Partial recursive functions are particular functions from some subset of vectors of NaturalNumber''''''s to NaturalNumber''''''s.  A partial recursive function may be undefined (divergent) at some points.

Every PrimitiveRecursiveFunction is a partial recursive function

Also if ''f''(''x'',''y''1,...,''yn'') is a partial recursive function of ''n''+1 variables then (µ''f'')(''y''1,...,''yn'') is a partial recursive function of ''n'' variables.

'''The µ operator'''

The µ operator performs an unbounded search on ''f''.

(µ''f'')(''y''1,...''y''''n'') is the least ''x'' such that ''f''(''x'',''y''1,...,''yn'')=0, or diverges if none exists.  Also if there is a y < x such that ''f''(''y'',''y''1,...,''yn'') diverges then (µ''f'')(''y''1,...''yn'') also diverges.

The idea is that the computation keeps trying values for x in increasing order until a value for ''f''(''x'',''y''1,...,''yn'') is found to be 0.  Therefore if some previous value for ''f'' diverges (think of diverging as stuck in an infinite loop) then the search also diverges.

PartialRecursiveFunctions provide a simple ModelOfComputation.