Infinite series representations for complex numbers

Abstract

We introduce two new algorithms that lead to finite or infinite series expansions for complex number in terms of ‘integral digits’ within the complex quadratic fields ℚ\(\left( {\sqrt { - m} } \right)\), form=1, 2,…, 11. In particular, we derive complex number representations as sums of reciprocal of Gaussion integers and as sums of reciprocals of algebraic integers in ℚ\(\left( {\sqrt { - m} } \right)\), form=2, 3, 7 and 11. In addition to convergence of the various algorithms we investigate the representation of ‘rationals’ relative to the fields ℚ\(\left( {\sqrt { - m} } \right)\), form=1, 2, 3, 7 and 11.