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A solution of the [[Hermite equation|Hermite equation]]
 
A solution of the [[Hermite equation|Hermite 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/h/h046/h046980/h0469801.png" /></td> </tr></table>
+
$$
 +
w  ^ {\prime\prime} - 2z w  ^  \prime  + 2 \lambda w  = 0 .
 +
$$
  
 
The Hermite functions have the form
 
The Hermite functions have the form
  
<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/h/h046/h046980/h0469802.png" /></td> </tr></table>
+
$$
 +
P _  \lambda  ( z)  =
 +
\frac{1}{\pi i }
 +
\int\limits _ {C _ {1} }  \mathop{\rm exp} (- t
 +
^ {2} + 2zt ) t ^ {- \lambda - 1 }  dt ,
 +
$$
  
<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/h/h046/h046980/h0469803.png" /></td> </tr></table>
+
$$
 +
Q _  \lambda  ( z)  =
 +
\frac{1}{\pi i }
 +
\int\limits _ {C _ {2} }
 +
\mathop{\rm exp} (- t  ^ {2} + 2zt ) t ^ {- \lambda - 1 }  dt ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469804.png" /> is the contour in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469805.png" />-plane consisting of the rays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469807.png" /> and the semi-circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h0469809.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698010.png" />. The half-sum of these solutions,
+
where $  C _ {1} $
 +
is the contour in the complex $  t $-
 +
plane consisting of the rays $  ( - \infty , - a ) $
 +
and $  ( a , \infty ) $
 +
and the semi-circle $  | t | = a > 0 $,  
 +
$  \mathop{\rm Im}  t \geq  0 $,  
 +
and $  C _ {2} = - C _ {1} $.  
 +
The half-sum of these solutions,
  
<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/h/h046/h046980/h04698011.png" /></td> </tr></table>
+
$$
 +
H _  \lambda  ( z)  =
 +
\frac{P _  \lambda  ( z) + Q _  \lambda  ( z) }{2}
 +
,
 +
$$
  
for an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698012.png" />, is equal to the Hermite polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698013.png" /> (cf. [[Hermite polynomials|Hermite polynomials]]). The name Hermite equation is also used for
+
for an integer $  \lambda = n \geq  0 $,  
 +
is equal to the Hermite polynomial $  H _ {n} ( x) $(
 +
cf. [[Hermite polynomials|Hermite polynomials]]). The name Hermite equation is also used for
  
<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/h/h046/h046980/h04698014.png" /></td> </tr></table>
+
$$
 +
y  ^ {\prime\prime} - x y  ^  \prime  + \nu y  = 0.
 +
$$
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698015.png" /> is an integer, this equation has the [[Fundamental system of solutions|fundamental system of solutions]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698017.png" /> are the Hermite polynomials and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698018.png" /> are the Hermite functions of the second kind, which can be expressed in terms of the [[Confluent hypergeometric function|confluent hypergeometric function]]:
+
When $  \nu $
 +
is an integer, this equation has the [[Fundamental system of solutions|fundamental system of solutions]] $  H _  \nu  ( x) , h _  \nu  ( x) $,  
 +
where $  H _  \nu  ( x) $
 +
are the Hermite polynomials and h _  \nu  ( x) $
 +
are the Hermite functions of the second kind, which can be expressed in terms of the [[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/h/h046/h046980/h04698019.png" /></td> </tr></table>
+
$$
 +
h _ {2n} ( x)  = (- 2)  ^ {n} n! _ {1} F _ {1} \left ( - n +
 +
\frac{1}{2}
 +
;
 +
\frac{3}{2}
 +
;
 +
\frac{x  ^ {2} }{2}
 +
\right ) ,
 +
$$
  
