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

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

+inf ty . It can be expressed in terms of confluent hypergeometric functions asD _{a(z) = (1)/(\sqrt{\pi)} }.Power series for this function have been obtained by Abadir (1993)