The Fourier expansion of a periodic function F(x) with period 2x is given by
[infinity] [infinity]
F(x)=a,+Σan cos(nx)+Σbn sin(nx)
n=1 n=1
where
x
an=1/π∫ f (x) cos(nx)dx
-x
x
ao=1/2π∫ f (x)dx
-x
x
bn=1/π∫ f (x) sin(nx)dx
-x
(a) Explain the modifications which occur to the Fourier expansion coefficients {an) and (bn) for even and odd periodic functions F(x).
(b) An odd square wave F(x) with period 2n is defined by
F(x) = 1 0≤x≤π
F(x)=-1 -π≤x≤0
Sketch this square wave on a well-labelled figure
. (c) Derive the first 5 terms in the Fourier expansion for F(x). (10 marks) (10 marks) (5 marks)