# [Maxima] complex expressions

Henry Baker hbaker1 at pipeline.com
Wed Jun 6 11:52:35 CDT 2012

```cabs() and abs() on complex numbers seem hopelessly broken.

I defined my own cabs() function:
myabs(a):=sqrt(a*conjugate(a));

(%i7) myabs(a)^2=realpart(a)^2+imagpart(a)^2;
2              2
(%o7)            a conjugate(a) = realpart (a) + imagpart (a)
(%i8) %,nouns;
2              2
(%o8)            a conjugate(a) = realpart (a) + imagpart (a)
(%i9) %,a=ar+%i*ay;
2                       2
(%o9) (ar - %i ay) (%i ay + ar) = realpart (%i ay + ar) + imagpart (%i ay + ar)
(%i10) %,expand;
2     2           2                       2
(%o10)     ay  + ar  = realpart (%i ay + ar) + imagpart (%i ay + ar)
(%i11) %,rectform;
2     2           2                       2
(%o11)     ay  + ar  = realpart (%i ay + ar) + imagpart (%i ay + ar)
(%i12) %,nouns;
2     2     2     2
(%o12)                       ay  + ar  = ay  + ar

At 09:31 AM 6/6/2012, Henry Baker wrote:
>Yes, but if I do the following, it doesn't work:
>
>(%i2) cabs(a)^2=realpart(a)^2+imagpart(a)^2;
>                           2           2              2
>(%o2)                abs(a)  = realpart (a) + imagpart (a)
>(%i3) %,a=r+%i*s;
>                      2           2                     2
>(%o3)       (%i s + r)  = realpart (%i s + r) + imagpart (%i s + r)
>(%i4) %,expand;
>          2               2           2                     2
>(%o4)  - s  + 2 %i r s + r  = realpart (%i s + r) + imagpart (%i s + r)
>(%i5) rectform(%);
>          2               2           2                     2
>(%o5)  - s  + 2 %i r s + r  = realpart (%i s + r) + imagpart (%i s + r)
>
>At 08:59 AM 6/6/2012, Richard Fateman wrote:
>>Henry:
>>
>>try this:
>>
>>a:  r+%i*s;
>>
>>cabs(a)^2= realpart(a)^2 + imagpart(a)^2;
>>
>>you can say declare(r,real); declare(s,real)  if you wish, but
>>it seems to be unnecessary.
>>
>>RJF
>>
>>On 6/6/12 8:37 AM, Stavros Macrakis wrote:
>>>I believe that cabs(a) used to return sqrt('realpart(a)^2+'imagpart(a)^2), though it now returns 'abs(a) (prints as |a|).  I think the former is more in the spirit of cabs (which is supposed to give you an explicit formula, not just punt to a nounform), so I would consider the current behavior to be a bug.
>>>
>>>If I remember correctly, a few years ago, someone tried to unify abs and cabs (I objected) -- this may be related to that?
>>>
>>>As a workaround, you can do cabs(rectform(a))^2 => 'realpart(a)^2+'imagpart(a)^2.  Unfortunately, rectform(abs(a)) currently returns 'abs(a) -- again, I think this is a bug.  If you're starting with the abs form, I suppose you could do subst(lambda([ex],cabs(rectform(ex))),'abs, ... ).
>>>
>>>                 -s
>>>
>>>On Wed, Jun 6, 2012 at 11:06 AM, Henry Baker <hbaker1 at pipeline.com> wrote:
>>>I'm trying to get maxima to prove that
>>>
>>>abs(a)^2=realpart(a)^2+imagpart(a)^2.
>>>
>>>I tell maxima:
>>>
>>>declare(a,complex);
>>>abs(a)^2=realpart(a)^2+imagpart(a)^2;
>>>
>>>What do I tell maxima to get it to simplify this?

```