\\% Regularized 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: 2011-August-08 (16:42)

\\read("eta.gpi");  \\ for Etax() and Etax_plus()


\\ The regularized ("improved") q-exponential as in
\\  Jan L. Cieslinski:
\\    "Improved q-exponential and q-trigonometric functions"
\\  http://arxiv.org/abs/1006.5652v1
\\


\\ ++++++++++++++++++++++++++++++++++++++++
\\ regularized q-exponential, second kind:
rexp_Q(q, x, S=-1)={ exp_Q(q, x/2, S) * Exp_Q(q, x/2, S); }
\\
\\ rexp_Q( q, x ) = 1 / rexp_Q(q, -x)
\\
\\ abs( rexp_Q( q, I*x ) ) = 1 (both q and x real)
\\
\\ rexp_Q( 1/q, x ) = rexp_Q(q, x)
\\ rexp_Q( q, q*x ) = (1-(1-q)*x/2)/(1+(1-q)*x/2) * rexp_Q(q, x)
\\ abs( rexp_Q( q, I*x ) ) = 1 (both q and x real)
\\ Qderiv( rexp_Q, q, x ) = rexp_Q(q, x) / (1+(1-q)*x/2)
\\ Qderiv( rexp_Q, q, x ) = (rexp_Q(q, q*x) + rexp_Q(q, x))/2

\\ ++++++++++++++++++++++++++++++++++++++++
\\ regularized q-cosine, q-sine, q-tangent, second kind:
rcos_Q(q, x, S=-1)={ ( rexp_Q(q, +I*x, S) + rexp_Q(q, -I*x, S) ) / 2; }
rsin_Q(q, x, S=-1)={ ( rexp_Q(q, +I*x, S) - rexp_Q(q, -I*x, S) ) / (2*I); }
\\ rcos_Q(q, x)^2 + rsin_Q(q, x)^2 = 1
\\ Qderiv( rsin_Q, q, x ) = +(rcos_Q(q, q*x) + rcos_Q(q, x))/2
\\ Qderiv( rcos_Q, q, x ) = -(rsin_Q(q, q*x) + rsin_Q(q, x))/2
\\ -1 <= rcos_Q(q, x) <= +1  (both q and x real)
\\ -1 <= rsin_Q(q, x) <= +1  (both q and x real)

rtan_Q(q, x, S=-1)={ rsin_Q(q, x, S) / rcos_Q(q, x, S); }
rcot_Q(q, x, S=-1)={ rcos_Q(q, x, S) / rsin_Q(q, x, S); }
\\ rcos_Q(q, 2*x) = (1-tan_Q(q, x)^2) / (1+tan_Q(q, x)^2)
\\ rsin_Q(q, 2*x) = (2*tan_Q(q, x)) / (1+tan_Q(q, x)^2)

\\ ++++++++++++++++++++++++++++++++++++++++
\\ regularized hyperbolic q-cosine, q-sine, q-tangent, second kind:
rcosh_Q(q, x, S=-1)={ ( rexp_Q(q, +x, S) + rexp_Q(q, -x, S) ) / 2; }
rsinh_Q(q, x, S=-1)={ ( rexp_Q(q, +x, S) - rexp_Q(q, -x, S) ) / 2; }
\\ rcosh_Q(q, x)^2 - rsinh_Q(q, x)^2 - 1

rtanh_Q(q, x, S=-1)={ rsinh_Q(q, x, S) / rcosh_Q(q, x, S); }
rcoth_Q(q, x, S=-1)={ rcosh_Q(q, x, S) / rsinh_Q(q, x, S); }


\\ ==== end of file ====