I saw Ahlfors's book Complex Analysis. It mentioned that analytic function $f(z)$ can be derived from a given real part $u(x,y)$, where $x$ and $y$ are real.
It said that $ u(x,y)=\frac{1}{2}[f(x+iy)+\bar{f}(x-iy)]. \tag{1} $
However, it mentioned that it is 'reasonable' that (1) holds even when $x$ and $y$ are 'complex'. Why?
I think that, if $x$ and $y$ are real, then real part $u(x,y)$ should be written down by $ u(x,y)=\frac{1}{2}[f(z)+\bar{f}(\bar{z})], \tag{2} $ where $z=x+iy$.
Hence, if $x$ and $y$ are complex, (2) should be equal to $ u(x,y)=\frac{1}{2}[f(x+iy)+\bar{f}(\bar{x}-i\bar{y})]. \tag{3} $ It confused me for a long time. Please help me.
Thanks!