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# fouries - Jan 7 Fourier Methods What How and Why In the...

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Jan 7. Fourier Methods - What, How, and Why? In the first decade of the 19th century, Jean Baptiste Joseph Fourier invented a technique using sums of trigonometric functions called ``Fourier Series'' to solve the differential equations of heat conduction. Today, physicists, astronomers, engineers, and many others use Fourier series to analyze functions of time and space, solve differential equations, and compress data, to name just a few applications. Periodic functions Definition. f ( x ) is periodic with period p if f ( x+p ) = f ( x ), for all x . For example. sin x and cos x both have period 2 π , sin λ x and cos ( λ x+ α ) both have period 2 π / λ , pulse functions are also periodic. Definition. Given any f ( x ) defined for a<x<b , we can define its periodic extension to all x by f ( x+nT ) = f ( x ), n = 0, ± 1, ± 2, ..., a<x<b. The extended function is not yet defined at x=a+nT ; if f ( a ) f ( b ), it will be discontinuous at x=a+nT . For allowing functions with discontinuities at any point x= x 0 , we introduce right-hand limit: f ( x 0 +0) = lim f(x 0 + ε 2 ) ε→ 0

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fouries - Jan 7 Fourier Methods What How and Why In the...

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