How to define a complex function with assumptions?

Question

I'm trying to define a complex function to be able to do analytic calculations. Specifically, I want to define a function $A(x,t)$ in the complex plane with argument $\theta(x,t)$ and modulus $R(x,t)>0$, both of which are real functions of real variables $x$ and $t\ge0$.

A[x_, t_] = R[x, t]*E^(I θ[x, t]);

Then I added some assumptions:

$Assumptions = Element[{R[x, t], θ[x, t], x, t}, Reals] && R[x, t] > 0 && t >= 0;

Test the definition:

Differentiation: D[A[x, t], t]

(E^(I θ[x,t]) (R^(0,1))[x,t]+I E^(I θ[x,t]) R[x,t] (θ^(0,1))[x,t]) checked!

Complex conjugation: Conjugate[A[X, T]]

(E^(-I Conjugate[θ[X,T]]) Conjugate[R[X,T]]) wrong!

What we expected from Conjugate is R[x, t]*E^(-I θ[x, t]) under the assumptions. What is wrong there? Please give me some suggestions or comments. Thank you.

Answer 1

Your assumptions aren't general enough, use R[__] etc.:

A[x_, t_] = R[x, t]*E^(I θ[x, t]);

$Assumptions = Element[{R[__], θ[__]}, Reals] && R[__] > 0;

Conjugate[A[X, T]] // FullSimplify

E^(-I θ[X, T]) R[X, T]