Factoring for exponents

Question

How can I factorise into powers of integer exponents?

An expression such as,

factored = 
  2^(-j - k) E^(-y/t) 
    (I (E^(-I y w) - E^(I y w)))^k (E^(-I y w) + E^(I y w))^j

can be expressed as,

unfactored = 
  I^k 2^(-j - k) E^(-y/t - I j y w - I k y w) 
    (1 - E^(2 I y w))^k (1 + E^(2 I y w))^j

My aim is to reduce unfactored to $e^{-y/t}\cos^j (y\,w)\sin^k (y\,w)$, but Mathematica doesn't seem to be able to do this unless the expression is factorised in terms of j, k as in factored

Attempts

• Collect[unfactored, {E^(y w), E^(- y w)}] and variants
• ExpToTrig[unfactored] // FullSimplify and variants

Working Example and a sanity check!

ExpToTrig[
  (I/2 E^(-I y w) (1 - E^(2 I y w)))^k 
    (1/2 E^(-I y w) (1 + E^(2 I y w)))^j] // FullSimplify

gives, $ \cos^j(y\,w) \sin ^k(y\,w) $

Answer 1

I have encountered problems of this sort when deriving plasma dispersion relations with Mathematica. The essence of the problem appears to be that FullSimplify does not have

tf[e_] := e //. 
    {Exp[z1_ - I z2_ z3_] (1 + Exp[2 I z3_])^z2_ -> Exp[z1] (Exp[-I z3] + Exp[I z3])^z2, 
     Exp[z1_ - I z2_ z3_] (1 - Exp[2 I z3_])^z2_ -> Exp[z1] (Exp[-I z3] - Exp[I z3])^z2}

among its TransformationFunctions. Adding it solves the problem.

FullSimplify[unfactored, (j | k) ∈ Integers, TransformationFunctions -> {Automatic, tf}]

(* E^(-(y/t)) Cos[w y]^j Sin[w y]^k *)

Note that, if j and k were specific integers, FullSimplify together with Expand would be sufficient. For instance,

FullSimplify[Expand[unfactored /. {j -> 3, k -> 4}]]

(*( E^(-(y/t)) Cos[w y]^3 Sin[w y]^4 *)

Answer 2

Try this:

unfactored /. 
  E^(a_ - I *p_* w y)*(1 + E^(2*I * w y))^p_ :> E^a*2*Cos[p w y]^p /. 
 E^(a_ - I *p_* w y)*(1 - E^(2*I * w y))^p_ :> -E^a*2*I*Sin[p w y]^p

(*  -I I^k 2^(2 - j - k) E^(-(y/t)) Cos[j w y]^j Sin[k w y]^k  *)

Have fun!