With and denoting here the general values of the incomplete gamma functions (§8.2(i)), we define
From §§8.2(i) and 8.2(ii) it follows that each of the four functions , , , and is a multivalued function of with branch point at . Furthermore, and are entire functions of , and and are meromorphic functions of with simple poles at and , respectively.
When (and when , in the case of , or , in the case of ) the principal values of , , , and are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). Elsewhere in the sector the principal values are defined by analytic continuation from ; compare §4.2(i).
From here on it is assumed that unless indicated otherwise the functions , , , and have their principal values.
Properties of the four functions that are stated below in §§8.21(iii) and 8.21(iv) follow directly from the definitions given above, together with properties of the incomplete gamma functions given earlier in this chapter. In the case of §8.21(iv) the equation
(obtained from (5.2.1) by rotation of the integration path) is also needed.
When and ,
When with (),