Source: https://dlmf.nist.gov/13.14
Timestamp: 2019-04-23 15:11:02+00:00

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See Olver (1997b, Chapter 7, §§9–11) and Buchholz (1969, §2.3a) . For (13.14.8) and (13.14.9) take limiting values in (13.14.33), using (13.14.2) and (13.2.2). See also Slater (1960, §1.7.1) .
This equation is obtained from Kummer’s equation (13.2.1) via the substitutions W=e-12⁢z⁢z12+μ⁢w, κ=12⁢b-a, and μ=12⁢b-12. It has a regular singularity at the origin with indices 12±μ, and an irregular singularity at infinity of rank one.
except that Mκ,μ⁡(z) does not exist when 2⁢μ=-1,-2,-3,….
In general Mκ,μ⁡(z) and Wκ,μ⁡(z) are many-valued functions of z with branch points at z=0 and z=∞. The principal branches correspond to the principal branches of the functions z12+μ and U⁡(12+μ-κ,1+2⁢μ,z) on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i).
See Temme (1996b, §7.2) . See also Olver (1997b, Chapter 7, Ex. 11.2) .
In (13.14.11)–(13.14.13) m is any integer.
Except when z=0, each branch of the functions Mκ,μ⁡(z)/Γ⁡(2⁢μ+1) and Wκ,μ⁡(z) is entire in κ and μ. Also, unless specified otherwise Mκ,μ⁡(z) and Wκ,μ⁡(z) are assumed to have their principal values.
Combine §13.2(iii) and (13.14.4), (13.14.5).
For Wκ,μ⁡(z) with ℜ⁡μ<0 use (13.14.31).
See Olver (1997b, Chapter 7, §§9–11) .
When 2⁢μ is an integer we may use the results of §13.2(v) with the substitutions b=2⁢μ+1, a=μ-κ+12, and W=e-12⁢z⁢z12+μ⁢w, where W is the solution of (13.14.1) corresponding to the solution w of (13.2.1).
See Slater (1960, §2.4.2) .
For (13.14.32) and (13.14.33) combine (13.2.41) and (13.2.42) with (13.14.4) and (13.14.5). (13.14.31) follows from (13.14.33).

References: §2
 §1
 §4
 §7
 §13
 §13
 §2