A question about expanding complex functions of real arguments

2

I have the following problem. There is such an expression as:

P[x_,y_] := z[y] E^(I beta x) + Conjugate[z[y] E^(I beta x)];

The variables x, y, beta are real but the function z[y] is complex. I need to expand P[x, y]^2. So I try this:

ComplexExpand[P[x, y]^2, z[y]]

The output contains Re[z[y]], Im[z[y]], Cos[...] and Sin[...]. In my case, I need the output to be like this:

z[y]^2 E(2 I beta x) + 2 z[y] Conjugate[z[y]] + Conjugate[z[y]]^2 E^(-2 I beta x)

What should I change in my approach? I have already tried TrigToExpand, FullSimplify and Hold but I did not get the desirable result.

Vladimir

Posted 2014-10-06T11:43:44.183

Reputation: 25

You can get rid of the Re and Im by using the option TargetFunctions->{Conjugate}. But that still gives Sin and Cos for the exponential functions. – celtschk – 2014-10-06T11:54:43.067

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– Dr. belisarius – 2014-10-06T12:10:37.360

Answers

3

The following gives what you intended:

Refine[Expand[P[x, y]^2], (x|y|beta) \[Element] Reals]
(*
==> Conjugate[z[y]]^2/E^((2*I)*beta*x) + 2*Conjugate[z[y]]*z[y] + 
     E^((2*I)*beta*x)*z[y]^2
*)

In cases where you can live with expansion of complex exponentials into Sin and Cos you can also use

ComplexExpand[P[x, y]^2, z[y], TargetFunctions->{Conjugate}]
(*
==> Conjugate[z[y]]^2*Cos[beta*x]^2 - (2*I)*Conjugate[z[y]]^2*Cos[beta*x]*
      Sin[beta*x] - Conjugate[z[y]]^2*Sin[beta*x]^2 + 
     2*Conjugate[z[y]]*Cos[beta*x]^2*z[y] + 2*Conjugate[z[y]]*Sin[beta*x]^2*
      z[y] + Cos[beta*x]^2*z[y]^2 + (2*I)*Cos[beta*x]*Sin[beta*x]*z[y]^2 - 
     Sin[beta*x]^2*z[y]^2
*)

celtschk

Posted 2014-10-06T11:43:44.183

Reputation: 18 543

Nice one. I think the question title is misleading b/c the problem isn't about ComplexExpand[] but about expressing a Complex expression in terms of certain things. Can you find a better title? I'm not sure about the better option. – Dr. belisarius – 2014-10-06T12:03:41.407

@celtschk Thank you a lot! – Vladimir – 2014-10-06T12:06:42.010

@belisarius I agree with you that the title does not express the very problem. I will rename it. – Vladimir – 2014-10-06T12:07:39.013