Macdonald function

modified cylinder function, Bessel function of imaginary argument

A function

where is an arbitrary non-integral real number and

is a cylinder function with pure imaginary argument (cf. Cylinder functions). They have been discussed by H.M. Macdonald [1]. If is an integer, then

The Macdonald function is the solution of the differential equation

(*)

that tends exponentially to zero as and takes positive values. The functions and form a fundamental system of solutions of (*).

For , has roots only when . If , then the number of roots in these two sectors is equal to the even number nearest to , provided that is not an integer; in the latter case the number of roots is equal to . For there are no roots if is not an integer.

Series and asymptotic representations are:

where is a non-negative integer;

where is the Euler constant;

where is an integer;

for large and .

Recurrence formulas:

References

[1]	H.M. Macdonald, "Zeroes of the Bessel functions" Proc. London Math. Soc. , 30 (1899) pp. 165–179
[2]	G.N. Watson, "A treatise on the theory of Bessel functions" , 1–2 , Cambridge Univ. Press (1952)