In this lesson, we take a step further and explore the possibilities of writing a Fourier series of f on [0,L] containing either just sine terms or just cosine terms, whicever we chooses. Such series are called half-range expansions. The key lies in a previous lemma.
FOURIER COSINE SERIES
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CONTRASTING POWER AND FOURIER SERIES
FOURIER SINE SERIES
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LESSON
Suppose that f is integrable on [0,L] and we want to expand f in a Fourier series on [0,L] containing just cosine terms. The idea is to extend f to a new function g defined on [-L,L] in such a way that g is an even function. The Fourier series of g on [-L,L] contains only cosine terms. Since f and g agree on [0,L], this gives a Fourier cosine series of f on [0,L]. To do this, define
A typical graph of g is obtained by folding the graph of g over the y-axis as shown below. Since g is an even function, its Fourier series on [-L,L] is
in which
Since g(x)=f(x) for If f is integrable on [0,L], the Fourier cosine series of f on [0,L], is
where
We remark that the function g we introduced had the sole purpose of suggesting how to define a pure cosine series on [0,L]. In computing a Fourier cosine series, we never mention g but simply calculate the coefficeint directly from the definition. As with all Fourier series, there is a convergence test for the Fourier cosine series. Let f be piecewise continuous on [0,L].
the average of the left and right limits of f at x. In particular, if f is also continuous at x, the series converges to f(x). Conclusion (1) should be no surprise. To understand conclusion (2), consider the convergence at zero for the Fourier series of g on [-L,L]. At zero, this series converges to
and
and so
and the cosine series of f converges to All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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and 
, we may think of the series as a Fourier cosine series of f(x) on [0,L]. Further, the coefficeints can be written in terms of f. This leads to a formal defintion of the Fourier cosine series. 
and 
, and f has both right and left derivatives at x, then at x the Fourier cosine series of f on [0,L] converges to 
,
exists, the cosine series converges at zero to 
.
exists, the cosine series converges at L to 
.
. But
,
at zero. A similar argument establishes convergence of the series to f(L-) at L.
