662 Chapter^9
- The function g(z) = f(l/z) = csc'l!"z has poles at z = 0, ±1, ±2, ....
It follows that z == oo is a cluster point of csc 'l!"Z.
- Let 'lfJ(z) = L,~;:_ 00 anz/(z-an)^2 , where a is a complex constant
such that lal > 1. This function has double poles at the points an, where
n is any integer. Since lal > 1, the poles 1, a, a^2 , ••• , have oo as an
accumulation point, while the poles a-^1 , a-^2 , ••• have 0 as an accumulation
point. Hence 0 and oo are cluster points of 'ljJ(z). We note that 'l/J(O) = 0
so that 'ljJ is defined, yet not regular, at z = 0.
Exercises 9 .1
Investigate whether each of the following functions is regular or singular at
the given points. If a point is singular, determine what kind of singularity
it is.
- f(z) = 1/(z + z^2 ); z ~ -1, z = oo
- f(z) = z^3 /(z -1)^2 ;z = 1,z = oo
- f(z) = z^2 e-z; z = oo
- f(z) = e^1 fz
2
;z = O,z = oo
- f(z) = cotz;z = O,z = oo
- f(z) = 1/(sinz - sina);z = a
- f(z) = etan{l/z); z = 0, z = oo
- f(z) = (1/z) + cos(l/z); z = 0, z = oo
- f(z) = sin(l/ sinz-^1 ); z = 0,; = oo
- f(z) = (ez - 1)/(ez + 1); z = i71", z = oo
- Give the general form of a function having only the following singular
points:
(a) A pole of order 3 at oo
(b) A pole of order 2 at z = i
( c) Simple poles at z = a and z = b, and a pole of order 2 at oo
( d) A simple pole at 0 and a simple pole at oo
( e) A pole of order h at 0 and a pole of order k at oo
(f) Simple poles at the points wk, where w = eitr/^3 (k = O, 1, 2, ... , 5)
12. Let f(z) = L,t,:_ 00 Ak(z - a)k valid in 0 < lz - al < 6. If f(z) is
regular at a prove that Ak = 0 for k = -1, -2, ...
- Prove that the conclusions of the Casorati-Weierstrass theorem hold
at a cluster point that is an accumulation point of poles (and not of
isolated essential singularities).
