Transcription of FourierSeries - Boston University Physics
{{id}} {{{paragraph}}}
Example: quiz answers
FourierSeries - Boston University Physics
Fourier Series and IntegralsFourier SeriesLetf(x) be a piecewise linear function on [ L, L] (This means thatf(x) may possessa finite number of finite discontinuities on the interval). Thenf(x) can be expanded in aFourier seriesf(x) =a02+ Xn=1ancosn xL+bnsinn xL,(1a)or, equivalently,f(x) = X cnein x/L(1b)withcn= (an ibn)/2n <0(an+ibn)/2n >0a0/2n= useful schematic form of the Fourier series is~f(x) =Xn(an cn+bn sn).(2)This emphasizes that the Fourier series can be viewed as an expansion of a vector~finHilbert space, in a basis that is spanned by the cn(cosine waves of different periodicities)and the sn(sine waves).To invert the Fourier expansion, multiply Eq. (1) by cosn xLor sinn xLand integrateover the interval. For this calculation, we need the basic orthogonality relation of the basisfunctions:ZL Lcosn xLcosm xLdx= mnL,(3)and similarly for the sin s.
FourierSeriesandIntegrals FourierSeries Let f(x) be a piecewise linear function on [−L,L] (This means that f(x) may possess a finite number of finite discontinuities on the interval).
Loading..
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: