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:
“Anyone who seeks for the true causes of miracles, and strives to understand natural phenomena as an intelligent being, and not to gaze at them like a fool, is set down and denounced as an impious heretic by those, whom the masses adore as the interpreters of nature and the gods. Such persons know that, with the removal of ignorance, the wonder which forms their only available means for proving and preserving their authority would vanish also.”
—Baruch (Benedict)