Modified Bessel Function of the Second Kind
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Modified Bessel Functions I and K." §9.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
25-3-2019
4930
Modified Bessel Function of the Second Kind
The modified bessel function of the second kind is the function 
which is one of the solutions to the modified Bessel differential equation. The modified Bessel functions of the second kind are sometimes called the Basset functions, modified Bessel functions of the third kind (Spanier and Oldham 1987, p. 499), or Macdonald functions (Spanier and Oldham 1987, p. 499; Samko et al. 1993, p. 20). The modified Bessel function of the second kind is implemented in the Wolfram Language as BesselK[nu, z].
is closely related to the modified Bessel function of the first kind 
and Hankel function 
,
(Watson 1966, p. 185). A sum formula for 
is
^k)/(k!(n+k)!),](http://mathworld.wolfram.com/images/equations/ModifiedBesselFunctionoftheSecondKind/NumberedEquation1.gif) |
(4)
|
where 
is the digamma function (Abramowitz and Stegun 1972). An integral formula is
 |
(5)
|
which, for 
, simplifies to
 |
(6)
|
Other identities are
 |
(7)
|
for 
and
The special case of 
gives 
as the integrals
(Abramowitz and Stegun 1972, p. 376).
REFERENCES:
Abramowitz, M. and Stegun, I. 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.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Modified Bessel Functions of Integral Order" and "Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions." §6.6 and 6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 229-245, 1992.
Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 20, 1993.
Spanier, J. and Oldham, K. B. "The Basset 
." Ch. 51 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 499-507, 1987.
Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.
الاكثر قراءة في التفاضل و التكامل
اخر الاخبار
اخبار العتبة العباسية المقدسة
