I'm reading about holomorphic functions in section 1.2 of Complex Analysis by Stein and Shakarchi, and I am pretty confused about the derivative notation that the authors employ. In this section the authors derive the Cauchy-Riemann equations for a complex-valued function $f: \Omega \to \mathbb{R}$, where $\Omega$ is an open subset of $\mathbb{C}$. The following passage from page 11 has me confused:
…consider the limit in (1) [$\lim_{h \to 0} \frac{f(z_0+h) – f(z_0)}{h}$] when $h$ is first real, say $h = h_1 + ih_2$ with $h_2 = 0$. Then, if we write $z = x + iy, \, z_0 = x_0 + iy_0$, and $f(z) = f(x,y)$, we find that
\begin{align*}
f'(z_0) &= \lim_{h_1 \to 0} \frac{f(x_0 + h_1, y_0) – f(x_0,y_0)}{h_1} \\[5pt]
&= \frac{\partial f}{\partial x}(z_0),
\end{align*}where $\partial/\partial x$ denotes the usual partial derivative in the $x$ variable. (We fix $y_0$ and think of $f$ as a complex-valued function of the single real variable $x$.) Now taking $h$ purely imaginary, say $h = ih_2$, a similar argument yields
\begin{align*}
f'(z_0) &= \lim_{h_2 \to 0} \frac{f(x_0, y_0 + h_2) – f(x_0,y_0)}{ih_2} \\[5pt]
&= \frac{1}{i} \frac{\partial f}{\partial y}(z_0),
\end{align*}where $\partial/\partial y$ is partial differentiation in the $y$ variable.
What exactly is the author's definition of $\frac{\partial f}{\partial x}(z_0)$ and $\frac{\partial f}{\partial y}(z_0)$? I find this very confusing as the author seems to be using $f$ to represent two different functions, one with domain $\Omega$ and the other with domain $\{(x,y): x + iy \in \Omega \}$…
Best Answer
I think you are overthinking the issue, but, in any case...
For a function $f$ on an open set $U$ in $\mathbb C$, using coordinates $x,y$ for $x+iy\in U$, for fixed $z_o\in U$, $$ {\partial\over \partial x}f(z_o) \;=\; \lim_{h\to 0} {f(z_o+h)-f(z_o)\over h} $$ where $h$ is real. Likewise, (now that I think about it, there may be an issue of conventions about a factor of $i$... sigh) $$ {\partial\over \partial y}f(z_o) \;=\; \lim_{h\to 0} {f(z_o+ih)-f(z_o)\over ih} $$ where $h$ is real. Yes, these are conventions, so it's fair to wonder how one could infer the implicit meanings.