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''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, [[#References|[1]]]).

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