Equations of the form
and is any linear combination of the modified Bessel functions and (§10.25(ii)).
when is real and positive, and by analytic continuation elsewhere. (All solutions of (9.13.1) are entire functions of .) When and become and , respectively.
Properties of and follow from the corresponding properties of the modified Bessel functions. They include:
where For real variables the solutions of (9.13.13) are denoted by , when is even, and by , when is odd. (The overbar has nothing to do with complex conjugates.) Their relations to the functions and are given by
Properties and graphs of , , are included in Olver (1977a) together with properties and graphs of real solutions of the equation
which are denoted by , .
The function on the right-hand side is recessive in the sector , and is therefore an essential member of any numerically satisfactory pair of solutions in this region.
where and . Solutions are , , where
and denotes the Bessel function (§10.2(ii)).
When is a positive integer the relation of these functions to , is as follows:
For properties of the zeros of the functions defined in this subsection see Laforgia and Muldoon (1988) and references given therein.
Each of the functions and satisfies the differential equation
and the difference equation
The are related by
Connection formulas for the solutions of (9.13.31) include
Further properties of these functions, and also of similar contour integrals containing an additional factor , in the integrand, are derived in Reid (1972), Drazin and Reid (1981, Appendix), and Baldwin (1985). These properties include Wronskians, asymptotic expansions, and information on zeros.