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. These functions have practical applications in several engineering problems related to transport of thermal or neutron, radiation in systems with special symmetries. W. G. Bickley was a British mathematician born in 1893.

Definition

The nth Bickley−Naylor function is defined by
and it is classified as one of the generalized exponential integral functions.
All of the functions for positive integer n are monotonously decreasing functions, because is a decreasing function and is a positive increasing function for.

Properties

The integral defining the function 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, .
Alternative ways to define the function include the integral, :
where is the modified Bessel function of the zeroth order. Also by definition we have.

Series expansions

The series expansions of the first and second order Bickley functions are given by:
where is the Euler constant and
is the th harmonic number.

Recurrence relation

The Bickley functions also satisfy the following recurrence relation:
where.

Asymptotic expansions

The asymptotic expansions of Bickley functions are given as
for.

Successive differentiation

Differentiating 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 Ki_n is shown below.


0	1.570796327	1.000000000	0.785398162
0.1	1.22863188	0.862521290	0.692543328
0.2	1.023679877	0.750458533	0.612064472
0.3	0.868832269	0.656147929	0.541862953
0.4	0.745203394	0.575660412	0.480375442
0.5	0.643693806	0.506373657	0.426358257
0.6	0.558890473	0.446366680	0.378791860
0.7	0.487198347	0.394159632	0.336825253
0.8	0.426061805	0.348575863	0.299739399
0.9	0.373578804	0.308659297	0.266921357
1.0	0.328286478	0.273620752	0.237845082
1.2	0.254888907	0.21564418	0.189162878
1.4	0.199050709	0.17049927	0.150734408
1.6	0.156156459	0.135163924	0.120310892
1.8	0.122960838	0.107392071	0.096165816
2.0	0.097120592	0.085490579	0.076963590
2.5	0.054422478	0.048670845	0.044307124
3.0	0.030848237	0.027924583	0.025646500
3.5	0.017634408	0.016117448	0.014909740
4.0	0.010146756	0.009346971	0.008698789
4.5	0.005868829	0.005441695	0.005090280
5.0	0.003408936	0.003178387	0.002986247
6.0	0.001161774	0.001092877	0.001034238
7.0	0.000400052	0.000378912	0.000360620
8.0	0.000138841	0.000132222	0.000126417
9.0	0.000048484	0.000046377	0.000044509
10	0.000017015	0.000016336	0.000015728

Computer code

Computer code in Fortran is made available by Amos.