parabolic cylinder function

There are independent even and odd solutions of the form ({{EquationNote|A}}). These are given by (following the notation of Abramowitz and Stegun (1965)):

$$y\_1(a;z)\; =\; \backslash exp(-z^2/4)\; \backslash ;\_1F\_1$$

\left(\tfrac12a+\tfrac14; \;

\tfrac12\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{even})

and

$$y\_2(a;z)\; =\; z\backslash exp(-z^2/4)\; \backslash ;\_1F\_1$$

\left(\tfrac12a+\tfrac34; \;

\tfrac32\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{odd})

where $\backslash ;\_1F\_1\; (a;b;z)=M(a;b;z)$ is the confluent hypergeometric function.

Other pairs of independent solutions may be formed from linear combinations of the above solutions. One such pair is based upon their behavior at infinity:

$$$$

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]

$$$$

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 $$\backslash xi\; =\; \backslash frac\{1\}\{2\}a+\backslash frac\{1\}\{4\}\; .$$

The function {{math|U(a, z)}} approaches zero for large values of {{mvar|z}} and {{math|{{!}}arg(z){{!}} < π/2}}, while {{math|V(a, z)}} diverges for large values of positive real {{mvar|z}}.

and

$$\backslash lim\_\{z\backslash to\backslash infty\}V(a,z)/\backslash left(\backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}e^\{z^2/4\}z^\{a-1/2\}\backslash right)=1\backslash ,\backslash ,\backslash ,\backslash ,(\backslash text\{for\}\backslash ,\backslash arg(z)=0)\; .$$

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 {{math|1=D_p(x)}} (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions:

$$\backslash begin\{align\}$$

U(a,x) &= D_{-a-\tfrac12}(x), \\

V(a,x) &= \frac{\Gamma(\tfrac12+a)}{\pi}[\sin( \pi a) D_{-a-\tfrac12}(x)+D_{-a-\tfrac12}(-x)] .

\end{align}

Function {{math|1=D_a(z)}} was introduced by Whittaker and Watson as a solution of eq.~({{EquationNote|1}}) with $\backslash tilde\; a=-\backslash frac14,\; \backslash tilde\; b=0,\; \backslash tilde\; c=a+\backslash frac12$ bounded at $+\backslash infty$. It can be expressed in terms of confluent hypergeometric functions as

:$D\_a(z)=\backslash frac\{1\}\{\backslash sqrt\{\backslash pi\; \}\}\{2^\{a/2\}\; e^\{-\backslash frac\{z^2\}\{4\}\}\; \backslash left(\backslash cos\; \backslash left(\backslash frac\{\backslash pi\; a\}\{2\}\backslash right)\; \backslash Gamma\; \backslash left(\backslash frac\{a+1\}\{2\}\backslash right)\; \backslash ,\; \_1F\_1\backslash left(-\backslash frac\{a\}\{2\};\backslash frac\{1\}\{2\};\backslash frac\{z^2\}\{2\}\backslash right)+\backslash sqrt\{2\}\; z\; \backslash sin\; \backslash left(\backslash frac\{\backslash pi\; a\}\{2\}\backslash right)\; \backslash Gamma\; \backslash left(\backslash frac\{a\}\{2\}+1\backslash right)\; \backslash ,\; \_1F\_1\backslash left(\backslash frac\{1\}\{2\}-\backslash frac\{a\}\{2\};\backslash frac\{3\}\{2\};\backslash frac\{z^2\}\{2\}\backslash right)\backslash right)\}.$

Power series for this function have been obtained by Abadir (1993).

Integrals along the real line,

$$$$

U(a,z)=\frac{e^{-\frac14 z^2}}{\Gamma\left(a+\frac12\right)}

\int_0^\infty e^{-zt}t^{a-\frac12}e^{-\frac12 t^2}dt

\,,\; \Re a>-\frac12 \;,

$$$$

U(a,z)=\sqrt{\frac2{\pi}}e^{\frac14 z^2}

\int_0^\infty \cos\left(zt+\frac{\pi}{2}a+\frac{\pi}{4}\right)

t^{-a-\frac12}e^{-\frac12 t^2}dt

\,,\; \Re a<\frac12 \;.

The fact that these integrals are solutions to equation ({{EquationNote|A}}) can be easily checked by direct substitution.

Differentiating the integrals with respect to $z$ gives two expressions for $U\text{'}(a,z)$,

$$$$

U'(a,z)=-\frac{z}{2}U(a,z)-

\frac{e^{-\frac14 z^2}}{\Gamma\left(a+\frac12\right)}

\int_0^\infty e^{-zt}t^{a+\frac12}e^{-\frac12 t^2}dt

=-\frac{z}{2}U(a,z)-\left(a+\frac12\right)U(a+1,z) \;,

$$$$

U'(a,z)=\frac{z}{2}U(a,z)-

\sqrt{\frac2{\pi}}e^{\frac14 z^2}

\int_0^\infty \sin\left(zt+\frac{\pi}{2}a+\frac{\pi}{4}\right)

t^{-a+\frac12}e^{-\frac12 t^2}dt

= \frac{z}{2}U(a,z)-U(a-1,z) \;.

Adding the two gives another expression for the derivative,

$$$$

2U'(a,z) = -\left(a+\frac12\right)U(a+1,z)-U(a-1,z)

\;.

