ABSTRACT
R(z) = c1Iν(z) + c2Kν(z) (B.5) Iν(z) and Kν(z) are real and positive when ν > −1 and z > 0. The Bessel function Iν(z) in series form is given by
Iν(z) = (
1 2 z
[(1 / 2)z]2k k!Γ(ν + k + 1) (B.6)
When ν is neither zero nor a positive integer, the general solutions B.2 and B.5 can be taken, respectively, in the form
R(z) = c1Jν(z) + c2J−ν(z) (B.7a) R(z) = c1Iν(z) + c2I−ν(z) (B.7b)
T&F Cat # K10695, Appendix B, Page 484, 12-6-2010
First Ten Roots of Jn(z) = 0; n = 0, 1, 2, 3, 4, 5
When ν = n is a positive integer, the solutionsJn(z) andJ−n(z) are not independent (see Tables B.1 through B.5); they are related by
Jn(z) = (−1)nJ−n(z) and J−n(z) = Jn(−z) (B.8) (n = integer). Similarly, when ν = n is a positive integer, the solutions In(z) and I−n(z) are not independent.
