Does the Im function work with symbolic arguments?

Question

Does the Im function work with symbolic arguments?

Clear[ y, t, k, ω ]

A ( Cos[ k y ] + I Sin[ k y ] ) 2I Sin[ ω t ] //ComplexExpand

(* Output: 2 I A Cos[ k y] Sin[ t ω ] - 2 A Sin[ k y ] Sin[ t ω ] *)

Im[ % ]

(* Output: -2 Im[ A Sin[ k y ] Sin[ t ω ] ] + 2 Re[ A Cos[ k y ] Sin[ t ω ] ] *)

Expected output: -2 A Sin[ k y ] Sin[ t ω ]

Answer 1

You should assume that your variables are real, (if you want M to proceed further) because Mathematica treats variables in general as complex. One of many ways to do it :

 expr = A ((Cos[k y] + I Sin[k y]) 2 I Sin[t ω]); 
 Refine[ Im[ expr], (A | k y | t ω) ∈ Reals]

2 A Cos[k y] Sin[t ω]

We needn't use ComplexExpand defining expr, but in this case it is the simplest approach (pointed out by Heike) :

ComplexExpand @ Im @ expr

Some other ways of imposing assumptions :

Assuming[(A | k y | t ω) ∈ Reals, Refine[ Im[ expr] ] ]

another way yielding the same result :

Block[{$Assumptions = A ∈ Reals && k y ∈ Reals && t ω ∈ Reals},
       Refine @ Im[ expr] ]

 2 A Cos[k y] Sin[t ω]