TITLE: Difference of $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ and as $f: \mathbb{C} \rightarrow \mathbb{C}$.
QUESTION [6 upvotes]: I want to know the difference of differentation as $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ and $f: \mathbb{C} \rightarrow \mathbb{C}$.
What are their differences, $f$ as two real variables, or $f$ as differentiation as a complex function?
This question arose when I took the youtube lectures by "Richard E. BORCHERDS" on complex analysis.

First treatment of real analysis :
In multivariable calculus, when we set $f(x,y)$ its total derivatives is written as
\begin{align}
df =f_x dx + f_y dy 
\end{align}
where $f_x, f_y$ are partial derivatives with respect to $x,y$
Formally, we say that a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is differentiable at $a \in \mathbb{R}^2$ if it exists a continuous linear map $\nabla f(a) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that
\begin{align}
\lim_{h \rightarrow 0} \frac{f(a+h) - f(a) - \nabla f(a) \cdot h}{\|h\|} =0
\end{align}
so when we consider multivariable calculus, we have to see whether the multivariable function have a partial derivatives(or directional derivatives) and then see the above limit holds[In the calculus, we learn that a function having a partial derivatives but not differentiable, i.e., $f(x,y) = \frac{xy}{\sqrt{x^2+y^2}}$ at $(x,y) \neq (0,0)$ but $0$ at $(x,y)=(0,0)$. ]

In the complex analysis, we treat $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ or $f: \mathbb{C} \rightarrow \mathbb{C}$ and define complex derivatives analogus to real derivatives and obtain Cauchy Riemann equation.
For example $w=u+iv$,
\begin{align}
  \begin{pmatrix}
    u(x,y) \\
    v(x,y)
  \end{pmatrix} = \begin{pmatrix}
    u(x_0, y_0) \\
    v(x_0, y_0)
  \end{pmatrix} + \begin{pmatrix}
     \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\
     \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}
  \end{pmatrix} \begin{pmatrix}
    x-x_0 \\
    y-y_0
  \end{pmatrix} + \epsilon
\end{align}
and doing $w$ as
\begin{align}
w=w_0 + A (z-z_0) + \epsilon, \quad A \in \mathbb{C}
\end{align}
[This is Borcherds treatment of differentiation as a linear approximation. Like real case he treats $w$ as $\mathbb{C}$ and does the linear approximation on $\mathbb{C}$] then identifying the component of $A$ he obtain Cauchy Riemann equation.

In complex cases, I feel Borcherds treat the differentiation as $x,y$ and $z$ equally, but in general case those two approaches are different am I?
For example, when dealing with complex analysis, differentiable at some open region (analytic) implies $C^{\infty}$ but I know in multi-variable calculus this does may not happen.

What are their differences, $f$ as two real variables, or $f$ as differentiation as a complex function?

REPLY [2 votes]: The derivative of a function $\mathbb R^2 \to \mathbb R^2$ is a $2\times2$ matrix. The complex derivative of a function $\mathbb C \to \mathbb C$ is a complex number. Application of the $2\times2$ matrix derivative is analogous to multiplication by the complex derivative.
If $a + bi$ is a complex number, then multiplying the complex number $x + yi$ by $a + bi$ sends it to $(ax -yb) + (ay + bx)i$. That means that the complex number can itself be thought of as the following $2\times2$ matrix.
$$
\begin{pmatrix}
a & -b \\ b & a
\end{pmatrix}
$$
This leads us to the following definition of complex differentiability. A function $f \colon \mathbb C \to \mathbb C$ is complex differentiable at a point $z$ if it is differentiable at $z$ when considered as a function $\mathbb R^2 \to \mathbb R^2$ and if its its derivative at $z$ is a matrix taking the form
$$
\begin{pmatrix}
a & -b \\ b & a
\end{pmatrix}\,.
$$
Therefore: any complex differentiable function is differentiable in the real sense, but not every function that is differentiable in the real sense is complex differentiable, since not every $2\times 2$ matrix takes the form given above.
Now, since the coefficients of the derivative of a function $f = \langle u,v\rangle$ are precisely the partial derivatives
$$
\begin{pmatrix}
\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}
\end{pmatrix}
$$
then this statement is precisely the Cauchy-Riemann equations.