Asymptotic Forms
The Bessel functions have the following asymptotic forms for non-negative α. For small arguments, one obtains:
where γ is the Euler–Mascheroni constant (0.5772...) and Γ denotes the gamma function. For large arguments, they become:
(For α = 1/2 these formulas are exact; see the spherical Bessel functions above.) Asymptotic forms for the other types of Bessel function follow straightforwardly from the above relations. For example, for large, the modified Bessel functions become:
Similarly, the last expressions are exact when α = 1/2.
For small arguments, they become:
Read more about this topic: Bessel Function
Famous quotes containing the word forms:
“The delicious faces of children, the beauty of school-girls, “the sweet seriousness of sixteen,” the lofty air of well-born, well-bred boys, the passionate histories in the looks and manners of youth and early manhood, and the varied power in all that well-known company that escort us through life,—we know how these forms thrill, paralyze, provoke, inspire, and enlarge us.”
—Ralph Waldo Emerson (1803–1882)