MATH307 December 2010
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Question 06 (a)
Determine the coeﬃcients cn in the expansion
where ƒ(x) = x and .
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Recall that the functions
are an orthogonal basis for square integrable functions, i.e.
How can we use this orthogonality to get the coefficients in the series?
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so that we want such that
From Hint 1 (and setting L=1) we have that are orthogonal i.e.,
where the overbar indicates the complex conjugate
We can get the coefficients of using this orthogonality. Take the inner product with each
If we use integration by parts: with , to obtain
Here we see the importance of having , otherwise the coefficient diverges using this formula. If n = 0,
Therefore, we have all the coefficients for the series.
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