Theorem 3.3 (Cauchy-Riemann
Equations). Suppose that
is differentiable at the point . Then the partial derivatives of exist at the point ,
and can be used to calculate the derivative at . That is,
Equating the real and imaginary parts of Equations (3-14) and (3-15) gives the so-called Cauchy-Riemann Equations:
(3-16) and .
Proof of Theorem 3.3 is in the book.
Complex Analysis for Mathematics and Engineering