Real and Complex Analysis (Rudin)

232 REAL AND COMPLEX ANALYSIS
11.2 Theorem Suppose f is a complex function in a that has a differential at
every point ofa. Thenf E H(a) if and only if the Cauchy-Riemann equation
holds for every z E a. In that case we have
(af)(z) = 0 (1)
f'(z) = (af)(z) (z E a). (2)
Iff = u + iv, u and v real, (1) splits into the pair of equations
where the subscripts refer to partial differentiation with respect to the indicated
variable. These are the Cauchy-Riemann equations which must be satisfied by the
real and imaginary parts of a hoi om orphic function.
11.3 The Laplacian Let f be a complex function in a plane open set a, such that
fxx andfyy exist at every point ofa. The Laplacian offis then defined to be
Iffis continuous in a and if
A/=fxx + fyy· (1)
A/= 0 (2)
at every point of a, thenfis said to be harmonic in a.
Since the Laplacian of a real function is real (if it exists), it is clear that a
complex function is harmonic in a if and only if both its real part and its imaginary
part are harmonic in a.
Note that
A/= 4aaf (3)
provided that fxy = /'x, and that this happens for all f which have continuous
second-order derivatives.
Iff is holomorphic, then af = O,fhas continuous derivatives of all orders, and
therefore (3) shows:
11.4 Theorem H olomorphic functions are harmonic.
We shall now tum our attention to an integral representation of harmonic
functions which is closely related to the Cauchy formula for holomorphic functions.
It will show, among· other things, that every real harmonic function is
locally the real part of a holomorphic function, and it will yield information
about the boundary behavior of certain classes of holomorphic functions in open
discs.