We are now going to investigate the properties of a complex function the real and imaginary part of which are conjugate functions. In particular we define the complex potential
In the complex (Argand-Gauss) plane every point is associated with a complex number
In general we can then write
The fact that Cauchy-Riemann conditions hold for both and , or equivalently that these functions are conjugate, is a necessary and sufficient condition for the function f to be analytic.
Now, if the function f is analytic, this implies that it is also differentiable, meaning that the limit
is finite and independent of the direction of .
If then we pose it follows that
and the same result would be obtained posing, for example,
so that the derivative of the complex potential W in the complex z plane gives the complex conjugate of the velocity.
Thus, knowledge of the complex potential as a complex function of z leads to the velocity field through a simple derivative.