1550251515-Classical_Complex_Analysis__Gonzalez_

396 Chapter6

Bers says that w( z) has at Zo the (F, G)-derivative w( Zo) if the finite limit

.( )

1

. w(z)-,\oF(z)-μoG(z)

w zo = nn

z->zo z - Zo

exists.

The functions a(z), b(z), A(z), B(z) determined by the systems

F-z = aF + bi!' G z = aG + bG

and

Fz =AF+ BF

are called the characteristic coefficients of the generating pair ( F, G). Then

Bers proves the following theorem: If w( z) exists at Zo' then the partials

Wz, Wz exist, and the equations

w; =aw+ bw

w = Wz - Aw -Bw

(6.24-11)

(6.24-12)

hold. If Wz, Wz exist and are continuous in some neighborhood of z 0 , and

if (6.24-11) holds at zo, then w(zo) exists and (6.24-12) holds.
(g) E. F. Barnett in [4] develops the theory of a class of generalized
analytic functions satisfying a system of the form
Uy= -w(y)vx
Ux - Vy= ,,P(y)vx
where w and '1jJ are real functions of y of class C(l) at least.
(h) B. A. Case has discussed in [23] the class of generalized analytic
functions f(z) = u(r, 0) + iv(r, 0), z = rei^9 , with f E c<^1 >(D) and u and
v satisfying s system of the form

VIJ = ,\(r )ur

UIJ = - ,\(r)vr (6.24-13)

where ,\; E c<^1 >(D) and ,(r) =/; 0 in D. For f E c<^2 >(D) both u and v

satisfy the generalized Laplace equation

(6.24-14)

Real-valued functions 'lj;(r, 0) are said to be ,\-harmonic in a region D if

'1jJ E c<^2 >(D) and if they satisfy (6.24-14). ,\-harmonic functions u, v also

satisfying (6.24-13) are called ,\-harmonic conjugates. See Exercises 6.1,

problem 35.

(i) I. N. Vekua in several papers and in his book [123] has developed

the general theory and applications of complex functions w( z) satisfying