<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/h/h046/h046980/h04698020.png" /></td> </tr></table>
+
$$
 +
h _ {2n+} 1 ( x)  = - (- 2)  ^ {n} n! _ {1} F _ {1} \left ( - n
 +
-
 +
\frac{1}{2}
 +
;
 +
\frac{1}{2}
 +
;
 +
\frac{x  ^ {2} }{2}
 +
\right ) .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''1''' , Interscience  (1953)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Krazer,  W. Franz,  "Transzendente Funktionen" , Akademie Verlag  (1960)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''1''' , Interscience  (1953)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Krazer,  W. Franz,  "Transzendente Funktionen" , Akademie Verlag  (1960)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The Hermite functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698022.png" /> are related to the parabolic cylinder functions (cf. [[Parabolic cylinder function|Parabolic cylinder function]]). See [[#References|[a1]]], Sect. 4b for some further results concerning the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698023.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046980/h04698024.png" /> is a non-negative integer.
+
The Hermite functions $  P _  \lambda  $
 +
and $  Q _  \lambda  $
 +
are related to the parabolic cylinder functions (cf. [[Parabolic cylinder function|Parabolic cylinder function]]). See [[#References|[a1]]], Sect. 4b for some further results concerning the functions $  H _  \nu  , h _  \nu  $
 +
when $  \nu $
 +
is a non-negative integer.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Durand,  "Nicholson-type integrals for products of Gegenbauer functions and related topics"  R.A. Askey (ed.) , ''Theory and Application of Special Functions'' , Acad. Press  (1975)  pp. 353–374</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Durand,  "Nicholson-type integrals for products of Gegenbauer functions and related topics"  R.A. Askey (ed.) , ''Theory and Application of Special Functions'' , Acad. Press  (1975)  pp. 353–374</TD></TR></table>

Latest revision as of 22:10, 5 June 2020


A solution of the Hermite equation

$$ w ^ {\prime\prime} - 2z w ^ \prime + 2 \lambda w = 0 . $$

The Hermite functions have the form

$$ P _ \lambda ( z) = \frac{1}{\pi i } \int\limits _ {C _ {1} } \mathop{\rm exp} (- t ^ {2} + 2zt ) t ^ {- \lambda - 1 } dt , $$

$$ Q _ \lambda ( z) = \frac{1}{\pi i } \int\limits _ {C _ {2} } \mathop{\rm exp} (- t ^ {2} + 2zt ) t ^ {- \lambda - 1 } dt , $$

where $ C _ {1} $ is the contour in the complex $ t $- plane consisting of the rays $ ( - \infty , - a ) $ and $ ( a , \infty ) $ and the semi-circle $ | t | = a > 0 $, $ \mathop{\rm Im} t \geq 0 $, and $ C _ {2} = - C _ {1} $. The half-sum of these solutions,

$$ H _ \lambda ( z) = \frac{P _ \lambda ( z) + Q _ \lambda ( z) }{2} , $$

for an integer $ \lambda = n \geq 0 $, is equal to the Hermite polynomial $ H _ {n} ( x) $( cf. Hermite polynomials). The name Hermite equation is also used for

$$ y ^ {\prime\prime} - x y ^ \prime + \nu y = 0. $$

When $ \nu $ is an integer, this equation has the fundamental system of solutions $ H _ \nu ( x) , h _ \nu ( x) $, where $ H _ \nu ( x) $ are the Hermite polynomials and $ h _ \nu ( x) $ are the Hermite functions of the second kind, which can be expressed in terms of the confluent hypergeometric function:

$$ h _ {2n} ( x) = (- 2) ^ {n} n! _ {1} F _ {1} \left ( - n + \frac{1}{2} ; \frac{3}{2} ; \frac{x ^ {2} }{2} \right ) , $$

$$ h _ {2n+} 1 ( x) = - (- 2) ^ {n} n! _ {1} F _ {1} \left ( - n - \frac{1}{2} ; \frac{1}{2} ; \frac{x ^ {2} }{2} \right ) . $$

References

[1] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 1 , Interscience (1953) (Translated from German)
[2] A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)

Comments

The Hermite functions $ P _ \lambda $ and $ Q _ \lambda $ are related to the parabolic cylinder functions (cf. Parabolic cylinder function). See [a1], Sect. 4b for some further results concerning the functions $ H _ \nu , h _ \nu $ when $ \nu $ is a non-negative integer.

References

[a1] L. Durand, "Nicholson-type integrals for products of Gegenbauer functions and related topics" R.A. Askey (ed.) , Theory and Application of Special Functions , Acad. Press (1975) pp. 353–374
How to Cite This Entry:
Hermite function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_function&oldid=47213
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article