Singularities/Residues/Applications(a) z^4 + 5z + 1 = 0
(b) z^7 - 4z^4 + z^3 + 1 = 0
( c) z^6 - 8z^2 + 3 = 0
(d) z^9 +z^5 +7z^4 -2 = 0- Let C be a simple closed contour and suppose that the inequality
lakzk I > lanzn + · · · + akHZk+l + ak-lZk-l + · · · + ao I
749holds for all z E C. Prove that the polynomial anzn + · · · + a 0 has
k zeros inside C if z = 0 lies within C, and has no zeros inside C if
z = 0 lies outside C.6. (a) If 0 < ae < 1, prove that the equation aez - z = 0 has exactly
one root in lzl < 1.
(b) Show that the equation sinz = 2z has exactly one root in lzl < 1.- Let F(z) = z - a - >..J(z), where f is analytic at z = a. Prove that
for IA.I sufficiently small, there is a disk lz - al < r in which F(z) has
just a simple zero. - Let g(z) be meromorphic in a region R but holomorphic on the simple
closed contour C homotopic to a point in R. If lg(z)I < 1 for z E C,
prove that the number of roots of the equation g(z) = 1 inside C is
equal to the number of poles of g(z) inside C. - Suppose that f and g are analytic in a region R and that C is a simple
closed contour homotopic to a point in R. If
IJ(z) + g(z)I <.IJ(z)I + g(z)I
for all z E C, then f and g have the same number of zeros inside C.
This symmetric form of RoU:che's theorem is due to I. Glicksberg [6].10. Show that for every 'T' > 0 there exists a positive integer N such that
for n;::: Nall the zeros of the polynomial 1 + z/1! + z^2 /2! + · · · + zn /n!
lie outside the circle lzl = r.- Consider the equation (z - l)nez = a, where n is a positive integer.
If lal < 1, show that this equation has exactly n distinct zeros in the
half-plane Rez > 0. If lal::; 1/2n, all zeros lie in the disk lz -11<^1 k
- Prove the following version of Hurwitz's theorem: Let Un(z)} be a
sequence of functions which are continuous on a compact set K and
analytic in Int K (Int K -:/:-0), and suppose that fn(z) :4 J(z) in K.
If J(z) -:/:- 0 on 8K, then for n large enough all the functions f n(z)
have in IntK the same number of zeros as J(z), counted with their
multiplicities.