Top Qs
Timeline
Chat
Perspective

Bickley–Naylor functions

Functions for thermal radiation in hot enclosures From Wikipedia, the free encyclopedia

Bickley–Naylor functions
Remove ads

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.[6]

Thumb
Remove ads

Definition

Summarize
Perspective

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 .

Remove ads

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, [a, π/2].

Alternative ways to define the function include the integral,[7] integral forms the Bickley-Naylor function:

where is the modified Bessel function of the zeroth order. Also by definition we have .

Remove ads

Series expansions

Summarize
Perspective

The series expansions of the first and second order Bickley functions are given by:

where γ is the Euler constant and

is the th harmonic number.

Remove ads

Recurrence relation

Summarize
Perspective

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

where .

Remove ads

Asymptotic expansions

Summarize
Perspective

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

for .

Remove ads

Successive differentiation

Summarize
Perspective

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 Kin is shown below.

More information , ...
Remove ads

Computer code

Computer code in Fortran is made available by Amos.[10]

See also

References

Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads