Higher Order Poles in theSMatrix

A method is developed for studying the possibility of a coincidence of more than one complex pole of the $S$ matrix so as to produce a higher order pole. It is shown thereby that complex higher order poles may be consistent with generalized unitarity, although a real higher order pole is not consistent with physical unitarity. Also discussed is the relevance of higher order poles, and of a group of simple poles, to the Wigner time-delay formula.