394 Chapter6
conjugate when the equation
. df dF
dn da
holds at every point of a certain region, where n is any direction whatever
in space and da lies in a plane perpendicular to n and having the same
relation to n as the plane yz has to the positive x-axis. In particular, the
preceding equation gives
8f dF
Bx = d(yz)'8f dF
-=--,
8y d(zx)8f dF
az ~ d(xy)(6.24-8)which are a generalized form of the Cauchy-Riemann equations.
From equations (6.24-8) it follows that
2 a^21 a^21 a^2 J
~ f = 8x2 + 8y2 + 8z2 = 0
so that f must be harmonic. Conversely, for every harmonic function f it
is always possible to find a functional F conjugate to f. For the definition
of dF / da and further developments of Volterra's generalization, we refer
the reader to his paper [124] and to his books, [125] and [126].
( c) E. Picard discussed the possibility of generalizing the Cauchy-
Riemann equations in two notes published in Comptes Rendus Acad. Sci.
Paris, 1891 ([91] and [92]). In the first note Picard considers a system of
equations of the form
Vx =aux+ buy
Vy= CUx + duy
where a, b, c, d are real analytic functions of x and y in. a region of the
plane where (a-d)^2 +4bc < 0. With this condition satisfied, the Jacobian
J ( u, v / x, y) will keep a fixed sign throughout that region. Then he observes
that u and v are determined in a compact subregion by the values that
u and v, respectively, assume on the boundary of the region. He also
suggests other generalizations of known properties in the theory of analytic
functions. However, that suggestion went largely unnoticed for several
years.
( d) L. Bers and A. Gelbart in several papers ([12], [13], and [14]) have
considered a system of partial differential equations of the form
a1(x)ux = r1(y)vy
a2(x)uy = -r2(y)vxand the class of complex functions f = u +iv associated with it.
(6.24-9)