9
Singularities. The Calculus of Residues.
Appli.catio1:1s
9.1 REGVLAR AND SINGULAR POINTSDefinition 9.1 A function f is said to be regular at z = a (a ~ oo) if
f is locally bounded at a (Definition 7.9). This may happen if either f is
analytic at a or if f is analytic and bounded in some deleted neighborhood
of a.
In the second case there is a unique number a 0 such that, if we define
f (a) = a 0 , the extended function becomes analytic at z = a (Riemann's
theorem). Hereafter we shall assume that the definition of f has been
extended in this manner. Hence. under such agreement the statement "f is
regular at a" is to be regarded as equivalent to the statement "f is analytic
at a" whenever a is a finite point. Then it follows that in some circular
neighborhood of a, say lz - al < r, f(z) has a Taylor representation
f(z) = ao + ai(z - a)+ a 2 (z - a)^2 + · · ·
with f(a) = ao.Definition 9.2 A function f is said to be regul~r at z = oo if f is defined
in some deleted neighborhood of co, namely, N'(oo) = {z: R < lzl < oo},
and if g(z) = f(l/z) is regular at z = 0.
The definition implies that g(z) is locally bounded in 0 < lzl < 8, for
some 8 > 0, so that f(z) must be also locally bounded in 1/8 < lzl < oo.
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