Differentiation 397an equation of the formw.z+Aw+Bw=F (6.24-15)
where A, B, and Fare of class LP with p > 2, i.e., of measurable functions
in the domain under consideration and such thatfv iflPdμ < 00
the integral being taken in the Lebesgue sense. Equation (6.24--15) is the
complex form of the system of real equationsu., - Vy + au + bv = f
Uy+ Vx +CU+ dv = g (6.24-16)Vekua introduces the concepts of generalized and of regular solutions
of equation (6.24-15). The scope of the method is extended further by
using the so-called generalized derivatives in the sense of Sobolev [111]
(distributions in the terminology of L. Schwartz [110]).
(j) E. Lammel ([76] and [77]), L. Sobrero [112], J. B. Diaz [33], and J.
Horvath [62] have constructed theories based on a generalization of the
Laplace equation.
Other authors, in particular, H. A. von Beckh-Widmanstetter [6], P.
Dentoni ([27] and [28]), A. Douglis [35], R. P. Gilbert and G. N. Hile [43],
P. W. Ketchun ([70], [71], and [72]), E. R. Lorch [81], G. B. Rizza ([102],
[103], and [104]), M. N. Rosculet ([105] and [106]), and M. See [109], have
extended the theory of analytic functions to functions in certain algebras.
A number of them consider also the relationship between those theories
and the theory of generalized harmonic functions.
(k) By using the conformality property or, equivalently, the constancy
of the magnification ratio, E. R. Hedrick and L. Ingold in [60] derive for a
mapping F: D -t R^3 defined by an equation ( = F( e), where e = ( x, y, z)
and ( = (u,v,w), or, by the system
the conditions
u=u(x,y,z)
v=v(x,y,z)
w = w(x,y,z)u2 xx + v2 + w2 :z: = u2 y + v2 y + w2 y = u2 z + v2 z + w2 z = E