Properties of functions satisfying the functional equation $f(x+iy)=f(x-iy)$

Question

Consider the analytic function $f(z)$ satisfying :

$$f(z)=f(\bar{z})$$

i.e. $$f(x+iy)=f(x-iy)$$

We can consider the case of real valued function, with real and complex domain, satisfy the above condition .

Question :

(1)What are some non trivial properties of such class of functions ?

(2)What are some noteworthy examples of such functions?

Answer 1

Define:$f:\mathbb{C}\to \mathbb{C}$ by $$f(z)=z.\bar{z}+iz.\bar{z}$$ and you'll get what you want

Answer 2

$f $ is analytic iff $\frac{df(z)}{d\bar{z}}=0$. So your property of interest implies $0=\frac{df(\bar{z})}{d\bar{z}}=f'(\bar{z})$, which means $f'$ is zero, ergo $f $ is constant.