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In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation

${\displaystyle {\frac {d^{2}f}{dz^{2}}}+\left({\tilde {a}}z^{2}+{\tilde {b}}z+{\tilde {c}}\right)f=0.}$

1

This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindrical coordinates.

The above equation may be brought into two distinct forms (A) and (B) by completing the square and rescaling z, called H. F. Weber's equations:^[1]

${\displaystyle {\frac {d^{2}f}{dz^{2}}}-\left({\tfrac {1}{4}}z^{2}+a\right)f=0}$

A

and

${\displaystyle {\frac {d^{2}f}{dz^{2}}}+\left({\tfrac {1}{4}}z^{2}-a\right)f=0.}$

B

If $${\displaystyle f(a,z)}$$ is a solution, then so are $${\displaystyle f(a,-z),f(-a,iz){\text{ and }}f(-a,-iz).}$$

If $${\displaystyle f(a,z)\,}$$ is a solution of equation (A), then $${\displaystyle f(-ia,ze^{(1/4)\pi i})}$$ is a solution of (B), and, by symmetry, $${\displaystyle f(-ia,-ze^{(1/4)\pi i}),f(ia,-ze^{-(1/4)\pi i}){\text{ and }}f(ia,ze^{-(1/4)\pi i})}$$ are also solutions of (B).

There are independent even and odd solutions of the form (A). These are given by (following the notation of Abramowitz and Stegun (1965)):^[2] $${\displaystyle y_{1}(a;z)=\exp(-z^{2}/4)\;_{1}F_{1}\left({\tfrac {1}{2}}a+{\tfrac {1}{4}};\;{\tfrac {1}{2}}\;;\;{\frac {z^{2}}{2}}\right)\,\,\,\,\,\,(\mathrm {even} )}$$ and $${\displaystyle y_{2}(a;z)=z\exp(-z^{2}/4)\;_{1}F_{1}\left({\tfrac {1}{2}}a+{\tfrac {3}{4}};\;{\tfrac {3}{2}}\;;\;{\frac {z^{2}}{2}}\right)\,\,\,\,\,\,(\mathrm {odd} )}$$ where ${\displaystyle \;_{1}F_{1}(a;b;z)=M(a;b;z)}$ is the confluent hypergeometric function.

Other pairs of independent solutions may be formed from linear combinations of the above solutions.^[2] One such pair is based upon their behavior at infinity: $${\displaystyle U(a,z)={\frac {1}{2^{\xi }{\sqrt {\pi }}}}\left[\cos(\xi \pi )\Gamma (1/2-\xi )\,y_{1}(a,z)-{\sqrt {2}}\sin(\xi \pi )\Gamma (1-\xi )\,y_{2}(a,z)\right]}$$ $${\displaystyle V(a,z)={\frac {1}{2^{\xi }{\sqrt {\pi }}\Gamma [1/2-a]}}\left[\sin(\xi \pi )\Gamma (1/2-\xi )\,y_{1}(a,z)+{\sqrt {2}}\cos(\xi \pi )\Gamma (1-\xi )\,y_{2}(a,z)\right]}$$ where $${\displaystyle \xi ={\frac {1}{2}}a+{\frac {1}{4}}.}$$

The function U(a, z) approaches zero for large values of z and |arg(z)| < π/2, while V(a, z) diverges for large values of positive real z. $${\displaystyle \lim _{z\to \infty }U(a,z)/\left(e^{-z^{2}/4}z^{-a-1/2}\right)=1\,\,\,\,({\text{for}}\,\left|\arg(z)\right|<\pi /2)}$$ and $${\displaystyle \lim _{z\to \infty }V(a,z)/\left({\sqrt {\frac {2}{\pi }}}e^{z^{2}/4}z^{a-1/2}\right)=1\,\,\,\,({\text{for}}\,\arg(z)=0).}$$

For half-integer values of a, these (that is, U and V) can be re-expressed in terms of Hermite polynomials; alternatively, they can also be expressed in terms of Bessel functions.

