## Parabolic Cylinder Function

These functions are sometimes called Weber Functions. Whittaker and Watson (1990, p. 347) define the parabolic cylinder functions as solutions to the Weber Differential Equation

 (1)

The two independent solutions are given by and , where

 (2) (3)
Here, is a Whittaker Function and are Confluent Hypergeometric Functions.

Abramowitz and Stegun (1972, p. 686) define the parabolic cylinder functions as solutions to

 (4)

This can be rewritten by Completing the Square,
 (5)

Now letting
 (6) (7)

gives
 (8)

where
 (9)

Equation (4) has the two standard forms
 (10) (11)

For a general , the Even and Odd solutions to (10) are
 (12) (13)

where is a Confluent Hypergeometric Function. If is a solution to (10), then (11) has solutions
 (14)

Abramowitz and Stegun (1972, p. 687) define standard solutions to (10) as
 (15) (16)

where
 (17) (18)

In terms of Whittaker and Watson's functions,

 (19) (20)

For Nonnegative Integer , the solution reduces to

 (21)

where is a Hermite Polynomial and He is a modified Hermite Polynomial.

The parabolic cylinder functions satisfy the Recurrence Relations

 (22)

 (23)

The parabolic cylinder function for integral can be defined in terms of an integral by
 (24)

(Watson 1966, p. 308), which is similar to the Anger Function. The result
 (25)

where is the Kronecker Delta, can also be used to determine the Coefficients in the expansion
 (26)

as
 (27)

For real,
 (28)

(Gradshteyn and Ryzhik 1980, p. 885, 7.711.3), where is the Gamma Function and is the Polygamma Function of order 0.

See also Anger Function, Bessel Function, Darwin's Expansions, Hh Function, Struve Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Parabolic Cylinder Function.'' Ch. 19 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 685-700, 1972.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1979.

Iyanaga, S. and Kawada, Y. (Eds.). Parabolic Cylinder Functions (Weber Functions).'' Appendix A, Table 20.III in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1479, 1980.

Spanier, J. and Oldham, K. B. The Parabolic Cylinder Function .'' Ch. 46 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 445-457, 1987.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.