Parabolic cylinder function explained

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

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, called H. F. Weber's equations:and

If $$f(a,z)$$ is a solution, then so are$$f(a,-z),\; f(-a,iz)\backslash textf(-a,-iz).$$

If $$f(a,z)\backslash ,$$ is a solution of equation, then $$f(-ia,ze^)$$ is a solution of, and, by symmetry,$$f(-ia,-ze^),\; f(ia,-ze^)\backslash textf(ia,ze^)$$are also solutions of .

Solutions

There are independent even and odd solutions of the form . These are given by (following the notation of Abramowitz and Stegun (1965)):$$y\_1(a;z)\; =\; \backslash exp(-z^2/4)\; \backslash ;\_1F\_1\; \backslash left(\backslash tfrac12a+\backslash tfrac14;\; \backslash ;\backslash tfrac12\backslash ;\; ;\; \backslash ;\; \backslash frac\backslash right)\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\; (\backslash mathrm)$$and $$y\_2(a;z)\; =\; z\backslash exp(-z^2/4)\; \backslash ;\_1F\_1\; \backslash left(\backslash tfrac12a+\backslash tfrac34;\; \backslash ;\backslash tfrac32\backslash ;\; ;\; \backslash ;\; \backslash frac\backslash right)\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\; (\backslash mathrm)$$where

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:$$U(a,z)=\backslash frac\backslash left[\backslash cos(\backslash xi\backslash pi)\backslash Gamma(1/2-\backslash xi)\backslash ,y\_1(a,z)\; -\backslash sqrt\{2\}\backslash sin(\backslash xi\backslash pi)\backslash Gamma(1-\backslash xi)\backslash ,y\_2(a,z)\; \backslash right]$$$$V(a,z)=\backslash frac\backslash left[\backslash sin(\backslash xi\backslash pi)\backslash Gamma(1/2-\backslash xi)\backslash ,y\_1(a,z)\; +\backslash sqrt\{2\}\backslash cos(\backslash xi\backslash pi)\backslash Gamma(1-\backslash xi)\backslash ,y\_2(a,z)\; \backslash right]$$where $$\backslash xi\; =\; \backslash fraca+\backslash frac\; .$$

The function approaches zero for large values of and, while diverges for large values of positive real .and$$\backslash lim\_V(a,z)/\backslash left(\backslash sqrte^z^\backslash right)=1\backslash ,\backslash ,\backslash ,\backslash ,(\backslash text\backslash ,\backslash arg(z)=0)\; .$$

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 (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions:

Function was introduced by Whittaker and Watson as a solution of eq.~ with $\backslash tilde\; a=-\backslash frac14,\; \backslash tilde\; b=0,\; \backslash tilde\; c=a+\backslash frac12$ bounded at

. It can be expressed in terms of confluent hypergeometric functions as }.Power series for this function have been obtained by Abadir (1993).

Parabolic Cylinder U(a,z) function

Integral representation

Integrals along the real line,$$U(a,z)=\backslash frac\backslash int\_0^\backslash infty\; e^t^e^dt\backslash ,,\backslash ;\; \backslash Re\; a>-\backslash frac12\; \backslash ;,$$The fact that these integrals are solutions to equation  can be easily checked by direct substitution.

Derivative

Differentiating the integrals with respect to

gives two expressions for ,$$U\text{'}(a,z)=-\backslash fracU(a,z)-\backslash frac\backslash int\_0^\backslash infty\; e^t^e^dt=-\backslash fracU(a,z)-\backslash left(a+\backslash frac12\backslash right)U(a+1,z)\; \backslash ;,$$$$U\text{'}(a,z)=\backslash fracU(a,z)-\backslash sqrte^\backslash int\_0^\backslash infty\; \backslash sin\backslash left(zt+\backslash fraca+\backslash frac\backslash right)t^e^dt=\; \backslash fracU(a,z)-U(a-1,z)\; \backslash ;.$$Adding the two gives another expression for the derivative,$$2U\text{'}(a,z)\; =\; -\backslash left(a+\backslash frac12\backslash right)U(a+1,z)-U(a-1,z)\backslash ;.$$

Recurrence relation

Subtracting the first two expressions for the derivative gives the recurrence relation,$$zU(a,z)\; =\; U(a-1,z)\; -\; \backslash left(a+\backslash frac12\backslash right)U(a+1,z)\; \backslash ;.$$

Asymptotic expansion

Expanding$$e^=1-\backslash frac12\; t^2+\backslash frac18\; t^4\; -\; \backslash dots\; \backslash ;$$in the integrand of the integral representationgives the asymptotic expansion of

,$$U(a,z)\; =\; e^z^\backslash left(1-\; \backslash frac\backslash frac+\; \backslash frac\backslash frac-\; \backslash dots\backslash right)\; .$$

Power series

Expanding the integral representation in powers of

gives$$U(a,z)=\backslash frac-\backslash fracz+\backslash fracz^2\; -\; \backslash dots\; \backslash ;.$$

Values at z=0

From the power series one immediately gets$$U(a,0)=\backslash frac\; \backslash ;,$$$$U\text{'}(a,0)=-\backslash frac\; \backslash ;.$$

Parabolic cylinder D_ν(z) function

Parabolic cylinder function

is the solution to the Weber differential equation,$$u$$+\left(\nu+\frac12-\frac z^2 \right)u=0 \,,that is regular at

\Rez\to+infty

with the asymptotics$$D\_\backslash nu(z)\; \backslash to\; e^z^\backslash nu\; \backslash ,.$$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

$$D\_\backslash nu(z)=\backslash sqrte^\backslash int\_0^\backslash infty\; \backslash cos\backslash left(zt-\backslash nu\; \backslash frac\backslash right)t^e^dt\backslash ,,\backslash ;\; \backslash Re\; \backslash nu\; >\; -1\; \backslash ;.$$

Asymptotic expansion

$$D\_\backslash nu(z)\; =\; e^z^\backslash left(1-\; \backslash frac\backslash frac+\; \backslash frac\backslash frac-\; \backslash dots\backslash right)\backslash ,,\backslash ;\; \backslash Re\; z\; \backslash to\; +\backslash infty\; .$$If

is a non-negative integer this series terminates and turns into a polynomial, namely the Hermite polynomial,$$D\_n(z)\; =\; e^\backslash ;2^H\_n\backslash left(\backslash frac\backslash right)\backslash ,,\; n=0,1,2,\backslash dots\; \backslash ;.$$

Connection with quantum harmonic oscillator

Parabolic cylinder

function appears naturally in the Schrödinger equation for the one-dimensional quantum harmonic oscillator (a quantum particle in the oscillator potential),$$\backslash left[-\backslash frac\{\backslash hbar^2\}\{2m\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}+\backslash frac12\; m\; \backslash omega^2\; x^2\; \backslash right]\backslash psi(x)=E\backslash psi(x)\; \backslash ;,$$where is the reduced Planck constant, is the mass of the particle, is the coordinate of the particle, is the frequency of the oscillator, is the energy,and is the particle's wave-function. Indeed introducing the new quantities$$z=\backslash frac\; \backslash ,,\backslash ;\; \backslash nu=\backslash frac-\backslash frac12\; \backslash ,,\backslash ;\; b\_o=\backslash sqrt\; \backslash ,,$$turns the above equation into the Weber's equation for the function ,$$u$$+\left(\nu+\frac12-\frac z^2 \right)u=0 \,.

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