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

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

You changed the problem. It works fine if you set *f: x+%i*y*.
What output would you like to see?

On Tue, Apr 9, 2024 at 1:26 PM <fi...@ig...> wrote:

> Dear Stavros,
>
> On 2024-04-09 11:50, Stavros Macrakis wrote:
> > 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*:
>
> Thank you for this advice.
>
> > 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 )
>
> unfortunately cabs(f) only works if f can be disassembled into real
> and imaginary part. If we only know, that f is complex, we get:
>
>  declare(f, complex, t, real); /* to be as clear as possible */
>                                 done
>  depends(f, t);
>                                [f(t)]
>  diff(cabs(f), t);
>                                   df
>                                 f --
>                                   dt
>                                ------
>                                abs(f)
>
>
> This in general is not correct, or am I wrong? The result is the same
> using abs(f).
>
>
> Best regards
>
>
> Torsten
>
>
>
> > > 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...
> > >
> ------------------------------------------------------------------------
> > >
> > >
> > > _______________________________________________
> > > Maxima-discuss mailing list
> > > Max...@li...
> > > https://lists.sourceforge.net/lists/listinfo/maxima-discuss
> > >
>
> --
> ------------------------------------------------------------------------
> Torsten Finke
> fi...@ig...
> ------------------------------------------------------------------------
>