# [Maxima] More work on the functions for the complex components

Dieter Kaiser drdieterkaiser at web.de
Tue May 5 13:30:09 CDT 2009

```I would like to suggest two further improvements of the functions for the
complex components:

1. Simplification of realpart/imagpart with conjugate as expression

As mentioned in the bug report ID: 1045531 "real/imagpart don't know about
conjugate" the following extension to risplit will give more correct
simplifications for expressions with the function conjugate. The results will
be:

(%i3) declare(z,complex)\$

(%i4) realpart(conjugate(z));
(%o4) realpart(z)
(%i5) imagpart(conjugate(z));
(%o5) -imagpart(z)

(%i6) realpart(conjugate(sin(z)));
(%o6) cosh(imagpart(z))*sin(realpart(z))
(%i7) imagpart(conjugate(sin(z)));
(%o7) -sinh(imagpart(z))*cos(realpart(z))

This is the code:

+          ((eq (caar l) '\$conjugate)
+           (cons (simplify (list '(%realpart) (cadr l)))
+                 (mul -1 (simplify (list '(%imagpart) (cadr l))))))

2. Change standard behavior of rectform

Maxima assumes that all unknown functions are real valued. See the bug report
ID: 1238141 "realpart(f(x+%i*y))".

We can change this in the function risplit in rpart.lisp. Now we get:

(%i22) rectform(f(x));
(%o22) realpart(f(x))+%i*imagpart(f(x))

(%i23) realpart(f(x));
(%o23) realpart(f(x))
(%i24) imagpart(f(x));
(%o24) imagpart(f(x))

This extension does not change the result for known functions:

(%i25) rectform(sin(x));
(%o25) sin(x)

(%i26) rectform(sin(z));
(%o26) cosh(?%imagpart(z))*sin(?%realpart(z))
+%i*sinh(?%imagpart(z))*cos(?%realpart(z))

This is the changed code:

@@ -356,7 +360,10 @@
(risplit-noun l))
((and (eq (caar l) '%product) (not (free (cadr l) '\$%i)))
(risplit-noun l))
-     (t (cons l 0)))))
+          (t
+;          (cons l 0))))) <-- This is the old standard behavior
+           (cons (list '(%realpart) l)
+                 (list '(%imagpart) l))))))

The testsuite has no problems with both extensions of the code.
Is there any problem I do not see? Should I commit the changes.

Hint:

We can add further code to look for real valued functions and add a property
like maps-reals-to-reals. With such extensions we can improve the handling of
functions further.

Dieter Kaiser

```