Simplify an expression with complex numbers

Question

How can I get a simplify expression of following when all a,b, .., f are real numbers?

FullSimplify[Abs[(a+I b)+(c+I d)(e+I f)]^2]

Use ComplexExpand. – Henrik Schumacher Jan 18 '18 at 22:28 — Henrik Schumacher, Jan 18 '18 at 22:28

kglr · Answer 1 · 2018-01-19T03:18:56.003

1

PowerExpand also works in this case:

FullSimplify[PowerExpand[Abs[(a + I b) + (c + I d) (e + I f)]^2], 
  Assumptions -> Element[{a, b, c, d, e, f}, Reals]]

$(b+d e+c f)^2+(a+c e-d f)^2$

edited Jan 19 '18 at 03:18

answered Jan 19 '18 at 02:55

kglr

score 0 · Answer 2 · answered Jan 19 '18 at 00:49

0

FullSimplify@Assuming[{a, b, c, d, e, f} \[Element] Reals,
  ComplexExpand[Abs[(a + I b) + (c + I d) (e + I f)]^2]]

$a^2+2 a (c e-d f)+b^2+2 b (c f+d e)+\left(c^2+d^2\right) \left(e^2+f^2\right)$

answered Jan 19 '18 at 00:49

David G. Stork

By default, ComplexExpand will assume that all variables are real. Consequently, the simpler FullSimplify@ComplexExpand[Abs[(a + I b) + (c + I d) (e + I f)]^2] is equivalent to your proposed solution. Further, you will get the same result with Simplify in the place of FullSimplify – Bob Hanlon Jan 19 '18 at 05:30

2 Answers2