Series Crimes

Abstract: Puiseux series are power series in which the exponents can be fractional and/or negative rational numbers. Several computer algebra systems have one or more built-in or loadable functions for computing truncated Puiseux series. Some are generalized to allow coefficients containing functions of the series variable that are dominated by any power of that variable, such as logarithms and nested logarithms of the series variable. Some computer algebra systems also have built-in or loadable functions that compute infinite Puiseux series. Unfortunately, there are some little-known pitfalls in computing Puiseux series. The most serious of these is expansions within branch cuts or at branch points that are incorrect for some directions in the complex plane. For example with each series implementation accessible to you: Compare the value of (z^2 + z^3)^3/2 with that of its truncated series expansion about z = 0, approximated at z = -0.01. Does the series converge to a value that is the negative of the correct value? Compare the value of ln(z^2 + z^3) with its truncated series expansion about z = 0, approximated at z = -0.01 + 0.1i. Does the series converge to a value that is incorrect by 2pi i? Compare arctanh(-2 + ln(z)z) with its truncated series expansion about z = 0, approximated at z = -0.01. Does the series converge to a value that is incorrect by about pi i? At the time of this writing, most implementations that accommodate such series exhibit such errors. This article describes how to avoid these errors both for manual derivation of series and when implementing series packages.