The functions U and V can also be related to the functions D_p(x) (a notation dating back to Whittaker (1902))^[3] that are themselves sometimes called parabolic cylinder functions:^[2] $${\displaystyle {\begin{aligned}U(a,x)&=D_{-a-{\tfrac {1}{2}}}(x),\\V(a,x)&={\frac {\Gamma ({\tfrac {1}{2}}+a)}{\pi }}[\sin(\pi a)D_{-a-{\tfrac {1}{2}}}(x)+D_{-a-{\tfrac {1}{2}}}(-x)].\end{aligned}}}$$

Function D_a(z) was introduced by Whittaker and Watson as a solution of eq.~(1) with ${\textstyle {\tilde {a}}=-{\frac {1}{4}},{\tilde {b}}=0,{\tilde {c}}=a+{\frac {1}{2}}}$ bounded at ${\displaystyle +\infty }$.^[4] It can be expressed in terms of confluent hypergeometric functions as

${\displaystyle D_{a}(z)={\frac {1}{\sqrt {\pi }}}{2^{a/2}e^{-{\frac {z^{2}}{4}}}\left(\cos \left({\frac {\pi a}{2}}\right)\Gamma \left({\frac {a+1}{2}}\right)\,_{1}F_{1}\left(-{\frac {a}{2}};{\frac {1}{2}};{\frac {z^{2}}{2}}\right)+{\sqrt {2}}z\sin \left({\frac {\pi a}{2}}\right)\Gamma \left({\frac {a}{2}}+1\right)\,_{1}F_{1}\left({\frac {1}{2}}-{\frac {a}{2}};{\frac {3}{2}};{\frac {z^{2}}{2}}\right)\right)}.}$

Power series for this function have been obtained by Abadir (1993).^[5]

Integrals along the real line,^[6] $${\displaystyle U(a,z)={\frac {e^{-{\frac {1}{4}}z^{2}}}{\Gamma \left(a+{\frac {1}{2}}\right)}}\int _{0}^{\infty }e^{-zt}t^{a-{\frac {1}{2}}}e^{-{\frac {1}{2}}t^{2}}dt\,,\;\Re a>-{\frac {1}{2}}\;,}$$ $${\displaystyle U(a,z)={\sqrt {\frac {2}{\pi }}}e^{{\frac {1}{4}}z^{2}}\int _{0}^{\infty }\cos \left(zt+{\frac {\pi }{2}}a+{\frac {\pi }{4}}\right)t^{-a-{\frac {1}{2}}}e^{-{\frac {1}{2}}t^{2}}dt\,,\;\Re a<{\frac {1}{2}}\;.}$$ The fact that these integrals are solutions to equation (A) can be easily checked by direct substitution.

Differentiating the integrals with respect to ${\displaystyle z}$ gives two expressions for ${\displaystyle U'(a,z)}$, $${\displaystyle U'(a,z)=-{\frac {z}{2}}U(a,z)-{\frac {e^{-{\frac {1}{4}}z^{2}}}{\Gamma \left(a+{\frac {1}{2}}\right)}}\int _{0}^{\infty }e^{-zt}t^{a+{\frac {1}{2}}}e^{-{\frac {1}{2}}t^{2}}dt=-{\frac {z}{2}}U(a,z)-\left(a+{\frac {1}{2}}\right)U(a+1,z)\;,}$$ $${\displaystyle U'(a,z)={\frac {z}{2}}U(a,z)-{\sqrt {\frac {2}{\pi }}}e^{{\frac {1}{4}}z^{2}}\int _{0}^{\infty }\sin \left(zt+{\frac {\pi }{2}}a+{\frac {\pi }{4}}\right)t^{-a+{\frac {1}{2}}}e^{-{\frac {1}{2}}t^{2}}dt={\frac {z}{2}}U(a,z)-U(a-1,z)\;.}$$ Adding the two gives another expression for the derivative, $${\displaystyle 2U'(a,z)=-\left(a+{\frac {1}{2}}\right)U(a+1,z)-U(a-1,z)\;.}$$

Subtracting the first two expressions for the derivative gives the recurrence relation, $${\displaystyle zU(a,z)=U(a-1,z)-\left(a+{\frac {1}{2}}\right)U(a+1,z)\;.}$$

Expanding $${\displaystyle e^{-{\frac {1}{2}}t^{2}}=1-{\frac {1}{2}}t^{2}+{\frac {1}{8}}t^{4}-\dots \;}$$ in the integrand of the integral representation gives the asymptotic expansion of ${\displaystyle U(a,z)}$, $${\displaystyle U(a,z)=e^{-{\frac {1}{4}}z^{2}}z^{-a-{\frac {1}{2}}}\left(1-{\frac {(a+{\frac {1}{2}})(a+{\frac {3}{2}})}{2}}{\frac {1}{z^{2}}}+{\frac {(a+{\frac {1}{2}})(a+{\frac {3}{2}})(a+{\frac {5}{2}})(a+{\frac {7}{2}})}{8}}{\frac {1}{z^{4}}}-\dots \right).}$$

