Re: [Maxima-discuss] derivative of abs(complex)

[Stavros M. <mac...@gm...>]

Maxima is not very smart about the *conjugate* function. If you want to
find a real expression for the absolute value of a complex expression, try
*cabs*:

depends([x,y],t) => [x(t), y(t)]

cabs(x+%i*y) =>
             2    2
       sqrt(y  + x )

diff(%,t) =>

          dy       dx
      2 y -- + 2 x --
          dt       dt
      ---------------
              2    2
      2 sqrt(y  + x )

On Tue, Apr 9, 2024 at 11:39 AM <fi...@ig...> wrote:

> Dear Maxima Users,
>
> I am a little confused about Maxima's understanding of the abs() function.
>
> Consider a complex function f = x + i y where x and y depend on some
> real parameter t. What should the first derivative df/dt be? Shouldn't
> it be purely real?
>
> (%i1) depends([x, y], t);
> (%o1)                            [x(t), y(t)]
> (%i2) f : x + %i * y;
> (%o2)                              %i y + x
> (%i3) diff(abs(f), t);
>                                            dy   dx
>                             (%i y + x) (%i -- + --)
>                                            dt   dt
> (%o3)                       -----------------------
>                                  abs(%i y + x)
> (%i4) expand(%);
>                 dy                 dy            dx              dx
>               y --            %i x --         %i -- y          x --
>                 dt                 dt            dt              dt
> (%o4)  (- -------------) + ------------- + ------------- + -------------
>           abs(%i y + x)    abs(%i y + x)   abs(%i y + x)   abs(%i y + x)
>
>
> Oops! Maxima tells me, that the first derivative is not real but has
> an imaginary part!
>
> What would be my mistake? Did I miss something in the documentation
> about complex derivatives? I would like to apologize in advance.
>
> For my understanding the following equation for some complex number z
> holds:
>
>    abs(z) = sqrt(z conj(z))
>
> right?
>
> So we try
>
> (%i5) diff(sqrt(f * conjugate(f)), t);
>                               dy   dx                dx      dy
>                (x - %i y) (%i -- + --) + (%i y + x) (-- - %i --)
>                               dt   dt                dt      dt
> (%o5)          -------------------------------------------------
>                          2 sqrt((x - %i y) (%i y + x))
> (%i6) expand(%);
>                                dy              dx
>                              y --            x --
>                                dt              dt
> (%o6)                    ------------- + -------------
>                                2    2          2    2
>                          sqrt(y  + x )   sqrt(y  + x )
>
>
> The expected result; and purely real.
>
> BTW.: For Maxima the following is not evaluated to zero:
>
> (%i7) abs(f) - sqrt(f * conjugate(f));
> (%o7)             abs(%i y + x) - sqrt((x - %i y) (%i y + x))
>
> Why this?
>
>
> Any advice appretiated and best regards
>
>
> Torsten
>
>
>
>
> --
> ------------------------------------------------------------------------
> Torsten Finke
> fi...@ig...
> ------------------------------------------------------------------------
>
>
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