq x leq 2pi ) and extended to be periodic.

Calculation of ( f_0 )

[ f_0 frac{1}{2pi} int_{pi}^{2pi} cos(x) , dx ]

[ f_0 frac{1}{2pi} left[ sin(x) right]_{pi}^{2pi} frac{1}{2pi} (sin(2pi) - sin(pi)) 0 ]

Calculation of ( a_n )

[ a_n frac{1}{pi} int_{pi}^{2pi} cos(x) cos(nx) , dx ]

For ( n 1 ):

[ a_1 frac{1}{pi} int_{pi}^{2pi} cos^2(x) , dx ]

[ a_1 frac{1}{2pi} int_{pi}^{2pi} (1 cos(2x)) , dx ]

[ a_1 frac{1}{2pi} left[ x frac{sin(2x)}{2} right]_{pi}^{2pi} 0 ]

For ( n eq 1 ):

[ a_n frac{1}{pi} int_{pi}^{2pi} cos(x) cos(nx) , dx ]

[ a_n frac{1}{2pi} left[ frac{sin((n 1)x)}{n 1} - frac{sin((n-1)x)}{n-1} right]_{pi}^{2pi} 0 ]

Calculation of ( b_n )

[ b_n frac{1}{pi} int_{pi}^{2pi} cos(x) sin(nx) , dx ]

[ b_n frac{1}{2pi} left[ -frac{cos((n 1)x)}{n 1} frac{cos((n-1)x)}{n-1} right]_{pi}^{2pi} ]

[ b_n -frac{1}{2pi} left[ frac{cos((n 1)2pi) - cos((n 1)pi)}{n 1} - frac{cos((n-1)2pi) - cos((n-1)pi)}{n-1} right] ]

[ b_n -frac{1}{2pi} left[ frac{1 (-1)^{n 1}}{n 1} - frac{1 (-1)^{n-1}}{n-1} right] ]

[ b_n -frac{1}{2pi} left[ frac{2}{n^2 - 1} (-1)^{n-1} right] ]

[ b_n -frac{1}{pi} left[ frac{1}{n^2 - 1} (-1)^{n-1} right] ]

Final Fourier Series Representation

The Fourier series representation of the function ( f(x) ) then becomes:

[ f(x) frac{1}{2} cos(x) - frac{1}{pi} sum_{n1}^{infty} frac{n}{n^2 - 1} (-1)^{n-1} sin(nx) ]

When only the first three series terms are considered, the series simplifies to:

[ f(x) approx frac{1}{2} cos(x) - frac{1}{pi} sum_{n1}^{3} frac{4n}{4n^2 - 1} sin(2nx) ]

This series provides a reasonable approximation of the original function over the interval [ pi leq x leq 2pi ].

Key Takeaways:

Fourier series is a powerful tool for representing periodic functions. The Fourier coefficients are calculated using specific integrals. The Fourier series representation helps approximate functions over a specific interval.

By understanding and applying the Fourier series, you can accurately represent and approximate a wide range of functions, extending their periodicity to the entire real line.