Bickley–Naylor functions

The nth Bickley−Naylor function Ki_n(x) is defined by

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

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

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 as a → 0 in the interval of integration, [a, π/2].

Alternative ways to define the function Ki_n include the integral[6]

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

All of the functions Ki_n for positive integer n are monotonously decreasing functions, because e^−x is a decreasing function and ${\displaystyle \sin x}$ is a positive increasing function for ${\displaystyle x\in (0,\pi /2)}$.

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.

• Exponential integral