information:classification
Independence of the complex conjugate

The above proof suggests another interpretation of the Cauchy–Riemann equations. The complex conjugate of z {\displaystyle z}, denoted z ¯ {\textstyle {\bar {z}}}, is defined by x + i y ¯:= x − i y {\displaystyle {\overline {x+iy}}:=x-iy} for real variables x {\displaystyle x} and y {\displaystyle y}. Defining the two Wirtinger derivatives as ∂ ∂ z = 1 2 (∂ ∂ x − i ∂ ∂ y), ∂ ∂ z ¯ = 1 2 (∂ ∂ x + i ∂ ∂ y), {\displaystyle {\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),} the Cauchy–Riemann equations can then be written as a single equation ∂ f ∂ z ¯ = 0, {\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0,} and the complex derivative of f {\textstyle f} in that case is d f d z = ∂ f ∂ z. {\textstyle {\frac {df}{dz}}={\frac {\partial f}{\partial z}}.} In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function f {\textstyle f} of a complex variable z {\textstyle z} is independent of the variable z ¯ {\textstyle {\bar {z}}}. As such, we can view analytic functions as true functions of one complex variable (z {\textstyle z}) instead of complex functions of two real variables (x {\textstyle x} and y {\textstyle y}).