I have a huge equations system and in some of them I have to average < > some terms. Let's assume that I have terms like $$A(r, t) \mathrm{e}^{i \tau},$$ where $i$ is complex unit. Now I have to average it by $t$. How can I do it in general form since I do not know the exact form of $A(r, t)$ ?
What is the result of $$ < A(r, t) \mathrm{e}^{i \tau}> =?$$
Thank you in advance!
EDIT ONE
$\tau$ and $t$ have nothing in common in this case
EDIT TWO
By averaging by $t$ I mean the standard procedure. Let's say we have to average $sin (t)$ on some interval $[0, T]$. It takes the following form: $$\frac{1}{T-0} \int_0^T sin(t) dt = - \frac{cos(T) - cos(0)}{T} = \frac{1 - cos(T)}{T} $$
EDIT THREE
Following the formal logic, complex amplitude can be represented in the form of:
$$A(r, t) = \rho \mathrm{e}^{i t} = \rho [cos(t) + i sin (t)] $$ and then simply integrate over the interval $[0, T]$: $$<A(r, t)> = < \rho [cos(t) + i sin (t)] > = \frac{\rho}{T} [\int_0^T cos(t) dt + i \int_0^T sin(t) dt ]= \frac{\rho}{T} [sin(T) + 1 - cos(T)]$$
and that shoud be an answer, right ..?
What is confusing me (well I probably look like a funny clown now) is my talk with my scientific supervisor. He said that in our case averaging procedure has to be a simple complex conjugation.. Am I missing something here? I do not see a sign of conjugation here..