11.2.1 | ||||
11.2.2 | ||||
Principal values correspond to principal values of ; compare §4.2(i).
The expansions (11.2.1) and (11.2.2) are absolutely convergent for all finite values of . The functions and are entire functions of and .
11.2.3 | ||||
11.2.4 | ||||
11.2.5 | ||||
11.2.6 | ||||
Principal values of and correspond to principal values of the functions on the right-hand sides of (11.2.5) and (11.2.6).
Unless indicated otherwise, , , , and assume their principal values throughout the DLMF.
11.2.7 | |||
Particular solutions:
11.2.8 | |||
11.2.9 | |||
Particular solutions:
11.2.10 | |||
In this subsection and are arbitrary constants.
When , , and , numerically satisfactory general solutions of (11.2.7) are given by
11.2.11 | ||||
11.2.12 | ||||
(11.2.11) applies when is bounded, and (11.2.12) applies when is bounded away from the origin.