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''Weber function, Weber–Hermite function''
 
''Weber function, Weber–Hermite function''
  
 
A solution of the differential equation
 
A 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/p/p071/p071190/p0711901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
 
 +
\frac{d  ^ {2} y }{dz  ^ {2} }
 +
+ \left ( \nu +
 +
\frac{1}{2}
 +
-  
 +
\frac{z  ^ {2} }{4}
 +
\right ) y
 +
= 0,
 +
$$
 +
 
 +
which is obtained as a result of separating the variables in the [[Wave equation|wave equation]]  $  \Delta u = k  ^ {2} u $
 +
in parabolic cylindrical coordinates (cf. [[Parabolic coordinates|Parabolic coordinates]]). Ordinarily one uses the solution
 +
 
 +
$$
 +
D _  \nu  ( z)  \equiv  U \left ( - \nu -  
 +
\frac{1}{2}
 +
, z \right )  = \
 +
2 ^ {( \nu - 1)/2 } e ^ {- z  ^ {2} /4 } \Psi \left (
 +
\frac{1 - \nu }{2}
 +
,
 +
\frac{3}{2}
 +
; \
 +
 
 +
\frac{z  ^ {2} }{2}
 +
\right ) ,
 +
$$
  
which is obtained as a result of separating the variables in the [[Wave equation|wave equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p0711902.png" /> in parabolic cylindrical coordinates (cf. [[Parabolic coordinates|Parabolic coordinates]]). Ordinarily one uses the solution
+
where  $  \Psi ( a, b;  z) $
 +
is the [[Confluent hypergeometric function|confluent hypergeometric function]]. The functions  $  D _  \nu  (- z) $
 +
and  $  D _ {- \nu - 1 }  (\pm  iz) $
 +
also satisfy equation (*). The functions  $  D _  \nu  ( z) $
 +
and  $  D _ {- \nu - 1 }  (\pm  iz) $
 +
are linearly independent for arbitrary  $  \nu $,
 +
and  $  D _  \nu  ( z) $
 +
and  $  D _  \nu  (- z) $
 +
are linearly independent for  $  \nu \neq 0, \pm  1 , .  . . $.  
 +
The parabolic cylinder functions are entire functions of  $  z $.
 +
The function  $  D _  \nu  ( z) $
 +
is real for real  $  \nu $
 +
and  $  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/p/p071/p071190/p0711903.png" /></td> </tr></table>
+
The differentiation formulas are,  $  n = 1, 2 ,\dots $:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p0711904.png" /> is the [[Confluent hypergeometric function|confluent hypergeometric function]]. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p0711905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p0711906.png" /> also satisfy equation (*). The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p0711907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p0711908.png" /> are linearly independent for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p0711909.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119011.png" /> are linearly independent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119012.png" />. The parabolic cylinder functions are entire functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119013.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119014.png" /> is real for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119016.png" />.
+
$$
  
The differentiation formulas are, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119017.png" />:
+
\frac{d  ^ {n} }{dz  ^ {n} }
 +
\left [ e ^ {z  ^ {2} /4 } D _  \nu  ( z) \right ]  = \
 +
(- 1)  ^ {n} (- \nu ) _ {n} e ^ {z  ^ {2} /4 } D _ {\nu - 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/p/p071/p071190/p07119018.png" /></td> </tr></table>
+
$$
  
<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/p/p071/p071190/p07119019.png" /></td> </tr></table>
+
\frac{d  ^ {n} }{dz  ^ {n} }
 +
\left [ e ^ {- z  ^ {2} /4 } D _  \nu  ( z)
 +
\right ]  = (- 1)  ^ {n} e ^ {- z  ^ {2} /4 } D _ {\nu + n }  ( z) .
 +
$$
  
 
The recurrence formulas are:
 
The recurrence formulas 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/p/p071/p071190/p07119020.png" /></td> </tr></table>
+
$$
 +
D _ {\nu + 1 }  ( z) - zD _  \nu  ( z) + \nu D _ {\nu - 1 }  ( z)  = 0,
 +
$$
 +
 
 +
$$
 +
D _  \nu  ^  \prime  ( z) +
 +
\frac{z}{2}
 +
D _  \nu  ( z) - \nu D _ {\nu - 1 }  ( z)  = 0,
 +
$$
  
<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/p/p071/p071190/p07119021.png" /></td> </tr></table>
+
$$
 +
D _  \nu  ^  \prime  ( z) -  
 +
\frac{z}{2}
 +
D _  \nu  ( z) + D _ {\nu + 1 }  ( z)  = 0.
 +
$$
  
