# Macdonald function

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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 . 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:  How to Cite This Entry:
Macdonald function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Macdonald_function&oldid=19172
This article was adapted from an original article by V.I. Pagurova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article