# Failure in the use of FullSimplify

My question is about an apparent failure of the function "FullSimplify" to simplify an easy algebraic expression.

This is the expression that I ask Mathematica to evaluate:

FullSimplify[Re[a^(I*b)] - Re[a^(-I*b)], Element[a, Reals] && a > 0 && Element[b, Reals]]


This should give the result 0. Instead Mathematica only restates my expression:

Re[a^(-I b) (-1 + a^(2 I b))]


Replacing a and b by actual numbers solves the problem.

What could be the cause of it? How to effectively use FullSimplify (and Simplify, Expand, Integrate and so...) with assumptions?

I read that the order of variables could play a role here, but I couldn't wrap my head around it.

I tried to check for similar problems on the website as well, but I couldn't find any answer that could explain this phenomenon.

• Have you tried ComplexExpand[]? Jun 2, 2015 at 15:33
• It looks like you need to put ComplexExpand under Re[] : FullSimplify[Re[(a^(I*b) - a^(-I*b)) // ComplexExpand], Element[a, Reals] && a > 0 && Element[b, Reals]] Jun 2, 2015 at 15:42
As pointed out in the comments by "Guess who it is" and "BlacKow", the solution is to use the function ComplexExpand[], or to add ComplexExpand under Re[].