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The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097850/w0978501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097850/w0978502.png" /> which are solutions of the [[Whittaker equation|Whittaker equation]]
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<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/w/w097/w097850/w0978503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097850/w0978504.png" /> satisfies the equation
+
The functions  $  M _ {\lambda , \mu }  ( z) $
 +
and  $  W _ {\lambda , \mu }  ( z) $
 +
which are solutions of the [[Whittaker equation|Whittaker 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/w/w097/w097850/w0978505.png" /></td> </tr></table>
+
$$ \tag{* }
 +
w ^ {\prime\prime} +
 +
\left (
  
The pairs of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097850/w0978506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097850/w0978507.png" /> are linearly independent solutions of the equation (*). The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097850/w0978508.png" /> is a branching point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097850/w0978509.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097850/w09785010.png" /> is an essential singularity.
+
\frac{ {1 / 4 } - \mu  ^ {2} }{z  ^ {2} }
 +
+
 +
{
 +
\frac \lambda {z}
 +
} -
 +
{
 +
\frac{1}{4}
 +
}
 +
\right ) w = 0.
 +
$$
 +
 
 +
The function  $  W _ {\lambda , \mu }  $
 +
satisfies the equation
 +
 
 +
$$
 +
W _ {\lambda , \mu }  ( z)  = \
 +
 
 +
\frac{\Gamma (- 2 \mu ) }{\Gamma \left ( {
 +
\frac{1}{2}
 +
} - \lambda - \mu \right ) }
 +
 
 +
M _ {\lambda , \mu }  ( z) +
 +
 
 +
\frac{\Gamma ( 2 \mu ) }{\Gamma \left ( {
 +
\frac{1}{2}
 +
} - \lambda + \mu \right ) }
 +
 
 +
M _ {\lambda , - \mu }  ( z).
 +
$$
 +
 
 +
The pairs of functions  $  M _ {\lambda , \mu }  ( z) , M _ {\lambda , - \mu }  ( z) $
 +
and  $  W _ {\lambda , \mu }  ( z) , W _ {- \lambda , \mu }  ( z) $
 +
are linearly independent solutions of the equation (*). The point $  z = 0 $
 +
is a branching point for $  M _ {\lambda , \mu }  ( z) $,  
 +
and $  z = \infty $
 +
is an essential singularity.
  
 
Relation with other functions:
 
Relation with other functions:
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with the [[Degenerate hypergeometric function|degenerate hypergeometric function]]:
 
with the [[Degenerate hypergeometric function|degenerate hypergeometric 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/w/w097/w097850/w09785011.png" /></td> </tr></table>
+
$$
 +
M _ {\lambda , \mu }  ( z)  = \
 +
z ^ {\mu + 1/2 }
 +
e  ^ {-} z/2
 +
\Phi \left ( \mu - \lambda +
 +
\frac{1}{2}
 +
; \
 +
2 \mu + 1; z \right ) ,
 +
$$
  
 
with the modified [[Bessel functions|Bessel functions]] and the [[Macdonald function|Macdonald function]]:
 
with the modified [[Bessel functions|Bessel functions]] and the [[Macdonald function|Macdonald 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/w/w097/w097850/w09785012.png" /></td> </tr></table>
+
$$
 +
M _ {0, \mu }  ( z)  = \
 +
2 ^ {2 \mu } \Gamma ( \mu + 1)
 +
\sqrt z I _  \mu  \left ( {
 +
\frac{z}{2}
 +
} \right ) ,
 +
$$
  
<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/w/w097/w097850/w09785013.png" /></td> </tr></table>
+
$$
 +
W _ {0, \mu }  ( z)  = \sqrt {
 +
\frac{z} \pi
 +
} K _  \mu  \left ( {
 +
\frac{z}{2}
 +
} \right ) ;
 +
$$
  
 
with the [[Probability integral|probability integral]]:
 
with the [[Probability integral|probability integral]]:
  
<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/w/w097/w097850/w09785014.png" /></td> </tr></table>
+
$$
 +
W _ {- {1 / 4 }  , {1 / 4 } } ( z)  = \
 +
2 z  ^ {1/4} e  ^ {z/2}
 +
\mathop{\rm Erfc} ( \sqrt z );
 +
$$
  
 
with the [[Laguerre polynomials|Laguerre polynomials]]:
 
with the [[Laguerre polynomials|Laguerre polynomials]]:
  
