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:
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(9.1a)
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(9.1b)
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(9.1c)
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where
fundamental angular frequency,
harmonic frequency, and the amplitudes and phases are given by the following equations where
, except for equation (9.1h) where
:
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(9.1d)
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(9.1e)
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(9.1f)
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(9.1g)
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(9.1h)
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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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Also in this chapter
External links
find literature about Fourier series
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