Bickley–Naylor functions

In mathematics, the n:th Bickley-Naylor function $Ki_{n}(x)$ is defined by:

$Ki_{n}(x)=\int _{0}^{\pi /2}e^{-{\frac {x}{sin\theta }}}\sin ^{n-1}\theta d\theta$

and these functions have practical applications in several engineering problems related to transport of thermal or neutron radiation in systems with special symmetries (e.g. spherical or axial symmetry).

Properties

The integral defining the function $Ki_{n}$ generally can't be evaluated analytically, but can be approximated to a desired accuracy with Riemann sums or other methods, taking the limit $a\rightarrow 0$ in the interval of integration, $[a,\pi /2]$.

Alternative ways to define the function $Ki_{n}$ include the integral:

$Ki_{n}(x)=\int _{0}^{\infty }e^{-x\cosh t}(\cosh t)^{-n}dx$

The values of these functions for different values of the argument $x$ were often listed in tables of special functions in the era when numerical calculation of integrals was slow.