Let $f(z)=u(x,y)+ i v(x,y)$ be a holomorphic function. I often find it difficult to deduce the form of $f$ as a function of $z=x+i y$ only (for example: $f=\sin x \cosh y + i \cos x \sinh y$ cas be written as $f(z)=\sin z$).
However, my book states that given $u$ and $v$ or even $u$ or $v$ only we can deduce the $z$-dependance of $f$ using the following relations:
$f(z)= 2 u (z/2, -i z/2)$
$f(z)= 2 i v(z/2, -iz/2)$
Can anyone help me understand how they come about?