Real and imaginary parts of a complex number

Question

I would like to find the real and imaginary parts of $x + y$ in terms of polar coordinates.

Here is code:

ComplexExpand[Re[x + y], {x, y}, TargetFunctions -> {Abs, Arg}]

How can I add the assumption that $Abs[x]$ equals to $Abs[y]$?

For example this code does not work:

ComplexExpand[Re[x + y], {x, y}, TargetFunctions -> {Abs, Arg}, 
Assumptions -> Abs[x] == Abs[y]]

By the above assumption, I expect to obtain: Re[x + y] = Abs[x] (Cos[Arg[x]] + Cos[Arg[y]]) — To Be, Jun 21 '14 at 13:23
Or, how can I change the code to obtain Re[x + y] = Abs[x] (Cos[Arg[x]] + Cos[Arg[y]]), by the assumption that Abs[x] = Abs[y]? — To Be, Jun 21 '14 at 13:24

m_goldberg · Accepted Answer · 2014-06-21 13:42:05Z

1

Would

Simplify[
  ComplexExpand[Re[x + y], {x, y}, TargetFunctions -> {Abs, Arg}], 
  Assumptions -> Abs[x] == Abs[y]]

Abs[y] (Cos[Arg[x]] + Cos[Arg[y]])

work for you?

answered Jun 21 '14 at 13:42

m_goldberg

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$\begingroup$ Very good. Thank you. $\endgroup$ – To Be Jun 21 '14 at 13:47
$\begingroup$ @To Be - The algorithms give Abs[y] rather than the equivalent Abs[x]; however, if the difference is important to you, you could try playing with the ComplexityFunction for FullSimplify or just use a rule /. Abs[y] -> Abs[x] $\endgroup$ – Bob Hanlon Jun 21 '14 at 14:09

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