Unformatted text preview: F , you can integrate the Fourier series of f term by term. Furthermore, the only assumption on f is that it is piecewise smooth and integrates to 0 over one period (to guarantee the periodicity of F .) Indeed, if you start with the Fourier series of f , f ( t ) = ∞ ± n =1 ² a n cos nπ p t + b n sin nπ p t ³ , and integrate term by term, you get F ( x ) = µ x f ( t ) dt = ∞ ± n =1 ² a n µ x cos nπ p tdt + b n µ x sin nπ p tdt ³ = ∞ ± n =1 ² a n ¶ p nπ · sin nπ p t ¸ ¸ ¸ x dt + b n ¶p nπ · cos nπ p t ¸ ¸ ¸ x ³ = p π ∞ ± n =1 b n n + ∞ ± n =1 ²p nπ b n cos nπ p x + p nπ a n sin nπ p x ³ , as derived earlier. See the following exercise for an illustration....
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 Fall '11
 StuartChalk
 Fourier Series, Periodic function, Continuous function, nπ

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