Relationship between real and Complex differentiability The most commonly used definition of differentiability of a function ${f: \mathbb{R}\rightarrow \mathbb{R}}$ is as follows. The function ${f}$ is (real) differentiable at ${x_0 \in \mathbb{R}}$ if the limit

$\displaystyle \begin{array}{rcl} \lim_{h\rightarrow 0}\frac{f(x_0+ h)- f(x_0)}{h} \end{array}$

exists. Equivalently, We say that ${f:\mathbb{R}\rightarrow \mathbb{R}}$ is differentiable at ${x_0}$ if there exists a linear transformation ${A: \mathbb{R}\rightarrow \mathbb{R}}$ satisfying

$\displaystyle \begin{array}{rcl} \lim_{h\rightarrow 0}\frac{f(x_0+ h)- f(x_0)- Ah}{h}= 0. \end{array}$

Such a linear transformation, if it exists, is unique and is called the derivative of ${f}$ at ${x_0}$. We then write ${A= f^{\prime}(x_0)}$. More generally, a function ${f: U \subset \mathbb{R}^n\rightarrow \mathbb{R}^m}$ is differentiable at ${\mathbf{x_0} \in \mathbb{R}^n}$ if there exists a linear transformation ${A: \mathbb{R}^n\rightarrow \mathbb{R}^m}$ such that

$\displaystyle \begin{array}{rcl} \lim_{h\rightarrow 0}\frac{||f(x_0+ h)- f(x_0)- Ah||}{|h|}= 0, \end{array}$

where the norms ${||\cdot||}$ and ${|\cdot |}$ are the standard norms on ${\mathbb{R}^n}$ and ${\mathbb{R}^m}$ respectively. The linear transformation, if it exists, is unique and is called the Jacobian of ${f}$ at ${x_0}$ and is denoted by ${Df(x_0)}$. If we write ${f= (f_1, f_2, \cdots, f_m)}$ and ${x= (x-1, x_2, \cdots, x_n0 \in \mathbb{R}^n}$, then we can write the Jacobian in matrix form as

$\displaystyle \begin{array}{rcl} \left[Df(x_0)\right]_{ij}= \left[\frac{\partial f_i}{\partial x_j}(x_0)\right] \end{array}$

Before looking at complex differentiability we recall Proposition: Let ${A: \mathbb{C}\rightarrow \mathbb{C}}$ be${\mathbb{R}}$-linear. The following statements are equivalent:

1. There exists ${\alpha \in \mathbb{C}}$ suchthat ${Az= \alpha z}$;
2. ${A}$ is ${\mathbb{C}}$-linear;
3. ${A(i)= iA(1)}$;
4. The matrix with respect to the canonical basis ${\{1+ 0i, 0+ i\}}$ has the form$\displaystyle \begin{array}{rcl} \left(\begin{array}{cc}a & -b\\b & a\end{array}\right), \quad a, b \in\mathbb{R}. \end{array}$

Now, a function ${V \subset \mathbb{C} \rightarrow \mathbb{C}}$ is complex-differentiable at ${z_0 \in \mathbb{C}}$ if the limit

$\displaystyle \begin{array}{rcl} \lim_{h\rightarrow 0}\frac{f(z_0+ h)- f(z_0)}{h}=: f^{\prime}(z_0) \end{array}$

exists. This can also be reformulated as follows. We say that ${f}$ is complex-differentiable at ${z_0}$ if there exists ${\alpha \in \mathbb{C}}$ such that

$\displaystyle \begin{array}{rcl} \lim_{h\rightarrow 0}\frac{|f(z_0+ h)- f(z_0)|}{|h|}= 0, \end{array}$

and we have ${\alpha= f^{\prime}(z_0)}$.

From the above discussion, it is clear that if ${f: V \subset \mathbb{C}\rightarrow \mathbb{C}}$ is complex-differentiable at ${z_0= x_0+ iy_0}$, then it is real-differentiable at ${(x_0, y_0)}$ and the Jacobian ${Df(x_0, y_0)}$ is ${\mathbb{C}}$-linear.