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Bickley–Naylor functions
Functions for thermal radiation in hot enclosures From Wikipedia, the free encyclopedia
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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]

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Definition
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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 .
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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 .
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Series expansions
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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.
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Recurrence relation
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The Bickley functions also satisfy the following recurrence relation:[8]
where .
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Asymptotic expansions
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The asymptotic expansions of Bickley functions are given as[9]
for .
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Successive differentiation
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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.
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Computer code
Computer code in Fortran is made available by Amos.[10]
See also
References
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