1550251515-Classical_Complex_Analysis__Gonzalez_

330 Chapter^6

Theorem 6.9 If w = g(z) is of class 'D(A) and if ( = f(w) is defined and

analytic in an open set B that contains the range of g, then

Proof We have

{

ozf(w): f:(w)OzW

Ozf(w) - f (w)ozw

(6.8-8)

Ozf(w) = % [ (! ) f(w) - i ( :y) f(w)]

= %[J'(w)wx - if'(w)wy]

= J'(w)OzW

The second formula in (6.8-8) follows in the same manner.

Theorem 6.10 Suppose that the complex functions w 1 = g 1 (z) and w 2 =

gz ( z) are both of class 'D( A). Let ( = f ( w1, Wz) be a complex function of

two variables w 1 , w 2 whose domain of definition contains the Cartesian

product of the ranges of 91 and g2, and assume that f is analytic with

respect to w1 and w 2 separately. Then

of of

ozf(wi,wz) = ~OzW1 + ~OzWz (6.8-9)

uw1 uWz

of of

Ozf(wi,wz) = ~OzW1 + ~O-zWz (6.8-10)

uw1 uWz

Proof Let W1 = U1 + iv1, Wz = Uz + ivz, and f( W1, Wz) = e + i77,

where Uj = Uj(x,y), Vj = Vj(x,y)(j = 1,2), and e = e(u1,V1,u2,V2),
77 = 77( ui, v 1 , u 2 , v 2 ). From the assumption and (6.5-4) it follows that

Hence we obtain

of oe. 077 077. oe

--=-+i-=--i-

OW1 ou1 ou1 ov1 ov1

of oe. 077 077. oe

--=-+i-=--i-

OW2 ouz ouz ov2 ovz