Page 40 - Calculus with Complex Numbers
P. 40
The lirst live expansions are valid for all z, whilst the last three are only valid for
I z I < 1. The exparlsion for (1 + zlaf is of course the binomial theorem which gives
a terminating series in the case a a positive integer. The particular case a = - 1
gives the geometric series
which on integrating term by term gives the series for logtl + z) (PV).
3 .6 C aI cu Iati ng M aclau rin expansions
We can either use the Maclaurin formula an = fçns (0)/rl! or we can combine the
standard expansions listed in Section 3.5.
For example, suppose f (z) = tan z. Then writing F = tan z, S = sec z and
observing that JF/JZ = Sl, dsldz = ST we have the following.
Hence we obtain
Alternatively, we can write
