Integration 4990 A
Fig. 7.27If we now let f(a) = b, then f will be defined and analytic in all of A.
Definition 7.12 A point such as a in Theorem 7.36 is called a removable
singularity of f.
Corollary 7.17 With the same assumptions as in Theorem 7.36, we have
lira f(z) = b
z-+a
(7.26-2)This is trivial since analyticity at a point ·implies continuity at the same
point.
Example Let f(z) = (z^2 - a^2 )/(z - a), z E <C - {a}. Then
b= _I J [((
2
-a
2)/((-a)]d( =_I J ((+a)d( = 2 a
27ri ( - a 27ri ( - a
7 7Thus f becomes analytic in <C by letting f (a) = 2a. More generally, if we
have f(z) = [g(z) - g(a)]/(z - a) where g(z) is analytic in A, then f is
analytic in A - {a} and locally bounded at a since limz-+a f (z) = g' (a).
According to (7.26-2), we must have b = g^1 (a). In fact,
b = _I J [g(() - g(a)]/(( - a) d( = _I J g(() -g(a) d(
27ri ( - a 27ri (( - a)^2
7 7
= g'(a)For instance, the function f(z) = (sinz)/z = (sinz - sinO)/(z - 0), which
is analytic in <C - {O} and locally bounded at the origin, becomes analytic
in <C by setting f (0) = cos 0 = 1.
7.27 Derivative of an Integral with Respect to a Parameter
Theorem 7 .3 7 Consider a function f ( z, () of the two complex variables
z and (, and suppose that: