1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation

M

i/--

1

I

v,

I I lr-------N

1 I DI

I -yy

0 1 P 10

• -u,-- 4 J


• 
  
  
  
  • ---Vy----__.,j
  
  
  
  

Fig. 6.21

381

The points A, B, C, D are the vertices of a rectangle inscribed in

the circle. Verify the following:

OA =VE, OC =VG, areaOMBQ-areaONDP = F

7. Show that the Jacobian of the ratio f(z)/z vanishes on the Kasner

circle of f at z.

8. The integral curves of the differential equation
   Fy'^2 + ( E - G)y' - F = 0

are called the characteristic lines of the nonanalytic function f E 'D( A).

The characteristic lines form an orthogonal family of curves in the

z-plane. Prove that the family of characteristic lines is the only or-

thogonal family of curves in the z-plane that is mapped by f into an

orthogonal family.

9. Given two functions g and h of class 'D(A), we may construct for each

z EA a circle with center at g(z) and radius lh(z)I. Find under what

condition on g and h the set of such circles is the family of Kasner

circles of a function f E C(^2 )(A).

10. For the mapping defined by f(z) = z + z-^1 , find the points at which
    the magnification ratio is a given constant.

6.20 THE MAXIMUM AND MINIMUM MODULUS

THEOREMS FOR FUNCTIONS OF CLASS 'D (A)

In this section we consider special cases of a more general proposition that
we have established in [52]. They are consequences of the finite increments
formula (6.12).