Infinite series representations for complex numbers |
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Authors: | Arnold Knopfmacher John Knopfmacher |
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Institution: | 1. Department of Mathematics, University of the Witwatersrand, 2050, Johannesburg, South Africa
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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 ? \(\left( {\sqrt { - m} } \right)\) , 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 ? \(\left( {\sqrt { - m} } \right)\) , form=2, 3, 7 and 11. In addition to convergence of the various algorithms we investigate the representation of ‘rationals’ relative to the fields ? \(\left( {\sqrt { - m} } \right)\) , form=1, 2, 3, 7 and 11. |
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