A complex function G(s)is said to be analyticin a region if G(s)and all its deriva-
tives exist in that region. The derivative of an analytic function G(s)is given by
Since can approach zero along an infinite number of different
paths. It can be shown, but is stated without a proof here, that if the derivatives taken
along two particular paths, that is, and are equal, then the deriva-
tive is unique for any other path and so the derivative exists.
For a particular path (which means that the path is parallel to the real
axis),
For another particular path (which means that the path is parallel to the
imaginary axis),
If these two values of the derivative are equal,
or if the following two conditions
are satisfied, then the derivative dG(s)/dsis uniquely determined. These two conditions
are known as the Cauchy–Riemann conditions. If these conditions are satisfied, the func-
tionG(s)is analytic.
As an example, consider the following G(s):
Then
G(s+jv)=
1
s+jv+ 1
=Gx+jGy
G(s)=
1
s+ 1
0 Gx
0 s
=
0 Gy
0 v
and
0 Gy
0 s
=-
0 Gx
0 v
0 Gx
0 s
+j
0 Gy
0 s
=
0 Gy
0 v
- j
0 Gx
0 v
d
ds
G(s)= lim
j¢vS 0a¢Gx
j¢v
+j
¢Gy
j¢v
b =-j
0 Gx
0 v
+
¢Gy
0 v
¢s=j¢v
d
ds
G(s)=¢limsS 0 a
¢Gx
¢s
+j
¢Gy
¢s
b =
0 Gx
0 s
+j
0 Gy
0 s
¢s=¢s
¢s=¢s+j¢v
¢s=¢s ¢s=j¢v,
¢s=¢s+j¢v, ¢s
d
ds
G(s)= lim
¢sS 0G(s+¢s)-G(s)
¢s
= lim
¢sS 0¢G
¢s
860 Appendix A / Laplace Transform TablesOpenmirrors.com