Unknown

4.5. The Fundamental Theorem: Functions of a Complex Variable 145

rcosO+irsinO) = r(cosO+isinO)(2rcosO+3)+

(b) Analyze the image D, of the curves C, for the values r > 2,

r = 2, r = 312, r = 1 and r < 1. Verify that, as r decreases, the

inner loop contracts to a point while Dr still makes at least one

circuit of the origin, and disappears for r = 312.

5. Let p(z) = z3 - 72 + 6.

(a) Verify that p( r cos 0 + ir sin 0) = r(cos 0 + i sin O)(4r2 cos2 0 -

r2 - 7) + (-2r3 cos 0 + 6).

(b) Show that p( r cos 0 + ir sin 0) is real exactly when 0 = 0, 0 = rr

or when cos2 0 = (l/4) + (7/[4r2]). Deduce that, when r is very

large, the image of the circle of center 0 and radius r crosses the

real axis when 0 is equal or close to one of the angles 0, z/3,

2~13, T, 4~13, 5~13.

(c) Suppose r2 < 713. Show that

(i) Im p(r cos O+ir sin 0) and Im (r cos O+ir sin 0) have opposite

signs;

(ii) the image of the circle of radius r meets the real axis only

twice;

(iii) the image does not make a circuit of the origin when r < 1

and makes one circuit of the origin when 1 < r.

(d) Suppose r2 > 713. Show that the image of the circle of radius r

crosses the real axis 6 times when 0 = 0, Or, ?r - Or, ?r, ?r + 0,. ,

27r - Or, where 0,. = arccos((l/4)(1 + 7re2))li2.

(e) Verify that the image of the circle Cr makes

(i) no circuit of the origin when 0 < r;

(ii) one circuit of the origin when 1 < r < 2;

(iii) two circuits of the origin when 2 < r < 3;

(iv) three circuits of the origin when 3 < r.

6. Let h(z) = P. Show that, if z makes one counterclockwise circuit of
   the origin along the circle ]z] = r, then h(z) makes n counterclockwise
   circuits of the origin.


7. Let p(z) = Z” + Q,,-~z”+~ + Q,+~.z”-~ +... + art + ao. Suppose that
   r is any positive real number exceeding 1 for which

f 2 2{lan-11 + Ian-2 I + I%-31 + a*.+ Ia11 + I(-Jol).

(a) Show that, if 1.~1 = r, then

Izn -p(z)1 5 {la,-ll+la,-2l+...+lall+ laol}r”-’ L (1/2)r”.