Subtracting the first two expressions for the derivative gives the recurrence relation,

$$$$

zU(a,z) = U(a-1,z) - \left(a+\frac12\right)U(a+1,z) \;.

Expanding

$$$$

e^{-\frac12 t^2}=1-\frac12 t^2+\frac18 t^4 - \dots \;

in the integrand of the integral representation

gives the asymptotic expansion of $U(a,z)$,

$$$$

U(a,z) = e^{-\frac14 z^2}z^{-a-\frac12}\left(1

- \frac{(a+\frac12)(a+\frac32)}{2}\frac{1}{z^2}

+ \frac{(a+\frac12)(a+\frac32)(a+\frac52)(a+\frac72)}{8}\frac{1}{z^4}

- \dots\right) .

Expanding the integral representation in powers of $z$ gives

$$$$

U(a,z)=\frac{\sqrt{\pi}\,2^{-\frac{a}{2}-\frac14}}{\Gamma\left(\frac{a}{2}+\frac34\right)}

-\frac{\sqrt{\pi}\,2^{-\frac{a}{2}+\frac14}}{\Gamma\left(\frac{a}{2}+\frac14\right)}z

+\frac{\sqrt{\pi}\,2^{-\frac{a}{2}-\frac54}}{\Gamma\left(\frac{a}{2}+\frac34\right)}z^2 - \dots \;.

From the power series one immediately gets

$$$$

U(a,0)=\frac{\sqrt{\pi}\,2^{-\frac{a}{2}-\frac14}}{\Gamma\left(\frac{a}{2}+\frac34\right)} \;,

$$$$

U'(a,0)=-\frac{\sqrt{\pi}\,2^{-\frac{a}{2}+\frac14}}{\Gamma\left(\frac{a}{2}+\frac14\right)} \;.

Parabolic cylinder function $D\_\backslash nu(z)$ is the solution to the Weber differential equation,

$$$$

u''+\left(\nu+\frac12-\frac{1}{4} z^2 \right)u=0 \,,

that is regular at $\backslash Re\; z\backslash to\; +\backslash infty$ with the asymptotics

$$$$

D_\nu(z) \to e^{-\frac14 z^2}z^\nu \,.

It is thus given as $D\_\backslash nu(z)=U(-\backslash nu-1/2,z)$ and its properties then directly follow from those of the $U$-function.

$$$$

D_\nu(z)=\frac{e^{-\frac14 z^2}}{\Gamma(-\nu)}

\int_0^\infty e^{-zt} t^{-\nu -1} e^{-\frac12 t^2}dt

\,,\; \Re \nu < 0 \,,\; \Re z > 0\;,

$$$$

D_\nu(z)=\sqrt{\frac2{\pi}}e^{\frac14 z^2}

\int_0^\infty \cos\left(zt-\nu \frac{\pi}{2}\right)

t^{\nu}e^{-\frac12 t^2}dt

\,,\; \Re \nu > -1 \;.

$$$$

D_\nu(z) = e^{-\frac14 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 $\backslash nu$ is a non-negative integer this series terminates and turns into a polynomial, namely the Hermite polynomial,

$$$$

D_n(z) = e^{-\frac14 z^2}\;2^{-n/2}H_n\left(\frac{z}{\sqrt{2}}\right)\,, n=0,1,2,\dots \;.

Parabolic cylinder $D\_\backslash nu(z)$ function appears naturally in the Schrödinger equation for the one-dimensional quantum harmonic oscillator (a quantum particle in the oscillator potential),

$$$$

\left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+\frac12 m \omega^2 x^2 \right]\psi(x)

=E\psi(x) \;,

where

$\backslash hbar$ is the reduced Planck constant,

$m$ is the mass of the particle,

$x$ is the coordinate of the particle,

$\backslash omega$ is the frequency of the oscillator,

$E$ is the energy,

and $\backslash psi(x)$ is the particle's wave-function. Indeed introducing the new quantities

$$$$

z=\frac{x}{b_o} \,,\; \nu=\frac{E}{\hbar\omega}-\frac12 \,,\; b_o=\sqrt{\frac{\hbar}{2m\omega}} \,,

turns the above equation into the Weber's equation for the function $u(z)=\backslash psi(zb\_o)$,

$$$$

u''+\left(\nu+\frac12-\frac{1}{4} z^2 \right)u=0 \,.

{{reflist | refs =

Abadir, K. M. (1993) "Expansions for some confluent hypergeometric functions." Journal of Physics A, 26, 4059-4066.

{{AS ref |19|686}}

{{citation |last=Weber |first=H.F. |date=1869 |title=Ueber die Integration der partiellen Differentialgleichung $\backslash partial^2u/\backslash partial\; x^2+\backslash partial^2u/\backslash partial\; y^2+k^2u=0$ |work=Math. Ann. |volume=1 |pages=1–36}}

Whittaker, E.T. (1902) "On the functions associated with the parabolic cylinder in harmonic analysis" Proc. London Math. Soc., 35, 417–427.

Whittaker, E. T. and Watson, G. N. (1990) "The Parabolic Cylinder Function." §16.5 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 347-348.

NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.2 of 2024-09-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

}}

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Category:Special hypergeometric functions

Category:Special functions