Singularities/Residues/ Applications 673Example Let f(z) = (z + 1)/(z^2 + 4). This function is analytic for 2 <
lzl < oo and has a simple zero at oo. Applying (9.8-7) we find that
. z(z + 1)
Res f(z) = - hm 2 = -1
z=oo z-+oo z + 4
( d) Residue at an isolated essential singularity. There are no simple rules
for the computation of the residue of a function at an isolated essential
singularity other than finding A_ 1 or B_i, i.e., the coefficients of (z -a)-^1
or of z-^1 in the Laurent expansion of f(z) valid in the vicinity of a or of
oo, respectively. However, in the following special cases the determination
of the residue is easily made.
1. If f is an entire function, then oo is either a pole or an essential
singularity off. Since f(z) has the representation
f(z) =Bo+ Biz+ B2z^2 + · · ·
valid for all finite values of z (with a finite number of terms in the case of
a polynomial), it follows that B-1 = 0. Hence
Resf=O
z=oo
(9.8-9)for any entire function.
2. If a "f; oo is an essential singularity of f, and its only singularity, the
transformation z' = (z-a)-1, or z = a+(l/z') gives f(a+(l/z')) = g(z'),
where g( z') is a function whose only singular point is an essential singularity
at oo. We have
A-1 = 2 ~i J f(z) dz= - 2 ~i J ( z~ 2 ) g(z') dz'
c+ ~-= - Res ( ~ 2 ) g(z')
z^1 =oo ZBut
g(z') =Bo +Biz'+ B 2 z^12 + · · ·
valid for all finite z', since g is entire. Then
(__!___) z'2 g(z') = z'2 Bo + Bi z' + B^2 + ...
Hence
that is,
Res J(z) = g'(O)
z=a
(9.8-10)