7.4 • SJNGULARJTIES, ZEROS, AND POLES 283•EXAMPLE 7.14 Locate the poles of g(z) = "co;~ul and specify their
order.
Solution The function f ( z) = z^2 sin 7r z has a zero of order 3 at z = 0 and
simple zeros at the points z = ±1, ±2, .... Corollary 7.5 implies that g has a
pole of order 3 at the point 0 and simple poles at the points ±1, ±2, ....-------.. EXERCISES FOR SECTION 7.4- Locate the zeros of the following functions and determine their order.
(a) (1 + z^2 }^4.
(b) sin^2 z.
(c) z^2 + 2z + 2.
(d) sin?.
(e) z^4 + 10z^2 + 0.
(f) 1 +expz.
(g) z6 + 1.
(h) z^3 exp (z - 1).
(i) z^6 + 2z^3 + 1.
(j) z^3 cos^2 z.
(k) z^8 + z^4 •
(1) z^2 cosh z.2. Locate the poles of the following functions and determine their order.(a) (z^2 + 1)-^3 (z -1)-^4 •(b) z-^1 (z^2 - 2z + 2}-^2 •
(c) (z6 + ir1.(d) (z^4 + z^3 - 2z^2 r^1.
(e) (3z^4 + 10z^2 + 3)-^1.
er> (i + ~rl (3 + ~) -1.
(g) zcotz.
(h) z-^5 sin z.
~i) ( z^2 sinz)-^1.
(j) z -^1 cscz.