Question

Why cosecx is not expanded as fourier series

Answer 1

Answer:

As it’s an odd function, you want a sine series. So the series is of the form

\cosec(x)=a1sin(x)+a2sin(2x)+a3sin(3x)+a4sin(4x)+… .

If you multiply by sin(nx) and integrate from −π to π you get

∫π−πsin(nx)sin(x)dx=an∫π−πsin2(nx)dx .

This gives 12an=∫π−πsin(nx)sin(x)dx .

If n=1 then this gives a1=4π and if n=2 then a2=0 .

Now sin(nx)=sin(x)cos((n−1)x)+cos(x)sin((n−1)x)

= sin(x)cos((n−1)x)+cos(x)sin(x)sin((n−2)x)+cos2(x)cos((n−2)x)

= sin(x)cos((n−1)x)+cos(x)sin(x)sin((n−2)x)+cos((n−2)x)−sin2(x)cos((n−2)x) .

Divide by sin(x) and integrate. When n=3 the integral is ∫π−πcot(x)dx .

I think this integral diverges. It’s the same as the integral of the tangent function over a full cycle and that diverges at π2 . So I don’t think the Fourier series exists.