Question

Show that the Fourier series of the function f(x)= cos(ax) on the interval [−π,π], when a is not an integer, is given by

[infinity]
cos(ax)=²asin(aπ)/π ​[1/2a2​+∑​ (−1)ⁿ⁺¹/n²−a²​cos(nx)]
ⁿ⁼¹
for −π≤x≤π

Answer 1

To show the Fourier series of f(x) = cos(ax) for a non-integer a, one calculates the Fourier coefficients by integrating cos(ax) cos(nx) over the interval [−π, π]. Since cos(ax) is even, b_n coefficients are zero. The final series expression is then derived, matching the given formula.

Step-by-step explanation:

The student is asked to show the Fourier series representation of the function f(x) = cos(ax), where a is not an integer, on the interval [−π, π]. According to Fourier series theory, any periodic function can be represented as an infinite sum of sines and cosines. Our goal is to find the coefficients of those sinusoidal components matching the function cos(ax).

To find the Fourier series, we start by calculating the Fourier coefficients. For a general function f(x), we have:

1. The a0 term, which gives the average value of the function over the period.
2. The an coefficients, which are found using the cosine terms of the integral.
3. The bn coefficients, which are found using the sine terms of the integral.

Since cos(ax) is an even function, all bn will be zero. By integrating cos(ax) times cos(nx) over the interval [−π, π] and applying orthogonality of cosine functions, we arrive at the expression for the Fourier series:

cos(ax) = (a0/2) + Σ ancos(nx)

Using the provided formula, it states that the Fourier series is:

[infinity]cos(ax) = (2a sin(aπ)/π) [1/2a2 + Σ (−1)n+1/(n2 − a2) cos(nx)]

Calculation of the coefficients a0 and an would show that they match the terms in the provided series representation.