where
It can be seen that, except at s=–1 (that is,s=–1,v=0),G(s)satisfies the
Cauchy–Riemann conditions:
HenceG(s)=1/(s+1)is analytic in the entire splane except at s=–1. The deriva-
tivedG(s)/ds, except at s=1, is found to be
Note that the derivative of an analytic function can be obtained simply by differentiat-
ingG(s)with respect to s. In this example,
Points in the splane at which the function G(s)is analytic are called ordinary
points, while points in the splane at which the function G(s)is not analytic are called
singularpoints. Singular points at which the function G(s)or its derivatives approach
infinity are called poles. Singular points at which the function G(s)equals zero are
calledzeros.
IfG(s)approaches infinity as sapproaches–pand if the function
forn=1,2,3,p
has a finite, nonzero value at s=–p, then s=–pis called a pole of order n. If n=1,
the pole is called a simple pole. If n=2,3,p, the pole is called a second-order pole, a
third-order pole, and so on.
To illustrate, consider the complex function
G(s)=
K(s+2)(s+10)
s(s+1)(s+5)(s+15)^2
G(s)(s+p)n,
d
ds
a1
s+ 1
b =-
1
(s+1)^2
=-
1
(s+jv+ 1 )^2
=-
1
(s+ 1 )^2
d
ds
G(s)=
0 Gx
0 s
+j
0 Gy
0 s
=
0 Gy
0 v
- j
0 Gx
dv
0 Gy
0 s
=-
0 Gx
0 v
=
2 v(s+1)
C(s+1)^2 +v^2 D^2
0 Gx
0 s
=
0 Gy
0 v
=
v^2 - (s+1)^2
C(s+1)^2 +v^2 D^2
Gx=
s+ 1
(s+1)^2 +v^2
and Gy=
- v
(s+1)^2 +v^2
Appendix A / Laplace Transform Tables 861