380 Chapter^6
- g 1 : (x,y,B) -t (~,77), a lineal element-to-point transformation. By a
lineal element is meant a point with an associated direction B. - g2: z -t fz, a point-to-point transformation mapping each point z EA
into the center of the corresponding, Kasner circle. - g3: z -t fz + f;:, a point-to-point transformation mapping each point
z E A into the point JMz) = fz + fz (sometimes called the principal
phase point of the circle). - g 4 : z -t Kz, a point-to-circle transformation mapping each point z EA
into the Kasner circle corresponding to that point. To the 002 points of
A corresp.ond in general 002 circles (a family or congruence of circles)
in the (-plane. There are, of course, some degenerated cases: If f is
analytic in A the circles reduce to points and g 4 = g 2 ; if f is conjugate
analytic in A, the family contains 001 circles, all with centers at (0, O);
finally, all circles may reduce to a single circle if f(z) = Az + Bz + C,
A, B, C complex constants, B f= 0.Other similar mappings may also be considered. If the existence of the
second derivative f 6: 82 ( z) is assumed, numerous other transformations are
induced.
EXERCISES 6.3
- (a) Check the conformality property of the mapping defined by w =
z^2 by considering the point z 0 = 1 + 2i, the lines z = 1 + it,
z = t + 2it(-oo < t < +oo ), and their images under w = z^2 •
(b) Same as part (a) but for the mapping defined by w = ez by consid-
ering the point z 0 = i7r, the lines z =it, z = t+i7r(-oo < t < +oo ),
and their images under w = e z.
- (a) Construct the graph of (6.16-1) for the function w = z^2 z at the
point z = i.
(b) Same as part (a) but for the function w = (1 - i)z + (1 + i)z + 2
at any point.- Show that the condition p = constant implies that E = G and F = 0.
4. Suppose the f E V(A) is not constant in A. Show that E = G f= 0 and
F f= 0 if ifz = kfz, k f= 0, ±1 being a real constant.
- Use formulas (6.16-13) to check (6.16-9) and (6.16-10).
6. Consider on the Kasner circle the points A and C representing f~ ( z) =
Ux + ivx and f~; 2 (z) =Vy - iuy, respectively (Fig. 6.21), as well as the
points B and D corresponding to Vy + ivx and Ux - iuy·