Rewrite a function of $x+iy$ in terms of $z$ only How can you rewrite a function like
$$f(x+iy)=e^{2xy}\cos(x^2-y^2)-i\left[e^{2xy}\sin(x^2-y^2)+C\right]$$
in terms of just $z$? Are there any tricks that you can use or is it just the fact that you need to recognize the functions when you see them?
 A: Suppose we have an expression for $f$ in terms of $z$. If we restrict $z$ to be real, then we are effectively replacing the complex variable $z$ by the expression $x + 0i$ where $x$ is a real variable. What we end up with  is an expression for $f$ along the real line in terms of $x$. What is miraculous is that we can reverse this process. That is, if we know an expression for $f$ when restricted to the real line in terms of the real variable $x$, we can recover an expression for $f$ on the complex plane in terms of the complex variable $z$. This follows from the Identity Theorem.
For example, given $f(x + iy) = e^{2xy}\cos(x^2-y^2) - i[e^{2xy}\sin(x^2-y^2) + C]$, when we restrict to the real axis (i.e. set $y = 0$), we obtain $f(x) = \cos(x^2)-i\sin(x^2) - iC = e^{-ix^2} -iC$. How do we get an expression for $f(z)$? Replace $x$ by $z$. That is, once we know that $f$ restricted to the real line is given by $f(x) = e^{-ix^2} - iC$ we know that $f(z) = e^{-iz^2} - iC$ for all $z \in \mathbb{C}$.

As Rob Arthan points out below, in order to use the method above, you need to know in advance that $f$ is holomorphic (that is, the expression you are trying to obtain is in terms of $z$ only rather than $z$ and $\bar{z}$); you can check this by verifying the Cauchy-Riemann equations. The reason you need to know that $f$ is holomorphic in advance is the use of the Identity Theorem.
