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Odd or Even? - Arbritrary Period Fourier Series
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[QUOTE="Ray Vickson, post: 5400817, member: 330118"] No software can tell you if the extended function is even, odd, or neither. YOU can decide to make it even, or you can extend it as an odd function (but not in any natural way). If you apply the Fourier series formulas to your function without further thought, you are tacitly assuming that the function is extended outside [0.3\ in some way. I did the same thing in Maple, and for expansion functions ##u_0 = \sqrt{1/3}## and ##u_n = \sqrt{2/3} \, \cos(2 \pi n x/3)##, ##v_n = \sqrt{2/3} \, \sin(2 \pi n x/3)## for ##n = 1, 2, 3, \ldots## (which are orthonormal on ##[0,3]##) the coefficients in [tex] f(x) = A_0 u_0 + \sum_{n=1}^{\infty}( A_n u_n(x) + B_n v_n(x) ) [/tex] are given by [tex] A_n = \int_0^3 f(x) u_n(x) \, dx, \;\; B_n= \int_0^3 f(x) v_n(x) \, dx [/tex] for all ##k##. Maple gets [tex] B_n = \frac{3 \sqrt{6}}{4 \pi^2 n^2} \left( \sin \left(\frac{2 \pi n}{3}\right) + \sin \left(\frac{4 \pi n}{3} \right) \right) [/tex] Although this does not look like 0, when you evaluate it at positive integer ##n## you do, actually, get 0! You can even prove this analytically, just by looking at the locations of the points at angles ##2 \pi n/3## and ##4 \pi n/3## on the unit circle. Therefore, the Fourier series of ##f(x)##[B] is a Fourier-cosine series[/B], exactly as you would expect of an even function. Is that what was bothering you? [/QUOTE]
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Odd or Even? - Arbritrary Period Fourier Series
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