Bessel Function - Asymptotic Forms

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: