fourier-square

fourier-square - Fourier Series An Example Formulas: Let...

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Fourier Series – An Example Formulas: Let f(x) be periodic with period 2 π . We want to write f ( x ) = A 0 2 + X 1 ( A k cos kx + B k sin kx ) . The Fourier coefficients are given by the formulas A k = 1 π Z π - π f ( x ) cos kx dx B k = 1 π Z π - π f ( x ) sin kx dx. Moreover one has the analogue of the “Pythagorean theorem” k f k 2 = π " A 2 0 2 + X 1 ( A 2 k + B 2 k ) # . Example: Consider the function f ( x ) = ± - 1 if - π < x 0 1 if 0 < x π To use the above formulas for the Fourier coefficients we split the integrals into two pieces A k = 1 π ²Z 0 - π ( - 1) cos kx dx + Z π 0 (+1) cos kx dx ³ . and B k = 1 π ²Z 0 - π ( - 1) sin kx dx + Z π 0 (+1) sin kx dx ³ . When one evaluates the A k , the two integrals cancel so A k = 0. Also, Z π 0 (+1) sin kx dx = - cos + 1 k = ± 0 if k is even 2 k if k is odd By a computation the first integral in B k has the same value as this second integral. Thus, B k = 2 π ± 0 if k is even 2 k if k is odd = ± 0 if k is even 4 if k is odd Consequently the desired Fourier series is
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