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Odd or Even? - Arbritrary Period Fourier Series
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[QUOTE="LCKurtz, post: 5400776, member: 198114"] [USER=573808]@Charge2[/USER]: You need to understand that the function you gave is neither even nor odd. In fact it doesn't make sense to ask whether it is even or odd because it is only defined on ##[0,3]##. And for the same reason, it isn't periodic either. What you [B]can[/B] do is talk about extending the definition of the function into an even or odd periodic function. For example, if you add to your definition that$$ f(x) = -3-x, ~-3\le x \le -2,~f(x) = -1,~-2\le x \le -1,~f(x) = x,~-1\le x \le 0$$ and periodic thereafter you would have an odd function of period ##6##. This is the function that a half range sine series of the given function would represent. Just repeating the graph as in post #2 would give an even extension of the function of period ##3##. If you use the half range cosine expansion on the given function it would represent the even extension and would be calculated as a function of period ##6## even though it is also of period ##3##. [Edit, added]: I just noticed that your OP states the given function on ##[0,3]## [B]represents one period[/B]. So it is defined on the whole line and periodic and is even. Nevertheless, I think you may find my comments above useful, at least I hope so. [/QUOTE]
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Odd or Even? - Arbritrary Period Fourier Series
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