of expressions such as 
,
where 
,
and
may be any number within the constraints of existence. We consider the
various cases that arise when some combination of poles occurs and demonstrate
the various simplified expressions for the derivatives in these cases.
When
is not an integer, there are no poles in either 
or 
and the usual derivative expressions hold. When
is an integer and 
,
the usual expressions also hold since 
and therefore 
.
The case where the usual expressions hold will be denoted Case 0.
When
is an integer and 
,
there are two cases are of interest. The first, Case 1, occurs when
and 
,
so that there are poles in the denominator of only. The second, Case
2, occurs when
and 
,
so that there are poles in both the numerator and the denominator.
In order to find expressions for the derivatives in cases Case 1
and Case 2 we use the following three facts. 1) The only singularities
of the gamma function 
are simple poles at 
with residues 
respectively. 2) 
whenever the expressions exist. (The identity 
may be used in deriving this.) 3) 
is the representation (away from the poles in the gamma functions) of an
analytic function (note that
is still assumed). Using these facts, the expressions for Case 1
and Case 2 are found by substituting 
for
(now restricted by the conditions of Case 1 and Case 2 to
be a nonnegative integer) in the corresponding case Case 0 expressions
and taking the limit 
.
Case 0:
non-integer, or
an integer with
and .
There are no poles in the numerator or denominator of .
The first derivative is given by
The rth derivative may be found by iteration, using equation
9.141
and the recursion relation
The second derivative is given by
Case 1:
an integer,
and .
The denominator contains a pole.
The zeroth derivative is 0.
Taking the appropriate limit in equation 9.141
gives us the first derivative
Taking the appropriate limit in equation 9.143
yields the second derivative
Case 2:
an integer,
and .
Both the numerator and denominator contain poles.
The zeroth derivative is simply
Taking the appropriate limit in equation 9.141
gives the first derivative
Taking the appropriate limit in equation 9.143
yields the second derivative
