“Parabolic Cylinder Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Parabolic Cylinder Function

释义

Parabolic Cylinder Function

These functions are sometimes called Weber Functions. Whittaker and Watson (1990, p. 347) define the paraboliccylinder 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 HypergeometricFunctions.

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 theexpansion

(26)

(27)

For

real,

(28)

(Gradshteyn and Ryzhik 1980, p. 885, 7.711.3), where

is the Gamma Function and

is thePolygamma 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.

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