 Home

Codex Wiki

Mathematics

Area of mathematics

Analysis

Bessel function

Modified Bessel function

For fixed real $\nu$ as $z\to+\infty$, the <Modified Bessel function of the first kind> and <Modified Bessel function of the second kind> have <asymptotic expansions>
$$
I_\nu(z)\sim\frac{e^z}{\sqrt{2\pi z}}\sum_{j=0}^\infty\frac{(-1)^ja_j(\nu)}{z^j},\qquad
K_\nu(z)\sim\sqrt{\frac\pi{2z}}e^{-z}\sum_{j=0}^\infty\frac{a_j(\nu)}{z^j},\qquad
a_0=1,\quad a_{j+1}=\frac{4\nu^2-(2j+1)^2}{8(j+1)}a_j.
$$
The expansion and its error bounds establish actual growing and decaying solutions, beyond formal matching of a <differential equation>. These formulas are documented in https://dlmf.nist.gov/10.40[NIST DLMF, equations 10.40.1–2].

Large-argument asymptotic expansion of a modified Bessel function

Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 30E / Solution

Large-argument asymptotic expansion of a modified Bessel function

 Ancestors (6)

 Incoming links (1)

 Discussion (0)

 Articles by others on the same topic (0)

Large-argument asymptotic expansion of a modified Bessel function

 Ancestors (6)

 Incoming links (1)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)