1550251515-Classical_Complex_Analysis__Gonzalez_

392 Chapter 6

The function u + iv may be considered as a complex function of the

point (p, q) on the surface S, and the system (6.24-2) as the generalized

Cauchy-Riemann equations. The elimination of v gives

~ [~(Fuq-Gup)] + :q [~(Fup-Euq)] =0

which is the generalized Laplace equation for the function u. If is easy to

verify that v satisfies the same equation.

If U + iV is another function of the same type on the surface S, we have

dU +zdV. = μ' ( Edp+ E+iH E dq )

Comparing (6.24-1) with (6.24-3), we obtain

I

dU + idV = !!:_(du+ idv)

μ

(6.24-3)

which shows that U + iV is either an analytic or a conjugate analytic

function of u +iv (Theorem 6.17).

We note that the Beltrami equations are equivalent (except for the

notation) to the equation

fz = 77fz (6.24-4)

where 77 is a complex parameter. In fact, letting 77 = a + i/3 equation
(6.24-4) gives

so that

or

Ux -Vy+ i(vx +Uy)= (a+ if3)[(ux +Vy)+ i(vx - uy)]

= [a(ux + vy)-/3(vx - uy)]

+ i[f3(ux + vy) + a(vx -uy)]

Ux -Vy= a(ux +Vy)-/3(vx - uy)

Vx +Uy= /3(ux + vy) + a(vx - uy)

f3vx -(1 + a)vy =(a - l)ux + /3uy

(1 -a)vx - /3vy = f3ux - (1 + a)uy

Solving (6.24-5) for Vx and Vy, we get

Vx = l -^1 [^2

17712

2/3ux - (1771 + 2a + l)uy]

Vy=

1

_

1

17712

[(1771^2 - 2a + l)ux - 2/3uy]

(6.24-5)

(6.24-6)