144 4. Equations
(c) Deduce from (b), that if 1.~1 < l/3, then p(z) is contained within
the circle of center 1 and radius 2/3. More generally, deduce
from (b) that the closer a circle is to the origin in the r-plane,
the closer its image in the w-plane will be to the point 1.
(d) Suppose r > l/2. Verify that, as 0 increases from 0, the point
p(r cos 0 + ir sine) starts from a point 1 + r + r2 on the real
axis and moves in a counterclockwise direction about ‘the origin,
staying above the real axis as long as 2r cos 0 + 1 is positive.
Show that the image of the circle of radius r under the mapping
z -+ p(z) crosses the real axis at the points 1 + r + r2, 1 - r2,
1 - r + r2 and again at 1 - r2 when the argument of z is 0, some
angle between (l/2) T and T, T, and some angle between ?r and
(3/2)x, respectively.
(e) Suppose r < l/2. Show that the real axis is intersected only
twice by the image of the circle of radius r, and argue that this
image is a small loop which does not intersect itself.
(f) Let C, be the circle of radius r in the z-plane and D, its image
in the w-plane under the mapping z - p(z). Sketch D, for
(i) 0 < r < l/2
(ii) r = l/2
(iii) l/2 < r < 1
(iv) r = 1
(v) 1 < r (say r = 4).
When l/2 < r, verify that D, has two loops.
(g) Verify that D4 lies in the annulus {w : 8 < 12~1 < 24).
(h) Let r be a fixed radius. Imagine a vector drawn in the w-plane
from 0 to a point on D,. As .z traces around C, in a counter-
clockwise direction, the vector joining 0 to p(z) will rotate. Verify
that, if 0 < r < 1, this vector will move back and forth without
completing even a single rotation around 0, while, if 1 < r, the
vector will make two complete circuits of the origin.- Let p(z) = z2 + 2% + 1.
(a) r;fT2;hat P( r cos^0 + ir sin 0) = Sr(cos^0 + i sin O)(r cos^0 + 1) +(b) Carry out an analysis of the image curves of C, as in Exercise
2, and verify that, as r decreases, the value for which the inner
loop of the image D, disappears is the same as the value for
which D, passes through the origin. Explain the significance of
this.- Let p(z) = z2 + 32 + 2.