On the Two q-Analogue Logarithmic Functions
Abstract
There is a simple, multi-sheet Riemann surface associated with eq(z)'s inverse function lnq(w) for 0< q < 1. A principal sheet for lnq(w) can be defined. However, the topology of the Riemann surface for lnq(w) changes each time "q" increases above the collision point of a pair of the turning points of eq(x). There is also a power series representation for lnq(1+w). An infinite-product representation for eq(z) is used to obtain the ordinary natural logarithm lneq(z) and the values of sum rules for the zeros "zi" of eq(z). For |z|<|z1|, eq(z)=expb(z) where b(z) is a simple, explicit power series in terms of values of these sum rules. The values of the sum rules for the q-trigonometric functions, sinq(z) and cosq(z), are q-deformations of the usual Bernoulli numbers.
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