Start with the identity
for the gamma function. Make the change of variables 
. With 
, independent of t, we find
Define the operator 
by
and define the transform T by 
. As defined, the transform T is closely related to the Mellin transform [54] (it is an inverse-Mellin transform) and this similarity may be used to establish the conditions for the existence of the transform and its inverse. Of interest in this work is the following: For 
independent of t, the functions 
and 
form a transform pair. That is,
When using the T transform, the order of evaluation of the integral over t and other integrals will need to be interchanged. See appendix 9.7.1 for the justification of the commutation of the integrals.