674 Chapter9
where g(z) = f(a + (1/z)).
Example f(z) = e^1 1z has z = 0 as an isolated essential singularity, and
it is its only singular point. Here g(z) = ez and g'(O) = 1. Hence
Rese^1 /z = 1
z=O- If f(z) = F(z)/(z - ar (m '::/: 0 an integer), and F(z) has at a an
isolated essential singularity, this point being its only singularity, then a
is also an essential singularity of f and its only singular point. Letting
z =a+ 1/z', f(a + 1/z') = g(z') and F(a + 1/z') = G(z'), we have
g(z') = z'mG(z') = z'm(Bo + B1z' + B2z'^2 + · · ·)
= Boz'm + B1z'm+l + B2z^1 m+^2 + Bkz'm+k + · · · (9.8-11)If m + k = -1 we obtafo.
so that
1
Res g(z') =-Bk= --G(k)(O)
z^1 =oo k!1
Resf(z) = -a<k>(o)
z=a k!
(9.8-12)Example If f(z) = z^3 e^1 /z
2
, we have m = -3, k = 2, G(z') = ez'2
,
G"(O) = 2, so that
1
~~gf(z) = - 21 (2) = -1If m ~ 0, the expansion (9.8-11) does not contain the power z^1 -^1 • Hence
in this case we have
Exercises 9.2
Find the following.
z2
- Res ---
z=l Z - 1
ez - Res --
z=O z^3 - Res cot z
z=O
z+2
- Res·--
z=oo z^2 + 1
Resf(z) z=a = 02z2. Res --
z=oo z - 3
z+l- Res z=2 ( Z - 2 ) 2
- Rescsc^2 z
z=O - Res sinz
z=oo