Bickley–Naylor functions

When solving physics or engineering problems where formulae for the thermal radiation intensities in hot enclosures, such as industrial furnaces or reactors, are sought, the solutions are often quite complicated unless the problem is essentially one-dimensional (such as the radiation field in a thin layer of gas between two parallel rectangular plates). Often the mathematical solutions contain unusual special functions, including the Bickley−Naylor functions.

Definition

In mathematics, the nth Bickley−Naylor function Ki_n(x) is defined by

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

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 cannot 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, π/2].

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

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

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.