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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 complex derivative of $x = a + jb \in \complex$, $a,b \in \reals$, is defined as   \begin{equation}  %\label{eqn:eqn1}  \begin{equation}  Dx = \frac{dx}{da} + j\frac{dx}{db}.  \end{equation} 
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Given $x \in \complex^n$ with $x_i = a_i + jb_i \in \complex$, $a_i,b_i \in \reals$,   What is $D\|x\|_{\ell_2}^2$? \\  We have  \begin{align*} %\begin{align*}  \begin{equation}  \|x\|_{\ell_2}^2 &= \sum_{i=1}^n |x_i|^2 = \sum_{i=1}^n x_i^*x_i \\   &= \sum_{i=1}^n (a_i +jb_i)^*(a_i +jb_i) \\  &= \sum_{i=1}^n (a_i -jb_i)(a_i +jb_i) \\   &= \sum_{i=1}^n (a_i^2 +b_i^2).  \end{align*} \end{equation}   %\end{align*}  %Applying Equation \ref{eqn:eqn1} with $f(x) = \|x\|_{\ell_2}^2$, we have