<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/p/p071/p071190/p07119022.png" /></td> </tr></table>
+
Asymptotically, for fixed  $  \nu $
 +
and  $  |  \mathop{\rm arg}  z | < 3 \pi /4 $,
 +
as  $  z \rightarrow \infty $,
 +
one has
  
Asymptotically, for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119024.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119025.png" />, one has
+
$$
 +
D _  \nu  ( z)  = \
 +
z  ^  \nu  e ^ {- z  ^ {2} /4 } \left [ \sum _ { k= } 0 ^ { N } 
 +
\frac{(- \nu /2) _ {k} ( 1/2
 +
- \nu /2) _ {k} }{k!}
 +
\left (
 +
\frac{z  ^ {2} }{-}
 +
2 \right )  ^ {-} k\right . +
 +
$$
  
<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/p/p071/p071190/p07119026.png" /></td> </tr></table>
+
$$
 +
+ \left .
 +
O ( | z | ^ {- 2N- 2 } )  \right ] ,
 +
$$
  
<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/p/p071/p071190/p07119027.png" /></td> </tr></table>
+
and, for bounded  $  | z | $
 +
and  $  |  \mathop{\rm arg} ( - \nu ) | \leq  \pi /2 $,
 +
as  $  | \nu | \rightarrow \infty $,
 +
one has
  
and, for bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119029.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119030.png" />, one has
+
$$
 +
D _  \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/p/p071/p071190/p07119031.png" /></td> </tr></table>
+
\frac{1}{\sqrt 2 }
 +
  \mathop{\rm exp} \left [
 +
\frac \nu {2}
 +
  \mathop{\rm ln} (- \nu ) -
 +
\frac \nu {2}
 +
- \sqrt {- \nu } z
 +
\right ] \times
 +
$$
  
<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/p/p071/p071190/p07119032.png" /></td> </tr></table>
+
$$
 +
\times
 +
\left [ 1 + O \left (
 +
\frac{1}{\sqrt {| \nu | } }
 +
\right )  \right ] .
 +
$$
  
Parabolic cylinder functions are related to other functions as follows (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119033.png" />): To the [[Hermite polynomials|Hermite polynomials]] by
+
Parabolic cylinder functions are related to other functions as follows ( $  n = 0, 1 , . . . $):  
 +
To the [[Hermite polynomials|Hermite polynomials]] 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/p/p071/p071190/p07119034.png" /></td> </tr></table>
+
$$
 +
D _ {n} ( z)  = \
 +
2  ^ {-} n/2 e ^ {- z  ^ {2} /4 } H _ {n} \left (
 +
\frac{z}{\sqrt 2}
 +
\right ) ,
 +
$$
  
 
to the [[Probability integral|probability integral]] (error function) by
 
to the [[Probability integral|probability integral]] (error function) 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/p/p071/p071190/p07119035.png" /></td> </tr></table>
+
$$
 +
D _ {-} n- 1 ( z)  = \
 +
 
 +
\frac{(- 1)  ^ {n} \sqrt 2 }{n!}
 +
e ^ {- z  ^ {2} /4 }
 +
\frac{d  ^ {n} }{dz  ^ {n} }
 +
\left
 +
( e ^ {z  ^ {2} /3 }  \mathop{\rm erfc} 
 +
\frac{z}{\sqrt z }
 +
\right ) ,
 +
$$
  
 
and to the [[Bessel functions|Bessel functions]] by
 
and to the [[Bessel functions|Bessel functions]] 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/p/p071/p071190/p07119036.png" /></td> </tr></table>
+
$$
 +
D _ {-} 1/2 ( z)  = \
 +
\sqrt {
 +
\frac{\pi z }{2}
 +
} K _ {1/4} \left (
 +
\frac{z  ^ {2} }{4}
 +
\right ) .
 +
$$
  
 
====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">  J.C.P. Miller,  "Giving solutions of the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119037.png" />, tables of Weber parabolic cylinder functions" , H.M. Stationary Office  (1955)</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">  J.C.P. Miller,  "Giving solutions of the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119037.png" />, tables of Weber parabolic cylinder functions" , H.M. Stationary Office  (1955)</TD></TR></table>

