Subject: A question on exponentialize and demoivre
From: Nikolaos I. Ioakimidis
Date: Thu, 13 Mar 2003 03:15:06 +0200
Dear Stavros,
Many thanks for your prompt reply and advice. Excellent idea!
Now my ode is being solved in terms of the hyperbolic cosine
and sine functions:
(C1) ode : 'diff(x,t,2)-x = 0$
(C2) subst(t=t/%i,demoivre(subst(t=%i*t,ode2(ode,x,t)))), ratsimp;
(D2) x = (%K1 - %K2) SINH(t) + (%K2 + %K1) COSH(t)
and this result is completely satisfactory for me.
Many sincere thanks again and best regards,
Nikos
----- Original Message -----
> --- "Stavros Macrakis" <stavros.macrakis@verizon.net> wrote:
> > 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