284 CHAPTER 7 • TAYLOR AND LAURENT SERIES
(k) (1 - expz)-^1 •
(1) z-^6 sinh z.- L-0cate t he singularities of the following functions and determine their type.
(a) z-~~n z ·
(b) sin(~).
(c) zexp (~).
(d) tan z.(e) (z^2 +zr
1
sinz.
(f) SJ: ,t
(g) (exp ; >- 1(h) ccu-cos(2•)
••
4. Suppose that f has a removable singularity at zo. Show that the function ] has
either a removable singularity or a pole at z 0.- Let f be analytic and have a zero of order le at zo. Show that f' has a zero of
order k - 1 at zo. - Let f and g be analytic at zo and have zeros of order m and n, respectively, at zo.
What can you say about the zero of f + 9 at zo? - Let f and 9 have poles of order m and n, respectively, at zo. Show that f + g has
either a pole or a removable singularity at zo - Let f be analytic and have a zero of order k at zo. Show that the function Lj-has
a simple pole at zo. - Let f have a pole of order k at zo. Show that f' has a pole of order le+ 1 at zo.
- Prove the following corollaries.
(a) Corollary 7.4.
(b) Corollary 7.5.
(c) Corollary 7.6.
(d) Corollary 7 .7.
(e) Corollary 7.8.- Find the singularities of the following functions.
(a) sin(~/>)·
(b) Log z^2 •
(c) cotz- ~·
12. How are t he definitions of singularity in complex analysis and asymptote in calculus
different? How are they similar?