1. Some basic theorems. Before proceeding to the study of various specific
problems connected with the zeros of polynomials, we shall find it useful to
consider certain general theorems to which we shall make frequent reference.
The first of these theorems provides an intuitively obvious sufficient condition
for the nonvanishing of a sum of complex numbers. It requires that each term
in the sum be a vector drawn from the origin to a point on the same side of some
line through the origin. This theorem may be stated as follows.
(1,1). If each complex number wj9j = 1, 2, • • •, p, has the properties
that Wj T£ 0 and
(1.1) y ^ a r g w , y + 7T, j = 1, 2, • • •,/?,
where y is a real constant, then their sum w = 2?
w, cannot vanish.
In proving Th. (1,1), we begin with the case y = 0 when the Wj are vectors drawn
from the origin to points on the positive axis of reals or in the upper half-plane.
If arg Wj = 0 for ally, then
0 for ally and hence 5R(w) 0. If arg w, 5* 0
for some value of y, then 3(w,) 0 for that j and hence 3(w) 0. Thus, if
= 0, w s* 0.
In the case that y ^ 0, we may consider the quantities wj = e~yiWj. These
satisfy ineq. (1,1) with y = 0 and consequently their sum wf does not vanish.
As w' = e~yiw, it follows that iv^O .
This proof establishes not merely that w 7* 0, but also the following. The
point w lies inside the convex sector consisting of the origin and all the points z
for which y ^ arg z ^ y + d, d TT, if all the points Wj lie in the same sector.
Our second theorem expresses the so-called Principle of Argument.
(1,2). Let f(z) be analytic interior to a simple closed Jordan curve C
and continuous and different from zero on C. Let K be the curve described in the
w-plane by the point w =f(z) and let A
arg/(z) denote the net change in arg/(z)
as the point z traverses C once over in the counterclockwise direction. Then the
number p of zeros off(z) interior to C, counted with their multiplicities, is
That is, it is the net number of times that K winds about the point w = 0.