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''Thomson functions''
 
''Thomson functions''
  
The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k0551801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k0551802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k0551803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k0551804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k0551805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k0551806.png" />, defined by
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k0551807.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ber} _  \nu  ( z) \pm  \mathop{\rm bei} _  \nu  ( z)  = J _  \nu  ( z e ^ {\pm  3 i \pi / 4 } ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k0551808.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm her} _  \nu  ( z) + i  \mathop{\rm hei} _  \nu  ( z)  = H _  \nu  ^ {( 1)} ( z e ^ {3 i \pi / 4 } ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k0551809.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm her} _  \nu  ( z) - i  \mathop{\rm hei} _  \nu  ( z)  = H _  \nu  ^ {( 2)} ( z e ^ {- 3 i \pi / 4 } ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518010.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ker} _  \nu  ( z) + i  \mathop{\rm kei} _  \nu  ( z)  =
 +
\frac{i \pi }{2} H _  \nu  ^ {( 1)} ( z e ^ {3 i \pi / 4 } ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518011.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ker} _  \nu  ( z) - i  \mathop{\rm kei} _  \nu  ( z)  = -
 +
\frac{i \pi }{2} H _  \nu  ^ {( 2)} ( z e ^ {- 3 i \pi / 4 } ) ,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518012.png" /> are the [[Hankel functions|Hankel functions]] and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518013.png" /> are the [[Bessel functions|Bessel functions]]. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518014.png" /> the index is omitted. The Kelvin functions form a [[Fundamental system of solutions|fundamental system of solutions]] of the equation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518015.png" /></td> </tr></table>
+
$$
 +
z  ^ {2} y  ^ {\prime\prime} + z y  ^  \prime  - ( i z  ^ {2} + \nu  ^ {2} ) y  = 0 ,
 +
$$
  
which for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518016.png" /> turns into the Bessel equation.
+
which for $  z = \sqrt i x $
 +
turns into the Bessel equation.
  
 
The series representations are:
 
The series representations are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518017.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ber} ( z)  = \
 +
\sum _ {k=0}^  \infty 
 +
 
 +
\frac{( - 1 )  ^ {k} z  ^ {4k} }{2  ^ {4k} [ ( 2 k ) ! ]  ^ {2} }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518018.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ber} ( z)  = \sum _ {k=0} ^  \infty 
 +
\frac{( - 1 )  ^ {k} z  ^ {4k+} 2 }{2  ^ {4k+} 2 [ ( 2 k + 1 ) ! ]  ^ {2} }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518019.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ker} ( z)  = \left (  \mathop{\rm ln} 
 +
\frac{2}{z}
 +
- C \right
 +
)  \mathop{\rm ber} ( z) +
 +
\frac \pi {4}
 +
  \mathop{\rm bei} ( z) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518020.png" /></td> </tr></table>
+
$$
 +
+
 +
\sum _ {k=0} ^  \infty  ( - 1 )  ^ {k}
 +
\frac{z  ^ {4k} }{
 +
2  ^ {4k} [ ( 2 k ) ! ]  ^ {2} }
 +
\sum _ { m= 1} ^ { 2k } 
 +
\frac{1}{m}
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518021.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm kei} ( z)  = \left (  \mathop{\rm ln} 
 +
\frac{2}{z}
 +
- C \right
 +
)  \mathop{\rm bei} ( z) -  
 +
\frac \pi {4}
 +
  \mathop{\rm ber} ( z) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518022.png" /></td> </tr></table>
+
$$
 +
+
 +
\sum _ {k=0}^  \infty  ( - 1 )  ^ {k}
 +
\frac{z  ^ {4k+} 2 }{2  ^ {4k+} 2 [ ( 2 k + 1 ) ! ]  ^ {2} }
 +
\sum _ { m= 1} ^ { 2k+  1 }
 +
\frac{1}{m}
 +
.
 +
$$
  
 
The asymptotic representations are:
 
The asymptotic representations are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518023.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ber} ( z)  = \
 +
 
 +
\frac{e ^ {\alpha ( z) } }{\sqrt {2 \pi z } }
 +
\
 +
\cos  \beta ( z) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518024.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ber} ( z)  =
 +
\frac{e ^ {\alpha ( z) } }{
 +
\sqrt {2 \pi z } }
 +
  \sin  \beta ( z) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518025.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ker} ( z)  = \sqrt {
 +
\frac \pi {2z}
 +
} e ^ {\alpha (
 +
- z ) }  \cos  \beta ( - z ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518026.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm kei} ( z)  = \sqrt {
 +
\frac \pi {2z}
 +
} e ^ {\alpha (
 +
- z ) }  \sin  \beta ( - z ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518027.png" /></td> </tr></table>
+
$$
 +
|  \mathop{\rm arg}  z |  <
 +
\frac{5}{4}
 +
\pi ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518028.png" /></td> </tr></table>
+
$$
 
+
\alpha ( z)  \sim \
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055180/k05518029.png" /></td> </tr></table>
 
  
These functions were introduced by W. Thomson (Lord Kelvin, [[#References|[1]]]).
+
\frac{z}{\sqrt 2 }
 +
+
  
====References====
+
\frac{1}{8 z \sqrt 2 }
<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>
+
  -
  
 +
\frac{25}{384 z  ^ {3} \sqrt 2 }
 +
-
  
 +
\frac{13}{128 z  ^ {4} }
 +
- \dots ,
 +
$$
  
====Comments====
+
$$
 +
\beta ( z)  \sim 
 +
\frac{z}{\sqrt 2}
 +
-
 +
\frac \pi {8}
 +
-
 +
\frac{1}{8 z
 +
\sqrt 2 }
 +
-
 +
\frac{1}{384 z  ^ {3} \sqrt 2 }
 +
+ \dots .
 +
$$
  
 +
These functions were introduced by W. Thomson (Lord Kelvin, [[#References|[1]]]).
  
 
====References====
 
====References====
<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>
+
<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>

Latest revision as of 08:25, 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=15392
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article