Bickley–Naylor functions

The nth Bickley−Naylor function ${\displaystyle \operatorname {Ki} _{n}(x)}$ is defined by

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

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

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

The integral defining the function ${\displaystyle \operatorname {Ki} _{n}(x)}$ 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 ${\displaystyle \operatorname {Ki} _{n}(x)}$ include the integral,[7] integral forms the Bickley-Naylor function:

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

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

${\displaystyle \operatorname {Ki} _{n}(x)=\int _{1}^{\infty }{\frac {e^{-xt}dt}{t^{n}{\sqrt {t^{2}-1}}}}}$

${\displaystyle \operatorname {Ki} _{n}(x)={\frac {1}{(n-1)!}}\int _{x}^{\infty }(t-x)^{n-1}K_{0}(t)dt.}$

${\displaystyle {\frac {\operatorname {Ki} _{n}(x)}{x^{n}}}={\frac {1}{(n-1)!}}\int _{1}^{\infty }(t-1)^{n-1}K_{0}(xt)dt.}$

where ${\displaystyle \operatorname {K} _{0}(x)}$ is the modified Bessel function of the zeroth order. Also by definition we have ${\displaystyle \operatorname {Ki} _{0}(x)=\operatorname {K} _{0}(x)}$.

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

${\displaystyle \operatorname {Ki} _{1}(x)={\frac {\pi }{2}}+x\left(\gamma +\ln \left({\frac {x}{2}}\right)\right)\sum _{k=0}^{\infty }{\frac {(x^{2}/4)^{k}}{(k!)^{2}(2k+1)}}-x\sum _{k=0}^{\infty }{\frac {(x^{2}/4)^{k}}{(k!)^{2}(2k+1)^{2}}}-x\sum _{k=1}^{\infty }{\frac {(x^{2}/4)^{k}\Phi (k+1)}{(k!)^{2}(2k+1)}}}$

${\displaystyle \operatorname {Ki} _{2}(x)=1-{\frac {\pi }{2}}x-{\frac {x^{2}}{2}}\left(\gamma +\ln \left({\frac {x}{2}}\right)\right)\sum _{k=0}^{\infty }{\frac {(x^{2}/4)^{k}}{k!(k+1)!(2k+1)}}+{\frac {x^{2}}{4}}\sum _{k=0}^{\infty }{\frac {(4k+3)(x^{2}/4)^{k}}{k!(k+1)!(2k+1)^{2}}}+{\frac {x^{2}}{2}}\sum _{k=1}^{\infty }{\frac {(x^{2}/4)^{k}\Phi (k+1)}{k!(k+1)!(2k+1)}}}$

where γ is the Euler constant and

${\displaystyle \Phi (k+1)=1+{\frac {1}{2}}+{\frac {1}{3}}+...+{\frac {1}{k}}}$

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

${\displaystyle n\operatorname {Ki} _{n+1}(x)=(n-1)\operatorname {Ki} _{n-1}(x)-x\operatorname {Ki} _{n}(x)+x\operatorname {Ki} _{n-2}(x),~~~~~n\geq 2}$

where ${\displaystyle \operatorname {Ki} _{0}(x)=\operatorname {K} _{0}(x)}$.

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

${\displaystyle \operatorname {Ki} _{n}(x)\approx {\sqrt {\frac {\pi }{2x}}}e^{-x}\left\{1-{\frac {(1+4n)}{8x}}+{\frac {3(3+24n+16n^{2})}{128x^{2}}}\right\}}$

for ${\displaystyle x\gg 1}$

Differentiating ${\displaystyle Ki_{n+1}(x)}$ with respect to x gives

${\displaystyle {\frac {d}{dx}}\operatorname {Ki} _{n+1}(x)=-\operatorname {Ki} _{n}(x)}$

Successive differentiation yields

${\displaystyle {\frac {d^{n}}{dx^{n}}}\operatorname {Ki} _{n}(x)=(-1)^{n}\operatorname {K} _{0}(x)}$

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.

${\displaystyle x}$	${\displaystyle \operatorname {Ki} _{1}(x)}$	${\displaystyle \operatorname {Ki} _{2}(x)}$	${\displaystyle \operatorname {Ki} _{3}(x)}$
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 in Fortran is made available by Amos.[10]

• Exponential integral