# Complex conjugation of a complex function

I have a function with a constant term upfront and was wondering how to obtain the complex conjugate. The function is say:

$$(g(z))^* = (\frac{-i\Gamma}{2\pi}\times f(z))^*$$

Would the conjugation be conducted 'distributively' such that:

$$\ \ \ \ \ \ \ \ \ \ \ \ (g(z))^* = (\frac{-i\Gamma}{2\pi})^*\times f(z)^*$$ $$\rightarrow(g(z))^* = \frac{i\Gamma}{2\pi}\times f(z)^*$$

also in general for a complex function if a complex function is contained within another how is the conjugate computed, like this?

$$g(z) = h(f(z))$$ $$\rightarrow g(z)^* = (h(f(z)))^*$$ $$\rightarrow g(z)^* = h^*(f^*(z))$$

Sorry for the dual question , but thank you all for your time!

## 1 Answer

Concerning your first question, you are right. This follow from the equality $$\overline{z\times w}=\overline z\times\overline w$$.

Now, if $$f$$ is a function from $$\mathbb C$$ into itself, let us define $$\overline f(z)$$ as $$\overline{f(z)}$$. Then $$\overline{h\circ f}=\overline h\circ f$$, which, in general, is not the same thing as $$\overline h\circ\overline f$$.