7.128 Bug in simplify/conjugate in Maple 6 (17.7.00)

7.128.1 Bill Page

It seems there is something wrong with simplifying expressions containing conjugate and diff. I don’t find the abs(1,x) result very useful either. Is there a better way to deal with the derivative of the complex conjugate of a function?

> interface(version); 
 
  Maple Worksheet Interface, Maple 6, IBM INTEL NT, Jan 31 2000 
  Build ID 16401 
 
> diff(sqrt(x*conjugate(x)),x); 
 
                             ___     /d  ___\ 
                             (x) + x |-- (x)| 
                                     \dx    / 
                         1/2 ---------------- 
                                      ___ 
                               sqrt(x (x)) 
 
> simplify(%); 
Error, (in simplify/conjugate) wrong number (or type) of parameters in 
function diff 
 
> simplify(diff(conjugate(x),x)); 
 
                                d  ___ 
                                -- (x) 
                                dx 
 
> simplify(sqrt(x*conjugate(x))); 
 
                                | x | 
 
> diff(abs(x1),x1); 
 
                              abs(1, x1)
 

It is corrected with Maple 7. (U. Klein)

7.128.2 Robert Israel (20.7.00)

The best way is to avoid it! The derivative (in the complex sense) of the complex conjugate of a non-constant analytic function does not exist. So maybe it’s not surprising that Maple is confused (although there is a bug here too, and Maple should give you a more appropriate error message). In your example

f = sqrt(x*conjugate(x))
 

if you write x=s+I*t with \(s\) and \(t\) real, then you do have partial derivatives:

diff(f, s) = s/f and diff(f, t) = t/f
 

but these don’t fit together into a derivative with respect to the complex variable \(x\), which would require diff(f,x) = diff(f,s) = I*diff(f,t).