“Modified Bessel Function of the First Kind”的概念、定义、习题解题方法及翻译-数学辞典

单词

Modified Bessel Function of the First Kind

释义

Modified Bessel Function of the First Kind

A function which is one of the solutions to the Modified Bessel Differential Equation and is closelyrelated to the Bessel Function of the First Kind . The above plot shows for , 2, ..., 5. Interms of,

(1)

For a Real Number

, the function can be computed using

(2)

where

is the Gamma Function. An integral formula is

(3)

which simplifies for

an Integer

(4)

(Abramowitz and Stegun 1972, p. 376).

A derivative identity for expressing higher order modified Bessel functions in terms of is

(5)

where

is a Chebyshev Polynomial of the First Kind.

See also Bessel Function of the First Kind, Modified Bessel Function of the First Kind, Weber's Formula

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Modified Bessel Functions and .'' §9.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 374-377, 1972.

Arfken, G. ``Modified Bessel Functions, and .'' §11.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 610-616, 1985.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions.'' §6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 234-245, 1992.

Spanier, J. and Oldham, K. B. ``The Hyperbolic Bessel Functions and '' and ``The General Hyperbolic Bessel Function .'' Chs. 49-50 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 479-487 and 489-497, 1987.

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