Elementary Functions
( )z^3 - 6z^2 + llz - 6
cw=-------z3+1
271- Reduce w = (z^2 + z + 1)/(z - 2)^2 to the form W = Z^2 by suitable
changes of variables. - Reduce w = (z^2 - 4z + 1)/(z - l)(z + 2) to the form W =^1 / 2 (Z + z-^1 )
by suitable changes of variables.
6. Illustrate geometrically the mapping defined by w = z - z-^1.
- Let
f (z) = czm --'---~----'--'-----'-(z -a:1)(z - a:2) · · · (z -a:n)
( a1z - l)(a2z -1) · · · (anz -1)
where lei = 1. Show that IJ(z)I = 1 on izl = 1. Then prove that
given any rational function F(z) it is possible to construct another
rational function G(z) with no poles in the disk lzl < 1, and such that
IF(z)I = IG(z)I on the circle izl = 1.- Suppose that the rational function R( z) assumes real values on the
circle lzl = 1. Determine the relative location of its zeros and poles.
9. (a) If a: 1 is the zero of smallest modulus of the polynomial P(z) =
ao + aiz + · · · + anzn(an /= 0), show thatia:1 I ~ I :: 11/n
(b) Suppose that ia1I + ia2i + · · · + lanl < iaol. Prove that P(z) has
no zeros on izl ~ 1.(c) If iaol + Ja1J + · · · + ian-11 ~ Jani, show that all zeros of P(z) lie
in izJ ~ 1.(d) Let M =max Jakil/k (k = 1, 2, ... , n; ao i= 0). Show that all zeros
of Q(z) = znP(l/z) lie in izl < M(l + iaoJ-^1 ).
[Q. G. Mohammad, Amer. Math. Monthly, 12 (1965), 36](e) If r =max lak+i/ak I, ak /= 0 (k = O, 1, ... , n ), show that the zeros
of Q(z) = zn P(l/z) lie in izl ~ 2r.
[Q. G. Mohammad, Amer. Math. Monthly, 12 (1965), 37-38]- With P(z) = ao + aiz + · · · + anzn(an /= 0), let
A= max(Ja1I, Ja2I, ... , Jani)
B = max(laol, la1I, ... , Jan-11)
Show that every zero a: of P(z) satisfies