<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/w/w097/w097850/w09785015.png" /></td> </tr></table>
+
$$
 +
W _ {n + \mu + 1/2, \mu }  ( z)  = \
 +
n! (- 1)  ^ {n} z ^ {\mu + 1/2 }
 +
e  ^ {-} z/2 L _ {n} ^ {2 \mu } ( z).
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The Whittaker function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097850/w09785016.png" /> can be expressed in terms of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097850/w09785017.png" />-function introduced in [[Confluent hypergeometric function|confluent hypergeometric function]]:
+
The Whittaker function $  W _ {\lambda , \mu }  $
 +
can be expressed in terms of the $  \Psi $-
 +
function introduced in [[Confluent hypergeometric function|confluent hypergeometric 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/w/w097/w097850/w09785018.png" /></td> </tr></table>
+
$$
 +
W _ {\lambda , \mu }  ( z)  = e ^ {- z/2 } z ^ {\mu + 1/2 } \Psi
 +
( \mu - \lambda + 1/2;  2 \mu + 1;  z).
 +
$$
  
 
Thus, the special cases discussed in [[Confluent hypergeometric function|confluent hypergeometric function]] can be rewritten as special cases for the Whittaker functions. See also the references given there.
 
Thus, the special cases discussed in [[Confluent hypergeometric function|confluent hypergeometric function]] can be rewritten as special cases for the Whittaker functions. See also the references given there.

Revision as of 08:29, 6 June 2020


The functions $ M _ {\lambda , \mu } ( z) $ and $ W _ {\lambda , \mu } ( z) $ which are solutions of the Whittaker equation

$$ \tag{* } w ^ {\prime\prime} + \left ( \frac{ {1 / 4 } - \mu ^ {2} }{z ^ {2} } + { \frac \lambda {z} } - { \frac{1}{4} } \right ) w = 0. $$

The function $ W _ {\lambda , \mu } $ satisfies the equation

$$ W _ {\lambda , \mu } ( z) = \ \frac{\Gamma (- 2 \mu ) }{\Gamma \left ( { \frac{1}{2} } - \lambda - \mu \right ) } M _ {\lambda , \mu } ( z) + \frac{\Gamma ( 2 \mu ) }{\Gamma \left ( { \frac{1}{2} } - \lambda + \mu \right ) } M _ {\lambda , - \mu } ( z). $$

The pairs of functions $ M _ {\lambda , \mu } ( z) , M _ {\lambda , - \mu } ( z) $ and $ W _ {\lambda , \mu } ( z) , W _ {- \lambda , \mu } ( z) $ are linearly independent solutions of the equation (*). The point $ z = 0 $ is a branching point for $ M _ {\lambda , \mu } ( z) $, and $ z = \infty $ is an essential singularity.

Relation with other functions:

with the degenerate hypergeometric function:

$$ M _ {\lambda , \mu } ( z) = \ z ^ {\mu + 1/2 } e ^ {-} z/2 \Phi \left ( \mu - \lambda + \frac{1}{2} ; \ 2 \mu + 1; z \right ) , $$

with the modified Bessel functions and the Macdonald function:

$$ M _ {0, \mu } ( z) = \ 2 ^ {2 \mu } \Gamma ( \mu + 1) \sqrt z I _ \mu \left ( { \frac{z}{2} } \right ) , $$

$$ W _ {0, \mu } ( z) = \sqrt { \frac{z} \pi } K _ \mu \left ( { \frac{z}{2} } \right ) ; $$

with the probability integral:

$$ W _ {- {1 / 4 } , {1 / 4 } } ( z) = \ 2 z ^ {1/4} e ^ {z/2} \mathop{\rm Erfc} ( \sqrt z ); $$

with the Laguerre polynomials:

$$ W _ {n + \mu + 1/2, \mu } ( z) = \ n! (- 1) ^ {n} z ^ {\mu + 1/2 } e ^ {-} z/2 L _ {n} ^ {2 \mu } ( z). $$

References

[1] H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[2] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952)

Comments

The Whittaker function $ W _ {\lambda , \mu } $ can be expressed in terms of the $ \Psi $- function introduced in confluent hypergeometric function:

$$ W _ {\lambda , \mu } ( z) = e ^ {- z/2 } z ^ {\mu + 1/2 } \Psi ( \mu - \lambda + 1/2; 2 \mu + 1; z). $$

Thus, the special cases discussed in confluent hypergeometric function can be rewritten as special cases for the Whittaker functions. See also the references given there.

How to Cite This Entry:
Whittaker functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whittaker_functions&oldid=49213
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article