There is a single dominant singularity at , so that we must use a factor of in the construction of the second-order center series,
where
where we distributed the factor on the series and manipulated the indices of the resulting sums. Unlike the original series, with coefficients that behave as , this series has coefficients that go to zero as when , and therefore our evaluation of the set of dominant singularities of was in fact correct. We have therefore for the representation
In order to take the real and imaginary parts of on the unit circle, we observe now that since we have on the unit circle
If we write this in terms of we get
We also have that
and therefore we have for on the unit circle
where we collected the real and imaginary terms. The original DP function is given by the imaginary part,
and the corresponding FC function is given by the real part,