/* -*- gp-script -*- */ \\% q-exponential and q-trigonometric series of the second kind. \\ Author: Joerg Arndt \\ License: GPL version 3 or later \\ online at http://www.jjj.de/pari/ \\ version: 2014-October-16 (18:30) \\read("eta.gpi"); \\ for Etax() and Etax_plus() \\ ++++++++++++++++++++++++++++++++++++++++ \\ q-exponentials, second kind: \\ These definitions are used more frequently. exp_Q(q, x, S=-1)={ 1 / Etax(q, (1-q)*x, S); } Exp_Q(q, x, S=-1)={ Etax_plus(q, (1-q)*x, S); } \\ \\ lim(q-->1, exp_Q(q, x)) = lim(q-->1, Exp_Q(q, x) ) = exp(x) \\ \\ Exp_Q(q, x) * exp_Q(q, -x) = Exp_Q(q, -x) * exp_Q(q, x) = 1 \\ \\ Exp_Q(1/q, x) - exp_Q(q, x) \\ exp_Q(1/q, x) - Exp_Q(q, x) \\ \\ exp_Q(q, x) = exp_q(q, (1-q)*x) \\ Exp_Q(q, x) = Exp_q(q, (1-q)*x) \\ exp_Q(q, x) = exp_Q(q^2, x/(1+q)) * exp_Q(q^2, q*x/(1+q)) \\ Exp_Q(q, x) = Exp_Q(q^2, x/(1+q)) * Exp_Q(q^2, q*x/(1+q)) \\ Qderiv( exp_Q, q, x ) = (+exp_Q(q, x)) \\ Qderiv( Exp_Q, q, x ) = (+Exp_Q(q, q*x)) \\ ++++++++++++++++++++++++++++++++++++++++ \\ q-cosine, q-sine, q-tangent, second kind: cos_Q(q, x, S=-1)={ ( exp_Q(q, +I*x, S) + exp_Q(q, -I*x, S) ) / 2; } sin_Q(q, x, S=-1)={ ( exp_Q(q, +I*x, S) - exp_Q(q, -I*x, S) ) / (2*I); } Cos_Q(q, x, S=-1)={ ( Exp_Q(q, +I*x, S) + Exp_Q(q, -I*x, S) ) / 2; } Sin_Q(q, x, S=-1)={ ( Exp_Q(q, +I*x, S) - Exp_Q(q, -I*x, S) ) / (2*I); } \\ cos_Q(q, x)*Cos_Q(q, x) + sin_Q(q, x)*Sin_Q(q, x) = 1 \\ sin_Q(q, x)*Cos_Q(q, x) - cos_Q(q, x)*Sin_Q(q, x) = 0 \\ Qderiv( sin_Q, q, x ) = (+cos_Q(q, x)) \\ Qderiv( cos_Q, q, x ) = (-sin_Q(q, x)) \\ Qderiv( Sin_Q, q, x ) = (+Cos_Q(q, q*x)) \\ Qderiv( Cos_Q, q, x ) = (-Sin_Q(q, q*x)) tan_Q(q, x, S=-1)={ sin_Q(q, x, S) / cos_Q(q, x, S); } \\ == Sin_Q(q, x, S)/Cos_Q(q, x, s); cot_Q(q, x, S=-1)={ cos_Q(q, x, S) / sin_Q(q, x, S); } \\ ++++++++++++++++++++++++++++++++++++++++ \\ hyperbolic q-cosine, q-sine, q-tangent, second kind: cosh_Q(q, x, S=-1)={ ( exp_Q(q, +x, S) + exp_Q(q, -x, S)) / 2; } sinh_Q(q, x, S=-1)={ ( exp_Q(q, +x, S) - exp_Q(q, -x, S)) / 2; } Cosh_Q(q, x, S=-1)={ ( Exp_Q(q, +x, S) + Exp_Q(q, -x, S)) / 2; } Sinh_Q(q, x, S=-1)={ ( Exp_Q(q, +x, S) - Exp_Q(q, -x, S)) / 2; } tanh_Q(q, x, S=-1)={ sinh_Q(q, x, S) / cosh_Q(q, x, S); } coth_Q(q, x, S=-1)={ cosh_Q(q, x, S) / sinh_Q(q, x, S); } \\ ==== end of file ====