If an additional derivative is taken in the Cauchy - Riemann equations, we have

From the equality of mixed second partial derivatives we get

= | 0 | (68) | |

= | 0 | (69) |

which means that

= | (70) |

so that

u(x,y) |
= | (71) | |

v(x,y) |
= | (72) |

or, in other words, each central value is the average of its neighbors. No average can exceed all the values it is averaging, nor fall below all of them, either. This observation is a precursor of the maximum modulus principle, which holds that the critical points of an analytic function are saddle points, and that extreme values of such a function can only occur on the boundary of a region over which it is examined.