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.

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