I'm reading about complex differentiation. The authors first identify $\mathbb C$ with $\mathbb R^2$, and $f \in \mathbb C^X$ with $F = (u,v) \in (\mathbb R^2)^X$.
Then they present a theorem and its proof:
- The identification of $$\frac{\left|F\left(x_{0}+\xi, y_{0}+\eta\right)-F\left(x_{0}, y_{0}\right)-A(\xi, \eta)\right|}{|(\xi, \eta)|}$$ with $$\left |\frac{f\left(z_{0}+h\right)-f\left(z_{0}\right)-f^{\prime}\left(z_{0}\right) h}{h} \right |$$ makes sense to me.
However, I could not understand how their values when taking the limits are equal. We say that we identify $F\left(x_{0}+\xi, y_{0}+\eta\right)$ with $f(z_0+h)$, but we do not say that their values are equal. The value of the first expression is in $\mathbb R^2$, whereas the value of the second expression is in $\mathbb C$.
- We have $z_0 = x_0 + i y_0$ and
$$[\partial f (z_0)]=\left[\begin{array}{cc}{\alpha} & {-\beta} \\ {\beta} & {\alpha}\end{array}\right]$$ and $$\left[\partial F\left(x_{0}, y_{0}\right)\right]=\left[\begin{array}{cc}{\partial_{1} u\left(x_{0}, y_{0}\right)} & {\partial_{2} u\left(x_{0}, y_{0}\right)} \\ {\partial_{1} v\left(x_{0}, y_{0}\right)} & {\partial_{2} v\left(x_{0}, y_{0}\right)}\end{array}\right]$$
I could not understand how we go from the identification of $\mathbb C$ and $\mathbb R^2$ to $[\partial f (z_0)]=\left[\partial F\left(x_{0}, y_{0}\right)\right]$.
Could you please elaborate on these points? Thank you for your help!
Best Answer
The value of $F(x)$ and $f(x)$ might not be equal since as you point out they are only identified in some way, but once we take norms, $|F(x)|=|f(x)|$ is an equality of real numbers. This is because the identification between $\mathbb{C}$ and $\mathbb{R}^2$ that we have specified preserves the norm.
Edit: The fact that $A$ satisfies the above limit means that $[\partial F(x_0,y_0)]=A$. This is simply the definition of the total derivative. If we look at what $A$ we see that it precisely the matrix of partials of $u$ and $v$ that you wrote. The way we get this limit is from the equality of norms I mentioned above.