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Re: Complex variables
*To*: mathgroup at christensen.cybernetics.net
*Subject*: [mg1497] Re: [mg1478] Complex variables
*From*: Allan Hayes <hay at haystack.demon.co.uk>
*Date*: Mon, 19 Jun 1995 02:06:13 -0400
Stephen Resenbloom in [mg1478] Complex variables
Wrote
> In Mathematica is there a way to specify that given variables are
purely real?
i.e. if f= x + I y and one can specify that x and y are purely real
variables, then Mathematica should be able to calculate
Re[f] = x and Im[f]=y. So far I have found no way to do this.
>
Stephen,
I'll come to your specific query below, but you may find the system
function ComplexExpand useful: it assume that variable are real
unless told other wise.
f = x + I y;
ComplexExpand[{Re[f], Im[f], Conjugate[f], Abs[f]}]
2 2
{x, y, x - I y, Sqrt[x + y ]}
ComplexExpand[{Re[g], Im[g], Conjugate[g], Abs[g]},{g}]
{Re[g], Im[g], -I Im[g] + Re[g], Abs[g]}
ComplexExpand has an option TargetFunctions which lets you specify
the kind of reduction you would like:
ComplexExpand[ Conjugate[g], {g}, TargetFunctions -> {Abs}]//Simplify
2
Abs[g]
-------
g
The package
<< Algebra`ReIm`
Lets you specify which variable are real, as you want (note that
we use up definitions to localize the effect and avoid the need to
unprotect Im and Re).
Im[x]^= Im[y]^= 0;
{Re[f], Im[f], Conjugate[f], Abs[f]}
{x, y, x - I y, Abs[x + I y]}
The result for Abs is rather surprising.
Here is a little more
ComplexExpand[a^b, {a}]//Factor (*{a} says that a is complex*)
b
Abs[a] (Cos[b Arg[a]] + I Sin[b Arg[a]])
We can use
<<Algebra`Trigonometry`
to put this in a more compact form
TrigToComplex[%]//Simplify
I b Arg[a] b
E Abs[a]
Allan Hayes
hay at haystack.demon.co.uk
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