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## [Axiom-developer] [#47 complexForm(log(%i) - log(-%i))] (new)

 From: anonymous Subject: [Axiom-developer] [#47 complexForm(log(%i) - log(-%i))] (new) Date: Sat, 22 Jan 2005 04:20:27 -0600

```Is the following a case of total simplification failure (a certain
bug from the user's point... unless AXIOM is not targeted for
Martians who as known have no need in simplification whatsoever :)

or

it is still possible to get  %pi  out of the lengthy expression
%i*log(-%i) - %i*log(%i)

?

Given my previous posting about complexForm, I mean maybe just a way
to expel those nasty logarithms and see the real and imaginary parts
not intertwined rather than a correct answer.

Any, maybe even a very cumbersome way?

.....................................................................

-> simplify(%i*log(-%i) - %i*log(%i))

%i log(- %i) - %i log( %i)

-> normalize(%i*log(-%i) - %i*log(%i))

%i log(- %i) - %i log( %i)

-> complexNormalize(%i*log(-%i) - %i*log(%i))

%i log(- %i) - %i log( %i)

.....................................................................

======================================================================

AXIOM Version of Wednesday December 15, 2004 at 00:50:38

AXIOM ->         complexForm(log(%i) - log(-%i))

0

COMPARE:

Derive 6.1>      LOG(#i) - LOG(-#i)
Maple 9.5.2>     ln(I) - ln(-I);
Mathematica 5.1> Log[I] - Log[-I]
Maxima 5.9.1>    RECTFORM(LOG(%I) - LOG(-%I));

#i*pi
I*Pi
I*PI
I*Pi
%I %PI

======================================================================

> AXIOM ->         complexForm(log(%i) - log(-%i))
>
>                  0

could you please file this as a bug on issuetracker? The problem is in the
operation "argument":

(8) -> argument(%i)

%pi
(8)  ---
2
Type: Expression Integer
(9) -> argument(-%i)

%pi
(9)  ---
2
Type: Expression Integer

the latter should be -pi/2, of course. The problem is in gaussian.spad,
COMPCAT:

if R has TranscendentalFunctionCategory then
half := recip(2::R)::R

if R has RealNumberSystem then
atan2loc(y: R, x: R): R ==
pi1 := pi()\$R
pi2 := pi1 * half
x = 0 => if y >= 0 then pi2 else -pi2

-- Atan in (-pi/2,pi/2]
theta := atan(y * recip(x)::R)
while theta <= -pi2 repeat theta := theta + pi1
while theta >   pi2 repeat theta := theta - pi1

x >= 0 => theta      -- I or IV

if y >= 0 then
theta + pi1      -- II
else
theta - pi1      -- III

argument x == atan2loc(imag x, real x)

else
-- Not ordered so dictate two quadrants
argument x ==
zero? real x => pi()\$R * half
atan(imag(x) * recip(real x)::R)

so the problem occurs if R does not have "RealNumberSystem":

(15) -> argument(-%i*1.0)

(15)  - 1.5707963267 948966192

since Float does have RNS, but INT does not. I'm not quite sure what the
condition should be. OrderedSet?

--