Kelvin functions

Thomson functions

The functions $ \mathop{\rm ber} ( z) $ and $ \mathop{\rm bei} ( z) $, $ \mathop{\rm her} ( z) $ and $ \mathop{\rm hei} ( z) $, $ \mathop{\rm ker} ( z) $ and $ \mathop{\rm kei} ( z) $, defined by

$$ \mathop{\rm ber} _ \nu ( z) \pm \mathop{\rm bei} _ \nu ( z) = J _ \nu ( z e ^ {\pm 3 i \pi / 4 } ) , $$

$$ \mathop{\rm her} _ \nu ( z) + i \mathop{\rm hei} _ \nu ( z) = H _ \nu ^ {( 1)} ( z e ^ {3 i \pi / 4 } ) , $$

$$ \mathop{\rm her} _ \nu ( z) - i \mathop{\rm hei} _ \nu ( z) = H _ \nu ^ {( 2)} ( z e ^ {- 3 i \pi / 4 } ) , $$

$$ \mathop{\rm ker} _ \nu ( z) + i \mathop{\rm kei} _ \nu ( z) = \frac{i \pi }{2} H _ \nu ^ {( 1)} ( z e ^ {3 i \pi / 4 } ) , $$

$$ \mathop{\rm ker} _ \nu ( z) - i \mathop{\rm kei} _ \nu ( z) = - \frac{i \pi }{2} H _ \nu ^ {( 2)} ( z e ^ {- 3 i \pi / 4 } ) , $$

where the $ H _ \nu $ are the Hankel functions and the $ J _ \nu $ are the Bessel functions. When $ \nu = 0 $ the index is omitted. The Kelvin functions form a fundamental system of solutions of the equation

$$ z ^ {2} y ^ {\prime\prime} + z y ^ \prime - ( i z ^ {2} + \nu ^ {2} ) y = 0 , $$

which for $ z = \sqrt i x $ turns into the Bessel equation.

The series representations are:

$$ \mathop{\rm ber} ( z) = \ \sum _ {k=0}^ \infty \frac{( - 1 ) ^ {k} z ^ {4k} }{2 ^ {4k} [ ( 2 k ) ! ] ^ {2} } , $$

$$ \mathop{\rm ber} ( z) = \sum _ {k=0} ^ \infty \frac{( - 1 ) ^ {k} z ^ {4k+} 2 }{2 ^ {4k+} 2 [ ( 2 k + 1 ) ! ] ^ {2} } , $$

$$ \mathop{\rm ker} ( z) = \left ( \mathop{\rm ln} \frac{2}{z} - C \right ) \mathop{\rm ber} ( z) + \frac \pi {4} \mathop{\rm bei} ( z) + $$

$$ + \sum _ {k=0} ^ \infty ( - 1 ) ^ {k} \frac{z ^ {4k} }{ 2 ^ {4k} [ ( 2 k ) ! ] ^ {2} } \sum _ { m= 1} ^ { 2k } \frac{1}{m} , $$

$$ \mathop{\rm kei} ( z) = \left ( \mathop{\rm ln} \frac{2}{z} - C \right ) \mathop{\rm bei} ( z) - \frac \pi {4} \mathop{\rm ber} ( z) + $$

$$ + \sum _ {k=0}^ \infty ( - 1 ) ^ {k} \frac{z ^ {4k+} 2 }{2 ^ {4k+} 2 [ ( 2 k + 1 ) ! ] ^ {2} } \sum _ { m= 1} ^ { 2k+ 1 } \frac{1}{m} . $$

The asymptotic representations are:

$$ \mathop{\rm ber} ( z) = \ \frac{e ^ {\alpha ( z) } }{\sqrt {2 \pi z } } \ \cos \beta ( z) , $$

$$ \mathop{\rm ber} ( z) = \frac{e ^ {\alpha ( z) } }{ \sqrt {2 \pi z } } \sin \beta ( z) , $$

$$ \mathop{\rm ker} ( z) = \sqrt { \frac \pi {2z} } e ^ {\alpha ( - z ) } \cos \beta ( - z ) , $$

$$ \mathop{\rm kei} ( z) = \sqrt { \frac \pi {2z} } e ^ {\alpha ( - z ) } \sin \beta ( - z ) , $$

$$ | \mathop{\rm arg} z | < \frac{5}{4} \pi , $$

where

$$ \alpha ( z) \sim \ \frac{z}{\sqrt 2 } + \frac{1}{8 z \sqrt 2 } - \frac{25}{384 z ^ {3} \sqrt 2 } - \frac{13}{128 z ^ {4} } - \dots , $$

$$ \beta ( z) \sim \frac{z}{\sqrt 2} - \frac \pi {8} - \frac{1}{8 z \sqrt 2 } - \frac{1}{384 z ^ {3} \sqrt 2 } + \dots . $$

These functions were introduced by W. Thomson (Lord Kelvin, [1]).

References

[1]	W. Thomson, "Mathematical and physical papers" , 3 , Cambridge Univ. Press (1980) pp. 492
[2]	E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen" , Teubner (1966)
[3]	I.S. Gradshtein, I.M. Ryzhik, "Table of integrals, series and products" , Acad. Press (1973) (Translated from Russian)
[a1]	M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1965)