Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

version of 3 November 1997,

If you wish to print a nicely formatted version of this chapter, you may download the rtf file, which will be interpreted and opened by Microsoft Word or the pdf file, which will be interpreted and opened by Adobe Acrobat Reader.

Many of the calculations in this chapter are available in the format of a Mathematica notebook or Maple worksheet.

(Some remarks for the instructor).

One of the most important uses of Fourier series is as a tool in solving differential equations. Because of that we will frequently want to differentiate a Fourier series. Since the Fourier series, like any infinite series, is a limit, questions can arise about whether it is permissible to differentiate in x before summing in n. In your advanced calculus class you should have seen examples where interchanging the order of two limits leads to different answers. This can happen quite easily.

**Example V.1**. Let f[n,x] := 0 if -1/n <= x <= 1/n, and 1 otherwise
- i.e. the indicator function (as in chapter III)
of the complement of the
interval [-1/n, 1/n]. Then f[n,0] = 0 for all
n, so if we let x tend to zero before n tends to infinity,

whereas if n tends to infinity first,

One of the great things about Fourier series is that, despite a very reasonable worry, it is usually completely reliable to interchange limits. Indeed, one nice way to calculate Fourier series is to differentiate or integrate other Fourier series. We'll do some examples and see how reliable the answer is, as well as a couple of situations where we have to be careful. As usual, there is a Mathematica notebook paralleling the text of this chapter

With the formulae for the Fourier coefficients given in
Chapter II
it is a routine matter to
calculate the Fourier series for the function
x - x^{3} defined for -1 < x < 1.
We find that there are no cosine contributions, and

To get a feel for this series, let us plot the sum of the first
three terms and compare with
x-x^{3}:

Wonderful! The function and the truncated series match rather closely.

What happens if we differentiate?

**Model Problem V.1**. Differentiate the terms in the Fourier series for
this function,
and compare with 1 - 3 x^{2}.

Solution.

If we differentiate the series for x-x^{3}, we get the series

supposing that it is legitimate to differentiate the infinite series term by term.
(Sometimes it definitely is not.) We could calculate the
Fourier series for 1-3x^{2} directly with the formulae of
chapter IV,
either by hand or with software
as in the Mathematica
notebook or the
Maple worksheet, and compare, but before doing so,
let's evaluate the differentiated series on its own merits. First note that
the function 1-3x^{2} is even, so there will only be cosine contributions,
as we have found. Next, let us plot the sum of the first four terms in
the differentiated series and compare with the exact function:

A superb match! Just for fun, let's see the comparison outside the interval where we cut off the polynomials:

Remember - the
Fourier series always corresponds to the periodic extension of the function
on the basic interval. The formula 1 - 3x^{2} is not valid for the series
outside [-1,1].

Finally, let us calculate the Fourier series using the integral formulae from Chapter II. The result, for example using the Mathematica notebook or the Maple worksheet, is

a_{0} = 0,

a_{k} = 12 (-1)^{k+1}/(k)
^{2}, k = 1, 2, ...,

as expected. (Notice that this example has something to do with the Legendre polynomials.)

Suppose now that instead of differentiating, we integrate a Fourier series
term by term. If a_{0} (which = c)_{0} is different from 0, we get another Fourier
series! If a_{0} is not 0, then we would only get another Fourier series after
replacing the function x with a Fourier series, but we won't consider that
case now.

Does the integrated series converge to the integral of the original function?
According to our experiment with differentiation, it seems so. Let us try
again, by integrating the series for x - x^{3}term by term. We find:

What does this show about the constant of integration? When we integrated
the series we got the integral with average average a_{0}, chosen here as 0.
Remember that a_{0} is always the average of the function. The difference in
height between the two graphs is just the average of
x^{2}/2 - x^{4}/4, which
doesn't happen to be 0.

**Exercise V.1**. Use the tricks of this chapter to calculate the
Fourier
series for the functions x^{n}, n = 0, 1, ..., on the interval -1 < x < 1.

**Exercise V.2**. What series do you obtain by differentiating the
Fourier series for the square pulse (cf. Model
Problem IV.1)? If you know about the "Dirac"
*delta function*, comment on the relationship between the series you get for
the square pulse and the one for the delta function delta(x-a)
(periodically extended).

**Exercise V.3**. Consider what happens when a
Legendre series is differentiated term by term.
Is the result a Legendre series? Does the
result converge to the derivative of the original function?

**Exercise V.4**. Check the convergence of Exercise V.3 numerically
in a specific example such as f(x) = sin(¼x), -1 < x < 1.

Onward to chapter VI (red syllabus)

Onward to chapter XII (yellow syllabus)

Back to chapter IV

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