These can easily be evaluated using the Taylor series discussed in the
last section, expanded around the origin
, and are an alternative
way of seeing that
. In the case of exponential and
trig functions, the expansions converge for all
, not just small ones
(although they of course converge *faster* for small ones).

(77) | |||

(78) | |||

(79) |

Depending on where you start, these can be used to prove the relations above. They are most useful for getting expansions for small values of their parameters. For small x (to leading order):

(80) | |||

(81) | |||

(82) | |||

(83) |

We will use these fairly often in this course, so learn them.