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) |

(2) | |

(3) |

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

(4) |

(5) |

(6) | |||

(7) |

gives

(8) |

(9) |

(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) |

(15) | |||

(16) |

where

(17) | |||

(18) |

In terms of Whittaker and Watson's functions,

(19) | |||

(20) |

For Nonnegative Integer , the solution reduces to

(21) |

The parabolic cylinder functions satisfy the Recurrence Relations

(22) |

(23) |

(24) |

(25) |

(26) |

(27) |

(28) |

**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.

© 1996-9

1999-05-26