Trap rule error in periodic functions

Any function that has all derivatives at its beginning, t=0, equal to all derivatives at its end, t=T, can be expanded as a periodic function.

A periodic function can always be expanded as

Note that c_j is complex and that

Since the first exponential is always 1. The integral from 0 to T of f is

The value of the integral is given by the c₀ term alone. All other terms in the expansion integrate to zero.

Let so that the midpoint trap rule approximation to this integral is

Note the interchange of summation orders. Let , so that

The last term can be written as a sum of z^k and we can use the familiar relation for z¹1

So that

Note that so that the numerator in is always zero. The denominator is also zero for j=mN for which . In this case the last sum in is N so that

The terms that enter are not uniform. If for some accidental reason c₁₀ = 0, 10 point integration will seem exact up to c₂₀. Changing N from 10 to 20 will give a false accuracy reading. The correct change is from 10 to 11 or 13, Do not simply double the points.