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

[<fi...@ig...>]

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
> >

-- 
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Torsten Finke
fi...@ig...
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