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,

In conclusion

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.