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$$  
 
$$  
  \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  ^ {( 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 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)  =   
 
  \mathop{\rm ker} _  \nu  ( z) + i  \mathop{\rm kei} _  \nu  ( z)  =   
\frac{i \pi }{2}
+
\frac{i \pi }{2} H _  \nu  ^ {( 1)} ( z e ^ {3 i \pi / 4 } ) ,
 
 
H _  \nu  ^ {(} 1) ( z e ^ {3 i \pi / 4 } ) ,
 
 
$$
 
$$
  
 
$$  
 
$$  
 
  \mathop{\rm ker} _  \nu  ( z) - i  \mathop{\rm kei} _  \nu  ( z)  =  -  
 
  \mathop{\rm ker} _  \nu  ( z) - i  \mathop{\rm kei} _  \nu  ( z)  =  -  
\frac{i \pi }{2}
+
\frac{i \pi }{2} H _  \nu  ^ {( 2)} ( z e ^ {- 3 i \pi / 4 } ) ,
 
 
H _  \nu  ^ {(} 2) ( z e ^ {- 3 i \pi / 4 } ) ,
 
 
$$
 
$$
  
 
where the  $  H _  \nu  $
 
where the  $  H _  \nu  $
are the [[Hankel functions|Hankel functions]] and the  $  J _  \nu  $
+
are the [[Hankel functions]] and the  $  J _  \nu  $
are the [[Bessel functions|Bessel functions]]. When  $  \nu = 0 $
+
are the [[Bessel functions]]. When  $  \nu = 0 $
the index is omitted. The Kelvin functions form a [[Fundamental system of solutions|fundamental system of solutions]] of the equation
+
the index is omitted. The Kelvin functions form a [[fundamental system of solutions]] of the equation
  
 
$$  
 
$$  
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$$  
 
$$  
 
  \mathop{\rm ber} ( z)  = \  
 
  \mathop{\rm ber} ( z)  = \  
\sum _ { k= } 0 ^  \infty   
+
\sum _ {k=0}^  \infty   
  
 
\frac{( - 1 )  ^ {k} z  ^ {4k} }{2  ^ {4k} [ ( 2 k ) ! ]  ^ {2} }
 
\frac{( - 1 )  ^ {k} z  ^ {4k} }{2  ^ {4k} [ ( 2 k ) ! ]  ^ {2} }
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$$  
 
$$  
  \mathop{\rm ber} ( z)  =  \sum _ { k= } 0 ^  \infty   
+
  \mathop{\rm ber} ( z)  =  \sum _ {k=0} ^  \infty   
 
\frac{( - 1 )  ^ {k} z  ^ {4k+} 2 }{2  ^ {4k+} 2 [ ( 2 k + 1 ) ! ]  ^ {2} }
 
\frac{( - 1 )  ^ {k} z  ^ {4k+} 2 }{2  ^ {4k+} 2 [ ( 2 k + 1 ) ! ]  ^ {2} }
 
  ,
 
  ,
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$$  
 
$$  
 
+  
 
+  
\sum _ { k= } 0 ^  \infty  ( - 1 )  ^ {k}  
+
\sum _ {k=0} ^  \infty  ( - 1 )  ^ {k}  
 
\frac{z  ^ {4k} }{
 
\frac{z  ^ {4k} }{
 
2  ^ {4k} [ ( 2 k ) ! ]  ^ {2} }
 
2  ^ {4k} [ ( 2 k ) ! ]  ^ {2} }
Line 105: Line 101:
 
$$  
 
$$  
 
+  
 
+  
\sum _ { k= } 0 ^  \infty  ( - 1 )  ^ {k}  
+
\sum _ {k=0}^  \infty  ( - 1 )  ^ {k}  
 
\frac{z  ^ {4k+} 2 }{2  ^ {4k+} 2 [ ( 2 k + 1 ) ! ]  ^ {2} }
 
\frac{z  ^ {4k+} 2 }{2  ^ {4k+} 2 [ ( 2 k + 1 ) ! ]  ^ {2} }
 
  \sum _ { m= } 1 ^ { 2k+ }  1  
 
  \sum _ { m= } 1 ^ { 2k+ }  1  
Line 183: Line 179:
  
 
====References====
 
====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></table>
+
<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>
====Comments====
+
<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>
====References====
+
</table>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Abramowitz,  I.A. Stegun,  "Handbook of mathematical functions" , Dover, reprint  (1965)</TD></TR></table>
 

Revision as of 08:24, 6 January 2024


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)
How to Cite This Entry:
Kelvin functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kelvin_functions&oldid=47484
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article