Revision as of 08:05, 6 June 2020


Weber function, Weber–Hermite function

A solution of the differential equation

$$ \tag{* } \frac{d ^ {2} y }{dz ^ {2} } + \left ( \nu + \frac{1}{2} - \frac{z ^ {2} }{4} \right ) y = 0, $$

which is obtained as a result of separating the variables in the wave equation $ \Delta u = k ^ {2} u $ in parabolic cylindrical coordinates (cf. Parabolic coordinates). Ordinarily one uses the solution

$$ D _ \nu ( z) \equiv U \left ( - \nu - \frac{1}{2} , z \right ) = \ 2 ^ {( \nu - 1)/2 } e ^ {- z ^ {2} /4 } \Psi \left ( \frac{1 - \nu }{2} , \frac{3}{2} ; \ \frac{z ^ {2} }{2} \right ) , $$

where $ \Psi ( a, b; z) $ is the confluent hypergeometric function. The functions $ D _ \nu (- z) $ and $ D _ {- \nu - 1 } (\pm iz) $ also satisfy equation (*). The functions $ D _ \nu ( z) $ and $ D _ {- \nu - 1 } (\pm iz) $ are linearly independent for arbitrary $ \nu $, and $ D _ \nu ( z) $ and $ D _ \nu (- z) $ are linearly independent for $ \nu \neq 0, \pm 1 , . . . $. The parabolic cylinder functions are entire functions of $ z $. The function $ D _ \nu ( z) $ is real for real $ \nu $ and $ z $.

The differentiation formulas are, $ n = 1, 2 ,\dots $:

$$ \frac{d ^ {n} }{dz ^ {n} } \left [ e ^ {z ^ {2} /4 } D _ \nu ( z) \right ] = \ (- 1) ^ {n} (- \nu ) _ {n} e ^ {z ^ {2} /4 } D _ {\nu - n } ( z), $$

$$ \frac{d ^ {n} }{dz ^ {n} } \left [ e ^ {- z ^ {2} /4 } D _ \nu ( z) \right ] = (- 1) ^ {n} e ^ {- z ^ {2} /4 } D _ {\nu + n } ( z) . $$

The recurrence formulas are:

$$ D _ {\nu + 1 } ( z) - zD _ \nu ( z) + \nu D _ {\nu - 1 } ( z) = 0, $$

$$ D _ \nu ^ \prime ( z) + \frac{z}{2} D _ \nu ( z) - \nu D _ {\nu - 1 } ( z) = 0, $$

$$ D _ \nu ^ \prime ( z) - \frac{z}{2} D _ \nu ( z) + D _ {\nu + 1 } ( z) = 0. $$

Asymptotically, for fixed $ \nu $ and $ | \mathop{\rm arg} z | < 3 \pi /4 $, as $ z \rightarrow \infty $, one has

$$ D _ \nu ( z) = \ z ^ \nu e ^ {- z ^ {2} /4 } \left [ \sum _ { k= } 0 ^ { N } \frac{(- \nu /2) _ {k} ( 1/2 - \nu /2) _ {k} }{k!} \left ( \frac{z ^ {2} }{-} 2 \right ) ^ {-} k\right . + $$

$$ + \left . O ( | z | ^ {- 2N- 2 } ) \right ] , $$

and, for bounded $ | z | $ and $ | \mathop{\rm arg} ( - \nu ) | \leq \pi /2 $, as $ | \nu | \rightarrow \infty $, one has

$$ D _ \nu ( z) = \ \frac{1}{\sqrt 2 } \mathop{\rm exp} \left [ \frac \nu {2} \mathop{\rm ln} (- \nu ) - \frac \nu {2} - \sqrt {- \nu } z \right ] \times $$

$$ \times \left [ 1 + O \left ( \frac{1}{\sqrt {| \nu | } } \right ) \right ] . $$

Parabolic cylinder functions are related to other functions as follows ( $ n = 0, 1 , . . . $): To the Hermite polynomials by

$$ D _ {n} ( z) = \ 2 ^ {-} n/2 e ^ {- z ^ {2} /4 } H _ {n} \left ( \frac{z}{\sqrt 2} \right ) , $$

to the probability integral (error function) by

$$ D _ {-} n- 1 ( z) = \ \frac{(- 1) ^ {n} \sqrt 2 }{n!} e ^ {- z ^ {2} /4 } \frac{d ^ {n} }{dz ^ {n} } \left ( e ^ {z ^ {2} /3 } \mathop{\rm erfc} \frac{z}{\sqrt z } \right ) , $$

and to the Bessel functions by

$$ D _ {-} 1/2 ( z) = \ \sqrt { \frac{\pi z }{2} } K _ {1/4} \left ( \frac{z ^ {2} }{4} \right ) . $$

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] J.C.P. Miller, "Giving solutions of the differential equation , tables of Weber parabolic cylinder functions" , H.M. Stationary Office (1955)
How to Cite This Entry:
Parabolic cylinder function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_cylinder_function&oldid=13870
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