Expanding the integral representation in powers of ${\displaystyle z}$ gives $${\displaystyle U(a,z)={\frac {{\sqrt {\pi }}\,2^{-{\frac {a}{2}}-{\frac {1}{4}}}}{\Gamma \left({\frac {a}{2}}+{\frac {3}{4}}\right)}}-{\frac {{\sqrt {\pi }}\,2^{-{\frac {a}{2}}+{\frac {1}{4}}}}{\Gamma \left({\frac {a}{2}}+{\frac {1}{4}}\right)}}z+{\frac {{\sqrt {\pi }}\,2^{-{\frac {a}{2}}-{\frac {5}{4}}}}{\Gamma \left({\frac {a}{2}}+{\frac {3}{4}}\right)}}z^{2}-\dots \;.}$$

From the power series one immediately gets $${\displaystyle U(a,0)={\frac {{\sqrt {\pi }}\,2^{-{\frac {a}{2}}-{\frac {1}{4}}}}{\Gamma \left({\frac {a}{2}}+{\frac {3}{4}}\right)}}\;,}$$ $${\displaystyle U'(a,0)=-{\frac {{\sqrt {\pi }}\,2^{-{\frac {a}{2}}+{\frac {1}{4}}}}{\Gamma \left({\frac {a}{2}}+{\frac {1}{4}}\right)}}\;.}$$

Parabolic cylinder function ${\displaystyle D_{\nu }(z)}$ is the solution to the Weber differential equation, $${\displaystyle u''+\left(\nu +{\frac {1}{2}}-{\frac {1}{4}}z^{2}\right)u=0\,,}$$ that is regular at ${\displaystyle \Re z\to +\infty }$ with the asymptotics $${\displaystyle D_{\nu }(z)\to e^{-{\frac {1}{4}}z^{2}}z^{\nu }\,.}$$ It is thus given as ${\displaystyle D_{\nu }(z)=U(-\nu -1/2,z)}$ and its properties then directly follow from those of the ${\displaystyle U}$-function.

$${\displaystyle D_{\nu }(z)={\frac {e^{-{\frac {1}{4}}z^{2}}}{\Gamma (-\nu )}}\int _{0}^{\infty }e^{-zt}t^{-\nu -1}e^{-{\frac {1}{2}}t^{2}}dt\,,\;\Re \nu <0\,,\;\Re z>0\;,}$$ $${\displaystyle D_{\nu }(z)={\sqrt {\frac {2}{\pi }}}e^{{\frac {1}{4}}z^{2}}\int _{0}^{\infty }\cos \left(zt-\nu {\frac {\pi }{2}}\right)t^{\nu }e^{-{\frac {1}{2}}t^{2}}dt\,,\;\Re \nu >-1\;.}$$

$${\displaystyle D_{\nu }(z)=e^{-{\frac {1}{4}}z^{2}}z^{\nu }\left(1-{\frac {\nu (\nu -1)}{2}}{\frac {1}{z^{2}}}+{\frac {\nu (\nu -1)(\nu -2)(\nu -3)}{8}}{\frac {1}{z^{4}}}-\dots \right)\,,\;\Re z\to +\infty .}$$ If ${\displaystyle \nu }$ is a non-negative integer this series terminates and turns into a polynomial, namely the Hermite polynomial, $${\displaystyle D_{n}(z)=e^{-{\frac {1}{4}}z^{2}}\;2^{-n/2}H_{n}\left({\frac {z}{\sqrt {2}}}\right)\,,n=0,1,2,\dots \;.}$$

Parabolic cylinder ${\displaystyle D_{\nu }(z)}$ function appears naturally in the Schrödinger equation for the one-dimensional quantum harmonic oscillator (a quantum particle in the oscillator potential), $${\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {1}{2}}m\omega ^{2}x^{2}\right]\psi (x)=E\psi (x)\;,}$$ where ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle m}$ is the mass of the particle, ${\displaystyle x}$ is the coordinate of the particle, ${\displaystyle \omega }$ is the frequency of the oscillator, ${\displaystyle E}$ is the energy, and ${\displaystyle \psi (x)}$ is the particle's wave-function. Indeed introducing the new quantities $${\displaystyle z={\frac {x}{b_{o}}}\,,\;\nu ={\frac {E}{\hbar \omega }}-{\frac {1}{2}}\,,\;b_{o}={\sqrt {\frac {\hbar }{2m\omega }}}\,,}$$ turns the above equation into the Weber's equation for the function ${\displaystyle u(z)=\psi (zb_{o})}$, $${\displaystyle u''+\left(\nu +{\frac {1}{2}}-{\frac {1}{4}}z^{2}\right)u=0\,.}$$