Editing Cauchy–Riemann equations (section)

=== Independence of the complex conjugate ===
The above proof suggests another interpretation of the Cauchy–Riemann equations. The [[complex conjugate]] of <math>z</math>, denoted <math display="inline">\bar{z}</math>, is defined by
<math display="block">\overline{x + iy} := x - iy</math>
for real variables ''<math>x</math>'' and <math>y</math>. Defining the two [[Wirtinger derivatives]] as<math display="block"> \frac{\partial}{\partial z}
  = \frac{1}{2} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right), \;\;\; \frac{\partial}{\partial\bar{z}}
  = \frac{1}{2} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right),
</math>
the Cauchy–Riemann equations can then be written as a single equation
<math display="block">\frac{\partial f}{\partial\bar{z}} = 0,</math>
and the complex derivative of ''<math display="inline">f</math>'' in that case is <math display="inline">\frac{df}{dz}=\frac{\partial f}{\partial z}.</math> In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function ''<math display="inline">f</math>'' of a complex variable ''<math display="inline">z</math>'' is independent of the variable <math display="inline">\bar{z}</math>. As such, we can view analytic functions as true functions of ''one'' complex variable (''<math display="inline">z</math>'') instead of complex functions of ''two'' real variables (''<math display="inline">x</math>'' and ''<math display="inline">y</math>'').