Consider differentiating the integral 
with respect to t Theorem D.2 generalizes theorem 9.42 of [73] and establishes conditions general enough to allow the commutation of the derivative and integral for the functions appearing in this paper. Define 
to be the partial derivative of F with respect to its second argument, evaluated at (x,t).
: If
(1) F(x,t) and 
are defined for 
, where 
, and where 
is convex,
(2) 
exists 
,
(3) 
and b>0, 
with f(x)>0 for 
, and 
such that
and 
, 
, 
,
then 
on 
.
Proof: Let 
for 
. By (1) and the
mean value theorem, 
with 
, 
such that 
. Using this and (3) we have that
for any 
, 
, 
and a nowhere-negative
(in 
) f(x) obeying such
that if 
and 
, then 
. From this and (2) it follows that for all
b>0, , and a nowhere-negative (in ) f(x) obeying 
such that if , then 
. Taking the limit
, noting that 
, and finally taking
with 
, we arrive at the desired result. QED.
The functions F(x,t) of interest have the form 
with Re(t)>-1 and c>0. For these functions it may be shown that the conditions of theorem D.2 hold.