Solutions of this equation are the Scorer functions and can be found by the method of variation of parameters (§1.13(iii)). The general solution is given by
and are entire functions of .
is a numerically satisfactory companion to the complementary functions and on the interval . is a numerically satisfactory companion to and on the interval .
In , numerically satisfactory sets of solutions are given by
If or , and is the modified Bessel function (§10.25(ii)), then
where the last integral is a Cauchy principal value (§1.4(v)).
where the integration contour separates the poles of from those of .
As , and with denoting an arbitrary small positive constant,
where is Euler’s constant (§5.2(ii)).
All zeros, real or complex, of and are simple.
Neither nor has real zeros.
has no nonnegative real zeros and has exactly one nonnegative real zero, given by . Both and have an infinity of negative real zeros, and they are interlaced.
For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c).