[§U

SOME BASIC THEOREMS

5

EXERCISES.

Prove the following.

1. If each of p vectors Wj drawn from the origin lies in the closed half-plane

y ^ arg w ^ y + rr and if at least one of them lies in the open half-plane y

arg w y + IT, then the sum w = ^fjr=1 Wj 5^ 0.

2. Th. (1,1) and ex. (1,1) hold for convergent infinite sums 2 £ a WJ *n which

all the Wj satisfy ineq. (1,1); also for integrals J£ w(t) dt in which a and b are

real numbers and in which w(t) is a continuous function of the real variable t

and y ^ arg w(t) y + -n for a ^ t ^ b.

3. Let v v = 2?ii WJ • If P °f t n e points Wj lie in the circle \z\ ^ R0 and the

remaining point Wj lies in the annulus Rx ^ |z| ^ /?

2

, where Rx pR0, then the

point w lies in the annulus Rx — pR0 ^ |z| ^ 7?2 + P^o • Hence tv^O .

4. If the point z traverses a line L in a specified direction, then the net change in

arg (z — zx) is n or — 7r according as zx is to the left or to the right of L relative to

the specified direction.

5.

THEOREM

(1,6). Let L be a line on which a given nth degree polynomial f(z)

has no zeros. Let AL arg/(z) denote the net change in arg/(z) as point z traverses

L in a specified direction and let p and q denote the number of zeros off(z) to the

left and to the right of this direction of L, respectively. Then

(1.7) / - ? = (lMA/.arg/(z)

and thus

(1.8) /? = (l/2)[« + (l/7r)A^arg/(z)],

(1.9)

?

= (l/2)[A2-(l/7r)A^arg/(z)].

6. The polynomial g(z) =

z11

+

bYzn~l

+ • • • + bn has at least m zeros in an

arbitrary neighborhood of the point z = c if

\g{k)(c)\

^ e for k = 0, 1, • • •, m — 1

and for e a sufficiently small positive number [Kneser 1, Iglisch 1]. Hint: Use

Rouche's Theorem.

7. Rouche's Theorem is valid when \P(z)\ ^ \Q(z)\ for z e C provided F(z) =

P(z) + Q(z) * 0 for z e C.

8. Rouche's Theorem is valid when C is the circle \z\ = 1 and when \P(z)\ ^

\Q{z)\ on C, provided that at each zero Z of F(z) on C the function R{z) =

Iog(£?(z)//(z)) has the properties R\Z) ^ 0, SR(Z/*'(Z)) 0, 3(Z/fc'(Z)) = 0

[Lipka 3].

9. Let C be a closed Jordan curve inside which P(z) and Q(z) are analytic.

On C let P(z) and 0(z) be continuous, Q(z) 5* 0 and W[P(z)IQ(z)] 0. Then

inside C, P(z) has the same number of zeros as does Q(z).

10. Rouche's Theorem (1,3) follows from the continuity of the zeros of F(z) =

),P(z) + Q(z) as functions of A. Hint: Show that no zero of/ma y cross C as

A increases continuously from 0 to 1.

11. In F(z) = 1 + axz + ^ 2 + * ' • + bnzn, the quantities n, b2, b3, - - -, bn

may be so determined that all the zeros of Flie on the unit circle. Hint: Choose n