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?
Best Answer
$\mathbb{C}$ is literally $\mathbb{R}^2$ with additional vector multiplication. The complex $i$ is simply $(0,1)$. For $a,b\in\mathbb{R}$ we can easily then check that $a+bi$ is the same as $(a,b)\in\mathbb{C}$. And so a function $f:\mathbb{C}\to\mathbb{C}$ is literally the same as a function $f:\mathbb{R}^2\to\mathbb{R}^2$.
But complex and real differentiation is (somewhat) different. For starters their respective definitions are obviously different. But every complex differentiable function $f:\mathbb{C}\to\mathbb{C}$ is real differentiable. Moreover if $f(a+bi)=u(a+bi)+iv(a+bi)$, where $u,v:\mathbb{C}\to\mathbb{R}$ are real valued functions and $f$ is complex differentiable, then
$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$$ $$\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$
which are known as Cauchy-Riemann equations. It turns out that this is also a sufficient condition for $f$ to be complex differentiable, given that both $u,v$ are continuously real differentiable. In such situation both derivatives agree in the following sense: since $\nabla f(a):\mathbb{R}^2\to\mathbb{R}^2$ is a linear map, then it corresponds to a real $2\times 2$ matrix, which then corresponds to a complex number since in this situation our matrix has a specific form $\left[\begin{matrix}\alpha & \beta \\ -\beta & \alpha\end{matrix}\right]$. The $\alpha+\beta i$ complex number is our complex derivative at $a$ and vice versa.
And so you can think of complex differentiation as a very special case of real differentiation. In fact those two simple equations make the complex analysis much much more restrictive than its real counterpart.
For example as you said: for complex differentiation $C^1$ already implies $C^\infty$ (smooth) and even $C^\omega$ (analytic). Another difference is that every bounded complex differentiable function must be constant (Liouville's theorem). Even more: a complex differentiable function takes every possible complex value, except at most one, if non-constant (little Picard theorem), and so on.