Macdonald function

modified cylinder function, Bessel function of imaginary argument

A function

$$ K _ \nu ( z) = \frac \pi {2} \frac{I _ {- \nu } ( z) - I _ \nu ( z) }{\sin \nu \pi } , $$

where $ \nu $ is an arbitrary non-integral real number and

$$ I _ \nu ( z) = \ \sum _ {m=0}^ \infty \frac{\left ( \frac{z}{2} \right ) ^ {\nu + 2 m } }{m ! \Gamma ( \nu + m + 1 ) } $$

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

$$ K _ {n} ( z) = \lim\limits _ {\nu \rightarrow n } K _ \nu ( z) . $$

The Macdonald function $ K _ \nu ( z) $ is the solution of the differential equation

$$ \tag{* } z ^ {2} \frac{d ^ {2} y }{d z ^ {2} } + z \frac{d y }{d z } - ( z ^ {2} + \nu ^ {2} ) y = 0 $$

that tends exponentially to zero as $ z \rightarrow \infty $ and takes positive values. The functions $ I _ \nu ( z) $ and $ K _ \nu ( z) $ form a fundamental system of solutions of (*).

For $ \nu \geq 0 $, $ K _ \nu ( z) $ has roots only when $ \mathop{\rm Re} z < 0 $. If $ \pi / 2 < | \mathop{\rm arg} z | < \pi $, then the number of roots in these two sectors is equal to the even number nearest to $ \nu - 1 / 2 $, provided that $ \nu - 1 / 2 $ is not an integer; in the latter case the number of roots is equal to $ \nu - 1 / 2 $. For $ \mathop{\rm arg} z = \pm \pi $ there are no roots if $ \nu - 1 / 2 $ is not an integer.

Series and asymptotic representations are:

$$ K _ {n + 1 / 2 } ( z) = \ \left ( \frac \pi {2z} \right ) ^ {1/2} e ^ {-} z \sum _ {r=0} ^ { n } \frac{( n + r ) ! }{r ! ( n - r ) ! ( 2 z ) ^ {r} } , $$

where $ n $ is a non-negative integer;

$$ K _ {0} ( z) = \ - \mathop{\rm ln} \left ( \frac{z}{2} \right ) I _ {0} ( z) + \sum _ {m=0}^ \infty \left ( \frac{z}{2} \right ) ^ {2m} \frac{1}{( m ! ) ^ {2} } \psi ( m + 1 ) , $$

$$ \psi ( 1) = - C ,\ \psi ( m + 1 ) = 1 + \frac{1}{2} + \dots + \frac{1}{m} - C , $$

where $ C = 0. 5772157 \dots $ is the Euler constant;

$$ K _ {n} ( z) = \ \frac{1}{2} \sum _ {m=0} ^ { n- } 1 \frac{( - 1 ) ^ {m} ( n - m - 1 ) ! }{m ! ( z / 2 ) ^ {n - 2 m } } + $$

$$ + ( - 1 ) ^ {n-} 1 \sum _ {m=0} ^ \infty \frac{( z / 2 ) ^ {n + 2 m } }{m ! ( n + m ) ! } \left \{ \mathop{\rm ln} \left ( \frac{z}{2} \right ) - \frac{\psi ( m + 1 ) - \psi ( n + m + 1 ) }{2} \right \} , $$

where $ n \geq 1 $ is an integer;

$$ K _ {\nu\ } \sim $$

$$ \sim \ \left ( \frac \pi {2z} \right ) ^ {1/2} e ^ {-} z \left [ 1 + \frac{ 4 \nu ^ {2} - 1 ^ {2} }{1 ! 8 z } + \frac{( 4 \nu ^ {2} - 1 ^ {2} ) ( 4 \nu ^ {2} - 3 ^ {2} ) }{2 ! ( 8 z ) ^ {2} } + \dots \right ] , $$

for large $ z $ and $ | \mathop{\rm arg} z | < \pi / 2 $.

Recurrence formulas:

$$ K _ {\nu - 1 } ( z) - K _ {\nu + 1 } ( z) = - \frac{2 \nu }{z} K _ \nu ( z) , $$

$$ K _ {\nu - 1 } ( z) + K _ {\nu + 1 } ( z) = - 2 \frac{d K _ \nu ( z) }{d z } . $$

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)