272 Chapter^5real numbers a and b, with a^2 + b^2 f= 0, the polynomial af(x) + bg(x)
has only real zeros.
[J. Rainwater, Amer. Math. Monthly, 71 (1964), 322-323]- Prove that the polynomial P( z) that assumes real values for real z and
nonreal values for nonreal z must be linear.
[D. I. A. Cohen, Amer. Math, Monthly, 72 (1965), 550]
13. In Section 3.4 the paraconjugate of a function f, with domain symmet-
ric with respect to the y-axis, was defined by f"(z) = ](-z). Show
that:(a) If f is a polynomial with zeros zi, z2, ... , Zn, then J"(z) is a
polynomial with zeros -zi, -z 2 , ••• , -Zn.
(b) lf"(iy)j = lf(iy)j (y real).(c) The function w = J"(z)/f(z) maps Rez = 0 onto the circle lwl =
l.
- A Hurwitz polynomial is defined as a nonconstant polynomial having
all its zeros in the left half-plane Re z < 0. Let. f be a nonconstant
polynomial such that f and f 11 have not a common zero, and let
f"(z)
g(z) = f(z) 'h z _ f(z) - f"(z)
( ) - f(z) + f"(z)
Show that the following statements are equivalent.(a) f is a Hurwitz polynomial.
(b) lg(z)I < 1 for Rez > 0.
(c) Reh(z) > 0 for Rez > 0.
Hint: Show that if Rezk < 0, then lz - zkl > lz + zkl so lf(z)I >
lf"(z)j. Conversely, if the last inequality holds, then f(z) is a Hurwitz
polynomial.5.18 The Exponential Function
We have already defined (Definition 1. 7) the complex exponential to the
base e by the formula(5.18-1)
where z = x+iy. To each z E C the formula assigns a unique complex value
w, so it defines a single-valued function, called the exponential function,w = ez = expzin the whole finite complex plane. A number of elementary properties of
this function were discussed in Theorem 1.11. In particular, we recall the
following fundamental properties: