# Hermite function

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)

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.