Parabolic cylinder function

There are independent even and odd solutions of the form (). These are given by (following the notation of Abramowitz and Stegun (1965)): y_1(a;z) = \exp(-z^2/4) \;_1F_1 \left(\tfrac12a+\tfrac14; \; \tfrac12\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{even}) and y_2(a;z) = z\exp(-z^2/4) \;_1F_1 \left(\tfrac12a+\tfrac34; \; \tfrac32\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{odd}) where \;_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 \xi = \frac{1}{2}a+\frac{1}{4} . The function approaches zero for large values of and , while diverges for large values of positive real . \lim_{z\to\infty}U(a,z)/\left(e^{-z^2/4}z^{-a-1/2}\right)=1\,\,\,\,(\text{for}\,\left|\arg(z)\right| and \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 (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions: \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 was introduced by Whittaker and Watson as a solution of eq.~() with \tilde a=-\frac14, \tilde b=0, \tilde c=a+\frac12 bounded at +\infty. It can be expressed in terms of confluent hypergeometric functions as :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) \, _1F_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) \, _1F_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). == Parabolic Cylinder U(a,z) function ==

Integral representation 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 The fact that these integrals are solutions to equation () can be easily checked by direct substitution. Derivative Differentiating the integrals with respect to z gives two expressions for U'(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) \;. Recurrence relation 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) \;. Asymptotic expansion 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) . Power series 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 \;. === Values at z=0 === 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 Dν(z) function ==

Parabolic cylinder function D_\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 \Re z\to +\infty with the asymptotics D_\nu(z) \to e^{-\frac14 z^2}z^\nu \,. It is thus given as D_\nu(z)=U(-\nu-1/2,z) and its properties then directly follow from those of the U-function. Integral representation 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\;, 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 \;. Asymptotic expansion 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 \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 \;. Connection with quantum harmonic oscillator Parabolic cylinder D_\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 \hbar is the reduced Planck constant, m is the mass of the particle, x is the coordinate of the particle, \omega is the frequency of the oscillator, E is the energy, and \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)=\psi(zb_o), u''+\left(\nu+\frac12-\frac{1}{4} z^2 \right)u=0 \,. == References ==