modified cylinder function, Bessel function of imaginary argument
where is an arbitrary non-integral real number and
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 .
|||H.M. Macdonald, "Zeroes of the Bessel functions" Proc. London Math. Soc. , 30 (1899) pp. 165–179|
|||G.N. Watson, "A treatise on the theory of Bessel functions" , 1–2 , Cambridge Univ. Press (1952)|
Macdonald function. V.I. Pagurova (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Macdonald_function&oldid=19172