Derivation of $g_i$ using real and imaginary parts

$g_i$s are complex functions. One can therefore write $S$ in terms of $g_i^I$ and ${g_i^{\it p}}$, the real and imaginary parts of $g_i$ and minimize with respect to $g_i^I$ and ${g_i^{\it p}}$ separately. It is shown here that the complex arithmetic achieves exactly this and the results are same as that given by complex calculus. The superscripts $I$ and $R$ in the following are used to represent the real and imaginary parts of complex quantities.

Expanding Equation D.5, ignoring $w_{ij}$s and writing it in terms of real and imaginary parts we get

$$\begin{split}\sum\limits_{{i,j} \atop {i \ne j}}\left\vert X_... ...=\sum\limits_{{i,j} \atop {i \ne j}}& S_0 S_0^\star \end{split}$$ (15.14)

where

$$S_0=\left[X_{ij}^R- {g_i^{\it p}}{g_j^{\it p}}- g_i^Ig_j^I\right]... ...iota \left[X_{ij}^I+ {g_i^{\it p}}g_j^I- g_i^I{g_j^{\it p}}\right] %%Imag part$$ (15.15)

Taking partial derivative of $S$ with respect to ${g_i^{\it p}}$ and reintroducing $w_{ij}$, we get

$$\begin{split}{\partial S \over \partial {g_i^{\it p}}}=&\sum\... ...I}^2 - {g_i^{\it p}} {{g_j^{\it p}}}^2\right]w_{ij} \end{split}$$ (15.16)

Therefore,

$${\partial S \over \partial {g_i^{\it p}}}= -2\sum\limits_{j \atop... ...t[Re(X_{ij}g_j^\star ) - \left\vert g_j\right\vert^2 {g_i^{\it p}}\right]w_{ij}$$ (15.17)

Equating $\partial S \over \partial {g_i^{\it p}}$ to zero, we get

$${g_i^{\it p}}= {\sum\limits_{j \atop {j \ne i}}Re(X_{ij}g_j^\star... ...j}) \over {\sum\limits_{j \atop {j \ne i}}\left\vert g_j \right\vert^2 w_{ij}}}$$ (15.18)

Similarly

$${\partial S \over \partial g_i^I}=-2\sum\limits_{j \atop {j \ne i}}\left[Im(X_{ij}g_j^\star) - \left\vert g_j\right\vert^2 g_i^I\right] w_{ij}$$ (15.19)

Therefore the equivalent imaginary part of Equation D.18 is

$$g_i^I= {\sum\limits_{j \atop {j \ne i}}Im(X_{ij}g_j^\star w_{ij}) \over {\sum\limits_{j \atop {j \ne i}}\left\vert g_j \right\vert^2 w_{ij}}}$$ (15.20)

writing $g_i={g_i^{\it p}}+ \iota g_i^I$ and substituting for ${g_i^{\it p}}$ and $g_i^I$ from Equation D.18 and D.20 respectively, we get

$$g_i = {\sum\limits_{j \atop {j \ne i}}X_{ij}g_j^\star w_{ij} \over {\sum\limits_{j \atop {j \ne i}}\left\vert g_j \right\vert^2 w_{ij}}}$$ (15.21)

This is same as Equation D.8, which was arrived at by evaluating a complex derivative of Equation D.5 as $\partial S/\partial g_i^\star$, treating $g_i$ and $g_I^\star$ as independent variables. Evaluating ${\partial S \over \partial g_i}=0$ would give the complex conjugate of Equation D.21. Hence, $\partial S/\partial g_i$ gives no independent information not present in $\partial S/\partial g_i^\star$.