Real and Complex Analysis (Rudin)

CHAPTER
ELEVEN
HARMONIC FUNCTIONS
The Cauchy-Riemann Equations
11.1 The Operators a and 8 Suppose J is a complex function defined in a plane
open set n. RegardJas a transformation which maps n into R2, and assume that
J has a differential at some point Zo E n, in the sense of Definition 7.22. For
simplicity, suppose Zo = J(zo) = O. Our differentiability assumption is then equivalent
to the existence of two complex numbers IX and P (the partial derivatives of J
with respect to x and y at Zo = 0) such that
J(z) = IXX + py + '1(z)z (z = x + iy), (1)
where '1(z)-+ 0 as z-+ O.
Since 2x = z + z and 2iy = z - Z, (1) can be rewritten in the form
IX - iP IX + iP
J(z) = -2- z + -2- z + '1(z)z. (2)
This suggests the introduction of the differential operators
Now (2) becomes
J(z) - z
z
z
- = (aJ)(O) + (af)(O) . - + '1(z)
(3)
(z #: 0). (4)
For real z, z/z = 1; for pure imaginary z, z/z = - 1. Hence J(z)/z has a limit
at 0 if and only if (8f)(0) = 0, and we obtain the following characterization of
holomorphic functions:
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