Bickley–Naylor functions

In physics, engineering, and applied mathematics, the Bickley–Naylor functions are a sequence of special functions arising in formulas for thermal radiation intensities in hot enclosures.

The solutions are often quite complicated unless the problem is essentially one-dimensional[1] (such as the radiation field in a thin layer of gas between two parallel rectangular plates).

These functions have practical applications in several engineering problems related to transport of thermal[2][3] or neutron,[4][5] radiation in systems with special symmetries (e.g. spherical or axial symmetry).

W. G. Bickley was a British mathematician born in 1893.

is defined by and it is classified as one of the generalized exponential integral functions.

for positive integer n are monotonously decreasing functions, because

generally cannot be evaluated analytically, but can be approximated to a desired accuracy with Riemann sums or other methods, taking the limit as a → 0 in the interval of integration, [a, π/2].

Alternative ways to define the function

is the modified Bessel function of the zeroth order.

The series expansions of the first and second order Bickley functions are given by: where γ is the Euler constant and is the

The Bickley functions also satisfy the following recurrence relation:[8] where

The asymptotic expansions of Bickley functions are given as[9] for

with respect to x gives Successive differentiation yields 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.

A table that lists some approximate values of the three first functions Kin is shown below.

Computer code in Fortran is made available by Amos.