Fourier series

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Problem 9.1a

Show that the Fourier series coefficients, and in equation (9.1a), are given by equations (9.1d,e).

Background

If is a periodic function, that is, one that repeats exactly at intervals (called the period) and has a finite number of maxima, minima, and discontinuities in each interval , then it can be expressed in a Fourier series in any one of the three following equivalent forms, being an integer:


(9.1a)
(9.1b)
(9.1c)

where fundamental angular frequency, harmonic frequency, and the amplitudes and phases are given by the following equations where , except for equation (9.1h) where :


(9.1d)
(9.1e)
(9.1f)
(9.1g)
(9.1h)

Solution

We multiply both sides of equation (9.1a) by , being any nonzero positive integer, and integrate between the limits and . (We omit the limits on the integrals in the following proofs since they are all the same.) Equation (9.1a) now becomes



By using the identities



the first integrand reduces to the sum of two cosines of the form while the second integrand becomes the difference between two sines of the form If , the integrals are zero, because at the limits the arguments of the cosines and sines are equal to , integral. When , the first integrand becomes and the right-hand side equals . When , the second integrand becomes , which again gives zero upon integration.

Thus we are left with


Solving for gives equation (9.1d).

To verify equation (9.1e), we multiply both sides of equation (9.1a) by and integrate, all integrals on the right side of the equation giving zero except for the one with the integrand . Proceeding as before, we arrive at equation (9.1e).

When we set in equations (9.1d,e) we get and


Problem 9.1b

Verify equations (9.1f,g) for the Fourier series coefficients and .

Solution

To verify equation (9.1f) for , we multiply both sides of equation (9.1b) by , , and proceed as in part (a). All integrals vanish except when , and the result is equation (9.1f).

When , we have



To verify equation (9.1g) for , we compare terms in equations (9.1a) and (9.1b) that involve , . The result is


Equating coefficients of and , we get


When , , , .

Problem 9.1c

Verify the coefficients for the exponential form, equation (9.1h).

Solution

To derive equation (9.1h), we use Euler’s equations (Sheriff and Geldart, 1995, problem 15.12a), namely,

Multiplying both sides of equation (9.1c) by , , and integrating, we get a series of integrals with integrands such as . Using Euler’s formula, , so the integrals vanish for all values of except when ; for this value we get

which is equation (9.1h), . For ,


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