Continuing with half-range expansions, we now entertain the possibility of writing a Fourier sine series of f on [0,L] The method is similar to that just used.
NOTE: The video starts with a small 'Thank You' note for all my fans, which sorely has nothing to do with the lesson.
FOURIER SINE SERIES
May want to check out:
FOURIER COSINE SERIES
HALF RANGE EXPANSION OF f(x)=e^2x
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VIDEO LECTURE
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LESSON
We extend f to an odd function w defined on [-L,L] by letting
Below is a typical graph of w.
Since w is an odd function, its Fourier series on [-L,L] contains only sine terms and is
in which
Since g(x)=f(x) for
Moreover on the interval [0,L]. we may take the Fourier series of w to be a Fourier sine series of f, as stated in the following definition. If f is integrable on [0,L], the Fourier sine series of f on [0,L], is
where
As usual, we form a convergence test for the Fourier sine 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 (2) is immediate upon letting x=0 and x=L in the sine series; all of the terms vanish because
 for any integer n.All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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