It follows that over , i.e. is harmonic .

Now suppose that is analytic in a neighborhood of . Moreover suppose that the partial derivatives of and are differentiable and that the second partial derivatives are continuous functions on . From Cauchy-Riemann equations follows:

As the second partial derivatives are continuous on , we have and . It follows that:

and |

The computations that we performed before Definition 4.1 can be summarized in a theorem:

On the one hand, e have:

On the other hand, we have:

The functions and are both harmonic.

- (i)
- The function
is harmonic:

- (ii)
- If there exists
such that
is analytic over
, then
and
verify the Cauchy-Riemann equations (v.s. section 3):

where and are independent of and respectively. As these two formulas define the same function, and must be constant, i.e. , where . - (iii)
- Finally, we have:

We can now discover another important property of the analytic conjugates.

For general , then equation defines an equilateral hyperbola, whose symmetry axes are the coordinate axes (red curves in Figure 2), and the equation defines an equilateral hyperbola whose asymptotes are the coordinates axes (blue curves in Figure 2). For , we have the union of the angle bisectors of the coordinate axes and the union of the coordinate axes respectively.

Noah Dana-Picard 2007-12-24