There are two dominant singularities, located at and at , so that we must use factors of in the construction of the first-order center series,
where
where , and 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
where . 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
and therefore we have for on the unit circle
where , and where we collected the real and imaginary terms. The original DP function is given by the imaginary part,
where , and the corresponding FC function is given by the real part,
where .