33-formulas

33-formulas - Sometimes you have repeat doing integration...

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INTEGRATION TIPS FOR FINDING FOURIER SERIES Math 21b, O. Knill USEFUL TRIGONOMETRIC FORMULAS: 2 cos( nx ) cos( my ) = cos( nx - my ) + cos( nx + my ) 2 sin( nx ) sin( my ) = cos( nx - my ) - cos( nx + my ) 2 sin( nx ) cos( my ) = sin( nx + my ) + sin( nx - my ) THE FOURIER SERIES OF cos 2 ( t ) and sin 2 ( t ). cos(2 t ) = cos 2 ( t ) - sin 2 ( t ) = 2 cos 2 ( t ) - 1 = 1 - sin 2 ( t ) Leads to the formulas cos 2 ( t ) = (1 + cos(2 t ) / 2 sin 2 ( t ) = (1 - cos(2 t ) / 2 Note that these are the Fourier series of the function f ( t ) = cos 2 ( t ) and g ( t ) = sin 2 ( t )! SYMMETRY. If you integrate an odd function over [ - π, π ] you get 0. The product between an odd and an even function is an odd function. INTEGRATION BY PART. Integrating the di±erentiation rule ( uv ) 0 = u 0 v + vu 0 gives the partial integration formula: Z uv 0 dt = uv - Z u 0 v dt Examples: Z t sin( t ) dt = - t cos( t ) + Z cos( t ) dt = sin( t ) - t cos( t ) . Z t cos( t ) dt = t sin( t ) - Z sin( t ) dt = cos( t ) + t sin( t ) .
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Unformatted text preview: Sometimes you have repeat doing integration by part. For example, to derive the formulas Z t 2 sin( t ) dt = 2 t sin[ t ]-( t 2-2) cos[ t ] . Z t 2 cos( t ) dt = 2 t cos[ t ] + ( t 2-2) sin[ t ] . one has to integrate by part twice. THE LENGTH OF THE FOURIER BASIS VECTORS. A frequently occuring denite integral: Z - cos 2 ( nt ) dt = Z - sin 2 ( nt ) dt = These formulas can be derived also by noting that the two integrals must be the same because cos( nt ) = sin( nt + / 2). If one sums those two integrals, using cos 2 ( nt ) + sin 2 ( nt ) = 1 one gets 2 . So each integral must be ....
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