Dear Francois, > 5/ Do you prefer : > a-expand (sin (2*x+y)) gives an Expression in sin x, sin y, cos x... of corse. > b-expand (sin (3)) remains sin (3) [or do you prefer with sin 1 and cos 1] > c-expand (sin (3*expressions without variables)) remains the same.
To be honest, I find it very difficult to answer your question. I guess that in general there is no such thing as a "normal form" for trigonometric expressions. However, for special cases there may well be. I vaguely remember that maxima was very good at simplifying that stuff, they have trigsimp, trigrat, trigreduce,... See http://www.ma.utexas.edu/maxima/maxima_14.html I think it would be very useful if you would browse the web a little and see what has been done on the subject. For example, for the case of no variables, MathSciNet gave me ------------------------------------------------------------------------------- MR0981072 (90c:68040) Schorn, Peter(1-NC-C) A canonical simplifier for trigonometric expressions in the kinematic equation. Inform. Process. Lett. 29 (1988), no. 5, 241--246. 68Q40 Summary: "We present a new canonical simplification algorithm for trigonometric expressions in the kinematic equation. The expressions consist of rational numbers and the function symbols $+, -, *, /, \sin$ and $\cos$. Variables are not allowed and the arguments of $\sin$ and $\cos$ are limited to rational multiples of $\pi$. We present an algorithm which computes a simple normal form in the real domain, in contrast to the known methods which either produce unnecessarily complicated normal forms or are inefficient due to large integer arithmetic. The simplifier has successfully been applied in the field of robot kinematics." ------------------------------------------------------------------------------- ScienceDirect told me that within ScienceDirect this article was cited once, namely by: Trigonometric polynomials with simple roots Achim Schweikard* Technische Universitt Berlin, Berlin, Germany Abstract Trigonometric polynomials frequently occur applications in physics, numerical analysis and engineering, since each periodic function can be approximated by a trigonometric polynomial. Additionally, there are many analogies between trigonometric and standard algebraic polynomials. Algorithms in computer algebra depend on methods for the square-free decomposition of polynomials. These methods use polynomial division and cannot be applied directly to trigonometric polynomials. Let P denote the set of odd multiples of ?. A trigonometric polynomial T* is a reduced representation of a trigonometric polynomial T if the set of zeros of T in P is the same as the set of zeros of T* in P, and if all zero s of T* are simple zeros. It is shown that a reduced representation of a trigonometric polynomial with rational or algebraic coefficients can be found in polynomial time. Author Keywords: Analysis of algorithms; computer algebra; trigonometric and exponential polynomial; root multiplicity; greatest square-free divisor ------------------------------------------------------------------------------- it might also be helpful to send a message to Richard Fateman, or to math.sci.symbolic. I truly appreciate your work. I find it's a great pity that you cannot come to the workshop. Martin _______________________________________________ Axiom-developer mailing list [email protected] http://lists.nongnu.org/mailman/listinfo/axiom-developer
