The general values of the incomplete gamma functions and are defined by
without restrictions on the integration paths. However, when the integration paths do not cross the negative real axis, and in the case of (8.2.2) exclude the origin, and take their principal values; compare §4.2(i). Except where indicated otherwise in the DLMF these principal values are assumed. For example,
Normalized functions are:
In this subsection the functions and have their general values.
The function is entire in and . When , is an entire function of , and is meromorphic with simple poles at , , with residue .
(8.2.9) also holds when is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. For example, in the case we have
without restriction on .
If or , then
If , then