Complex Conjugate of a Function

[Niles]

Homework Statement

Hi

I have a complex function of the form
$$\frac{1}{1-Ae^{i(a+b)}}$$
I want to take the complex conjugate of this: The parameters a and b are complex functions themselves, but A is real. Am I allowed to simply say
$$\frac{1}{1-Ae^{-i(a^*+b^*)}}$$
where * denotes the c.c.? I seem to vaguely remember that $f^*(x) = f(x^*)$.

 

[christoff]

Be careful; the rule $f(z^*)=f(z)^*$ doesn't in general work when $z$ is a complex number.
Consider the following counterexample: take $f(z)=i|z|$, where $|\cdot |$ is the absolute value. Then $f(z^*)=i|z^*|=i|z|=f(z)$ but $f(z)^*=(i|z|)^*=-i|z|=-f(z)$, so we have $f(z^*)\neq f(z)^*$.

In general, your function $f$ can be very nice (the example above isn't complex-differentiable) but still fail to have this property.

For your problem, it's going to depend a lot on what your functions $a,b$ are.

 

[Niles]

Thanks for helping out. If a and b are just complex numbers, then it should be correct, no?

 

[christoff]

Thanks for helping out. If a and b are just complex numbers, then it should be correct, no?

Yes, if a and b are just complex numbers, then it works out like that.