Question

For each one of the following functions, obtain its Fourier series in exponential, trigonometric, and compact forms.

(a) f(t)=cos³(t)

Answer 1

The Fourier series of f(t) = cos^3(t) is found by expressing the cubed cosine in terms of cosines of multiple angles and then writing the series in exponential, trigonometric, and compact forms.

Step-by-step explanation:

The Fourier series for a function f(t) = cos^3(t) can be determined using trigonometric identities and the general formula for a Fourier series. First, the identity cos^3(t) can be expressed in terms of cosines of multiple angles by using the trigonometric identity cos^3(t) = (3 cos(t) + cos(3t))/4. Then the Fourier series can be written in exponential, trigonometric, and compact forms.

For the trigonometric form, the Fourier series would be written as a sum of cosines and sines with coefficients determined by the function's behavior over one period. For the exponential form, the series is a sum of terms involving e raised to the power of integer multiples of the imaginary unit times the argument t. The compact form simply uses a shorthand notation to represent the coefficients and terms in the series.