Subject: A question on exponentialize and demoivre
From: Stavros Macrakis
Date: Wed, 12 Mar 2003 19:05:50 -0500
> What should I do, please, for the transformation of an expression
> containing exponentials to hyperbolic sines and cosines?
...
> I would prefer to use an existing Maxima command (such
> as demoivre) avoiding to prepare my own simplification
> rules for the second case (that finally leading to hyperbolic
> functions).
The simplest way to do this is probably with a change of variable from
x->%i*x, then demoivre, then the reverse transformation. This even
catches cases where the hyperbolic functions are not completely obvious,
e.g.:
2 x - 2 x
%E x %E 1
(D13) ----- - %E + ------- + -
4 4 2
(C14) subst(%i*x,x,%);
2 %I x - 2 %I x
%E %I x %E 1
(D14) -------- - %E + ---------- + -
4 4 2
(C15) demoivre(%);
%I SIN(2 x) + COS(2 x) COS(2 x) - %I SIN(2 x)
1
(D15) ---------------------- + ---------------------- - %I SIN(x) -
COS(x) + -
4 4
2
(C16) subst(x/%i,x,%);
SINH(2 x) + COSH(2 x) COSH(2 x) - SINH(2 x)
1
(D16) --------------------- + --------------------- - SINH(x) -
COSH(x) + -
4 4
2
...
(C19) ratsimp(d16);
COSH(2 x) - 2 SINH(x) - 2 COSH(x) + 1
(D19) -------------------------------------
2