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''modified cylinder function, Bessel function of imaginary argument''
 
''modified cylinder function, Bessel function of imaginary argument''
  
 
A function
 
A function
  
<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/m/m062/m062010/m0620101.png" /></td> </tr></table>
+
$$
 +
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 ) }
 +
 
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m0620102.png" /> is an arbitrary non-integral real number and
+
is a cylinder function with pure imaginary argument (cf. [[Cylinder functions|Cylinder functions]]). They have been discussed by H.M. Macdonald [[#References|[1]]]. If  $  n $
 +
is an integer, then
  
<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/m/m062/m062010/m0620103.png" /></td> </tr></table>
+
$$
 +
K _ {n} ( z)  = \lim\limits _ {\nu \rightarrow n }  K _  \nu  ( z) .
 +
$$
  
is a cylinder function with pure imaginary argument (cf. [[Cylinder functions|Cylinder functions]]). They have been discussed by H.M. Macdonald [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m0620104.png" /> is an integer, then
+
The Macdonald function $  K _  \nu  ( z) $
 +
is the solution of the differential 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/m/m062/m062010/m0620105.png" /></td> </tr></table>
+
$$ \tag{* }
 +
z  ^ {2}
  
The Macdonald function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m0620106.png" /> is the solution of the differential equation
+
\frac{d  ^ {2} y }{d z  ^ {2} }
 +
+
 +
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/m/m062/m062010/m0620107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
\frac{d y }{d z }
 +
-
 +
( z  ^ {2} + \nu  ^ {2} ) y  = 0
 +
$$
  
that tends exponentially to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m0620108.png" /> and takes positive values. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m0620109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201010.png" /> form a [[Fundamental system of solutions|fundamental system of solutions]] of (*).
+
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|fundamental system of solutions]] of (*).
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201012.png" /> has roots only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201014.png" />, then the number of roots in these two sectors is equal to the even number nearest to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201015.png" />, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201016.png" /> is not an integer; in the latter case the number of roots is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201017.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201018.png" /> there are no roots if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201019.png" /> is not an integer.
+
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:
 
Series and 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/m/m062/m062010/m06201020.png" /></td> </tr></table>
+
$$
 +
K _ {n + 1 / 2 }  ( z)  = \
 +
\left (
 +
\frac \pi {2z}
 +
\right )  ^ {1/2} e  ^ {-} z \sum _ { r= } 0 ^ { n }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201021.png" /> is a non-negative integer;
+
\frac{( n + r ) ! }{r ! ( n - r ) ! ( 2 z )  ^ {r} }
 +
,
 +
$$
  
<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/m/m062/m062010/m06201022.png" /></td> </tr></table>
+
where  $  n $
 +
is a non-negative integer;
  
<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/m/m062/m062010/m06201023.png" /></td> </tr></table>
+
$$
 +
K _ {0} ( z)  = \
 +
- \mathop{\rm ln}  \left (
 +
\frac{z}{2}
 +
\right ) I _ {0} ( z) +
 +
\sum _ { m= } 0 ^  \infty 
 +
\left (
 +
\frac{z}{2}
 +
\right )  ^ {2m}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201024.png" /> is the Euler constant;
+
\frac{1}{( m ! )  ^ {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/m/m062/m062010/m06201025.png" /></td> </tr></table>
+
\psi ( m + 1 ) ,
 +
$$
  
<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/m/m062/m062010/m06201026.png" /></td> </tr></table>
+
$$
 +
\psi ( 1)  = - C ,\  \psi ( m + 1 )  = 1 +
 +
\frac{1}{2}
 +
+ \dots +
 +
\frac{1}{m}
 +
- C ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201027.png" /> is an integer;
+
where $  C = 0. 5772157 \dots $
 +
is the Euler constant;
  
<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/m/m062/m062010/m06201028.png" /></td> </tr></table>
+
$$
 +
K _ {n} ( 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/m/m062/m062010/m06201029.png" /></td> </tr></table>
+
\frac{1}{2}
  
for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062010/m06201031.png" />.
+
\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:
 
Recurrence formulas:
  
<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/m/m062/m062010/m06201032.png" /></td> </tr></table>
+
$$
 +
K _ {\nu - 1 }  ( z) - K _ {\nu + 1 }  ( z)  = -
 +
 
 +
\frac{2 \nu }{z}
 +
K _  \nu  ( 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/m/m062/m062010/m06201033.png" /></td> </tr></table>
+
$$
 +
K _ {\nu - 1 }  ( z) + K _ {\nu + 1 }  ( z)  = - 2
 +
\frac{d K _  \nu  ( z) }{d z }
 +
.
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.M. Macdonald,  "Zeroes of the Bessel functions"  ''Proc. London Math. Soc.'' , '''30'''  (1899)  pp. 165–179</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.N. Watson,  "A treatise on the theory of Bessel functions" , '''1–2''' , Cambridge Univ. Press  (1952)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.M. Macdonald,  "Zeroes of the Bessel functions"  ''Proc. London Math. Soc.'' , '''30'''  (1899)  pp. 165–179</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.N. Watson,  "A treatise on the theory of Bessel functions" , '''1–2''' , Cambridge Univ. Press  (1952)</TD></TR></table>

Revision as of 04:11, 6 June 2020


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)
How to Cite This Entry:
Macdonald function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Macdonald_function&oldid=47744
This article was adapted from an original article by V.I. Pagurova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article