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\newcommand{\reals}{\mathbb{R}}  \newcommand{\complex}{\mathbb{C}}  \newcommand{\dom}{\mathop{\bf dom}} % domain  \newcommand{\intr}{\mathop{\bf int}}  \subsubsection*{Complex derivative}  Here we provide a definition for the 'complex' derivative of a real-valued function $f : \complex^n \to \reals$ with respect to its complex variables.   The notation $f : \complex^n \to \reals$ means "$f$ is a mapping (or function) from the set of column vectors of size $n$ with complex components (denoted $\complex^n$) into the set of real numbers (denoted $\reals$)."  \\  The complex derivative of $x = a + jb \in \complex$, $a,b \in \reals$, is defined as   \begin{equation}  Dx = \frac{dx}{da} + j\frac{dx}{db}. 
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&=& 2a + j2b \nonumber\\  &=& 2x. \nonumber  \end{eqnarray}  Suppose $f: \complex^n \to \reals$ is a real-valued function defined for and  $x \in \complex^n$. \intr \dom f$.  The derivative $Df(x)$ is a $1 \times n$ matrix and (a \textit{row} vector),  defined as \\ by  \begin{equation}  \label{eqn:derivative}  Df(x) = \left[ \frac{\partial f}{\partial x_1}(x), \dots, \frac{\partial f}{\partial x_n}(x) \right].