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:
“Of the three forms of pride, that is to say pride proper, vanity, and conceit, vanity is by far the most harmless, and conceit by far the most dangerous. The meaning of vanity is to think too much of our bodily advantages, whether real or unreal, over others; while the meaning of conceit is to believe we are cleverer, wiser, grander, and more important than we really are.”
—John Cowper Powys (1872–1963)