(4) in (e z)/(z? - 1),
(e^z)/(z? - 1)=(e^z)/[(z- 1)(z+ 1)],
(e z) in | z- 1 | < δ and | z+ 1 | < δ,
∴z 1= 1 and z2 =- 1 are isolated singularities and both are first-order poles.
Residual definition and residual calculation rules 1:
Residual Res[f(z), z0] = z→ z0lim [(z-z0) f (z)].
When z0= 1, Res[f(z),1] = z→1lim [(z-1) (e z)/(z? - 1)]
=(e^z)/(z+ 1)
= e/2;
When z0 =- 1, Res[f(z),-1] = z →-1lim [(z+1) (e z)/(z? - 1)]
=(e^z)/(z- 1)
=- 1/2e;
∴ Residual Res[f(z), z0]=Res[f(z), 1,-1]
= e/2- 1/2e;
(7) In (z n)/(z- 1) n, n is a positive integer,
Z = 1 is an isolated singularity and an N-order pole.
Residual Res[f(z), z0] = z→ z0lim [(z-z0) f (z)], z0= 1,
∴ Residual Res[f(z),1] = z→1lim [(z-1) (z n)/(z-1) n]
=(z^n)/[(z- 1)^(n- 1)]