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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.22 Zeros

From (13.14.2) and (13.14.3) Mκ,μ(z) has the same zeros as M(12+μ-κ,1+2μ,z) and Wκ,μ(z) has the same zeros as U(12+μ-κ,1+2μ,z), hence the results given in §13.9 can be adopted.

Asymptotic approximations to the zeros when the parameters κ and/or μ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. For example, if μ(0) is fixed and κ(>0) is large, then the rth positive zero ϕr of Mκ,μ(z) is given by

13.22.1 ϕr=j2μ,r24κ+j2μ,rO(κ-32),

where j2μ,r is the rth positive zero of the Bessel function J2μ(x)10.21(i)). (13.22.1) is a weaker version of (13.9.8).