Good question. For that matter, log(-1) only simplifies to %i*%pi if
lognegint = true (default is false).
Here are the possible arguments I can think of for this behavior (I don't
guarantee that they correspond to the original rationale...):
1) log is multivalued, and the user may want to control whether log(-1) =>
%pi or -%pi etc.
2) Maxima assumes the user is working in the real domain, so shouldn't
introduce %i unless explicitly asked to. But domain:complex doesn't change
the behavior.
3) log(-1) may appear as an intermediate result of a real calculation which
goes away with simplification, e.g. logcontract(log(-1)+log(-x)), so why
confuse matters?
4) Once you have an expression like log(x)+%i*%pi, converting back to
log(-x) is slightly tricky -- actually, simply log(exp(EXPR)) will do it,
but there is no easy way to apply that globally, like logcontract.
I suspect that 3 and 4 are the most important.
So perhaps the default is OK, but log(-C) and log(%i*C) should expand out
when domain=complex or logexpand=ultra. Conversely, perhaps there should
be a logconmode=all which contracts things like log(x)+C so that arguments
3 and 4 go away....
Thoughts?
-s
On Mon, Feb 20, 2012 at 23:38, Henry Baker <hbaker1 at pipeline.com> wrote:
> How come log(-%i) doesn't already simplify by itself, instead of requiring
> rectform to do it?
>
> Is there some switch that Maxima uses to force log(constant) to simplify
> if it can?
>
> At 01:12 PM 2/20/2012, Stavros Macrakis wrote:
> >Henry,
> >
> >Yes, this derivation is essentially the same as the bottomup rectform.
> In fact:
> >
> > expr: logcontract(logarc(4*atan(1/5)-atan(1/239)));
> > rectform(substpart(e1:rectform(piece),expr,1,2,1)) => %pi/4
> >
> >You'll see that e1 here is -%i.
> >
> >What rectform does internally by default looks more like
> >
> > rectform(substpart(e2:polarform(piece),expr,1,2,1));
> >
> >where e2 isn't at all helpful.
> >
> >I agree that most mathematical functions (sin, etc.) and form-changing
> routines (factor etc.) should distribute over "=".
> >
> > -s
> >
> >On Mon, Feb 20, 2012 at 15:45, Henry Baker <hbaker1 at pipeline.com> wrote:
> >result=4*atan(1/5)-atan(1/239);
> > 1 1
> >(%o1) result = 4 atan(-) - atan(---)
> > 5 239
> >(%i2) lhs(%)=logcontract(logarc(rhs(%)));
> > 4
> > (%i - 5) (%i + 239)
> > %i log(- --------------------)
> > 4
> > (%i - 239) (%i + 5)
> >(%o2) result = ------------------------------
> > 2
> >(%i3) %*2/%i;
> > 4
> > (%i - 5) (%i + 239)
> >(%o3) - 2 %i result = log(- --------------------)
> > 4
> > (%i - 239) (%i + 5)
> >(%i4) exp(lhs(%))=exp(rhs(%));
> > 4
> > - 2 %i result (%i - 5) (%i + 239)
> >(%o4) %e = - --------------------
> > 4
> > (%i - 239) (%i + 5)
> >(%i5) lhs(%)=rectform(rhs(%));
> > - 2 %i result
> >(%o5) %e = - %i
> >(%i6) log(lhs(%))=log(rhs(%));
> >(%o6) - 2 %i result = log(- %i)
> >(%i7) rectform(lhs(%))=rectform(rhs(%));
> > %i %pi
> >(%o7) - 2 %i result = - ------
> > 2
> >(%i8) solve(%,result);
> > %pi
> >(%o8) [result = ---]
> > 4
> >
> >This example shows why rectform, log, exp, logcontract, logarc, etc.,
> should distribute over "=".
> >
> >At 11:20 AM 2/20/2012, Stavros Macrakis wrote:
> >>Consider the expression 4*atan(1/5)-atan(1/239) (Machin's formula).
> >>
> >>How can we use Maxima to simplify it to %pi/4?
> >>
> >>Well, here's one approach:
> >>
> >>atan(trigexpand(trigexpand(tan(ex))))
> >>
> >>And here's another way that doesn't depend on trig identities:
> >>
> >>(%i1) ex:4*atan(1/5)-atan(1/239);
> >>(%o1) 4*atan(1/5)-atan(1/239)
> >>(%i2) llex: logcontract(logarc(ex));
> >>(%o2) %i*log(-(%i-5)^4*(%i+239)/((%i-239)*(%i+5)^4))/2
> >>(%i3) rectform(llex);
> >>(%o3)
> (atan(sin(4*atan(1/5))/cos(4*atan(1/5)))-atan(sin(4*(%pi-atan(1/5)))/cos(4*(%pi-atan(1/5))))-2*atan(1/239))/2
> >> <<< oops, not so helpful
> >>(%i4) scanmap(rectform,llex,bottomup);
> >> <<< but reorganizing the rectform calculation gives a better
> result
> >> <<< makes me think that rectform could be improved...
> >>(%o4) %pi/4
> >>
> >>Are any of these techniques packaged into some analog of trigsimp?
> >>
> >> -s
>
>