Page 38 - Calculus with Complex Numbers
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Ffgure J.4
3 .5 M aclau ri n expansions
lt has always been important to be able to approximate functions by polynomials.
This is because polynomials are the only functions whose values can be calculated
arithmetically. For a calculator to calculate ex for given .x it has to evaluate the
series
to as many terms as are needed to achieve the required degree of accuracy. To
calculate the value of zr it is necessal'y to use the series
with .x = 1. ln practice lnoth of these calculations are done by more sophisticated
methods but they still have to make use of polynomial expansiorls in one form or
another.
Maclaurin (1742) gave the general form for expanding a function flx) in powers
of .x . The expansion is
where the rlth coeflicient an is given by the formula
y(n) (())
Jn = ,
n !
and where fçns (0) denotes the rlth derivative fçns @) of flx) evaluated at .x = 0.
We call this exparsiontheivlkc/tzrlWn expansion of f @) andwe call the coeflicient
an the rlth Maclaurin coeflicient of f(x).
