Sequences, Series, and Special Functions 531IC -zl = l(C -a) -(z - a)I 2:: IC -al -lz -al = 'T' - p. Hence, applying
Darboux's inequality to (8.3-4), we obtainpn+l M. Mr (p)n+l
IRn(z)I::; - 2 7l" - ( 'T' -p ) 'T'n +i (27rr) = -'T' - p 'T' -so Rn(z) ~ 0 as n ~ oo, since 0 ::; p/r < 1. Thus, letting n ~ oo in
(8.3-3), we get
00 j(k)( ) 00
f(z) = L T(z -a)k = _Lak(z -a)k (8.3-5)
k=O k=O
with ak = j(k)(a)/k! and ao = f(O)(a)/O! = f(a).
If we let z - a = h or z = a+ h in (8.3-5), we obtain the expansion
in the alternative form
valid for lhl < R.
If the origin is in D, we can take a = 0, and (8.3-5) becomes
00 j(k)( )
f(z) =I:--,0
-zk
k=O k.(8.3-6)(8.3-7)a series expansion of f(z) in powers of z, valid in a disk about the ori-
gin, traditionally called (by historical error) the Cauchy-Maclaurin series
expansion of f(z).t
It is clear that the derivation of (8.3-5) given above is valid as long as
there are no singularities of f on or inside C', i.e., the series expansion
converges to f(z) for any z such that lz - al < R, with R = Is - al,
s being a nonremovable singularity of f nearest a. It is also clear that
the radius of convergence of the series cannot be greater than R since the
function defined by a power series is analytic at every point inside the
circle of convergence, which, of course, means that no true singularity of
f may lie within that circle. In the case where f is analytic in the whole
(finite) complex plane, the radius of C' can be taken as large as we wish,
so R = oo in this case.
tThis expansion, together with (8.3-5)(for the real case), was first given by B. Tay-
lor in Methodus incrementorum (1715) and later reproduced by C. Maclaurin in
his Treatise of Fluxions (1742). The Taylor expansion was generalized to the
complex domain by Cauchy in 1831.