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# lec8-2 - Complex Fourier series Recalling the Euler formula...

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Complex Fourier series Recalling the Euler formula, we can write cos nx = e inx + e - inx 2 sin nx = e inx - e - inx 2 i This suggest that instead of using cos nx and sin nx , we might use e inx f ( x ) = X n = -∞ c n e inx To obtain the coefficients, multiply left and right sides by e - imx and integrate over one period c n = 1 2 π Z π - π f ( x ) e - inx dx

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Other intervals For a function with periodicity 2 π , we used functions with periodicity 2 π for the expansion, namely cos x , sin x , or e inx Consider now a function with periodicity 2 l , so that f ( x + 2 l ) = f ( x ) We can see that sin n π x l and cos n π x l have the desired periodicity, as does e in π x l We then have the relations f ( x ) = a 0 2 + X n =1 a n cos n π x l + X n =1 a n sin n π x l Or we might use the complex series, f ( x ) = X n = -∞ c n e n π x l
Fourier coefficients for more general intervals The coefficients then are found from a n = 1 l Z l - l f ( x ) cos n π x l dx b n = 1 l Z l - l f ( x ) sin n π x l dx c n = 1 2 l Z l - l f ( x ) e -

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lec8-2 - Complex Fourier series Recalling the Euler formula...

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