Fourier Series

gX ~ 1 2 a 0 ∑n1a n cos nXL b n sin nXLwitha n L 1 2LgXcos nX dX , etc.0 LReturn to the original variables by substituting X x − a and 2L b − a to givefx ~ 1 2 a 0 ∑n1a n cos2nx − ab − a b n sin2nx − ab − aanda n 2b − ab n 2b − a b 2nx − afxcosa b − ab 2nx − afxsinb − a adx ,dx .This provides the appropriate FS method for a function defined on a,b andassumedtopossess a period of b − a. This is not the only approach on an arbitrary interval and thehalf-range sine and cosine series can be defined in the same way as before.31