# How to define a complex function with assumptions?

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. • The $Assumptions aren't applied until you simplify your expression, for example with FullSimplify. – Roman Apr 6 at 15:58

## 1 Answer

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]