10.2.2 | |||
This solution of (10.2.1) is an analytic function of , except for a branch point at when is not an integer. The principal branch of corresponds to the principal value of (§4.2(iv)) and is analytic in the -plane cut along the interval .
When , is entire in .
For fixed each branch of is entire in .
10.2.3 | |||
When is an integer the right-hand side is replaced by its limiting value:
10.2.4 | |||
. | |||
Whether or not is an integer has a branch point at . The principal branch corresponds to the principal branches of in (10.2.3) and (10.2.4), with a cut in the -plane along the interval .
Except in the case of , the principal branches of and are two-valued and discontinuous on the cut ; compare §4.2(i).
Both and are real when is real and .
For fixed each branch of is entire in .
These solutions of (10.2.1) are denoted by and , and their defining properties are given by
10.2.5 | |||
as in , and
10.2.6 | |||
as in , where is an arbitrary small positive constant. Each solution has a branch point at for all . The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the -plane along the interval .
The principal branches of and are two-valued and discontinuous on the cut .
For fixed each branch of and is entire in .
Except where indicated otherwise, it is assumed throughout the DLMF that the symbols , , , and denote the principal values of these functions.
The notation denotes , , , , or any nontrivial linear combination of these functions, the coefficients in which are independent of and .
Table 10.2.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.2.1) for the stated intervals or regions in the case . When , is replaced by throughout.
Pair | Interval or Region |
---|---|
neighborhood of 0 in | |
neighborhood of in |