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A163931 Decimal expansion of the higher-order exponential integral E(x, m=2, n=1) at x=1. 72

%I

%S 0,9,7,8,4,3,1,9,7,2,1,6,6,7,0,1,7,9,3,2,5,5,3,7,7,8,9,0,4,5,2,8,0,0,

%T 8,2,7,6,9,5,8,2,2,6,9,5,3,0,2,6,5,7,6,5,5,7,4,4,2,1,2,4,2,4,5,4,4,7,

%U 1,3,7,6,2,6,1,4,0,9,0,4,8,8,7,3,6,9,6,0,4,8,9,1,8,5,5,5,0,8,9,4,5,4,6,7,0

%N Decimal expansion of the higher-order exponential integral E(x, m=2, n=1) at x=1.

%C We define the higher-order exponential integrals by E(x,m,n) = x^(n-1)*int(E(t,m-1,n)/t^n, t=x..infinity) for m => 0 and n => 1 with E(x,m=0,n) = exp(-x), see Meijer and Baken.

%C The properties of the E(x,m,n) are analogous to those of the well-known exponential integrals E(x,m=1,n), see Abramowitz and Stegun and the formulas.

%C The series expansions of the higher-order exponential integrals are dominated by the constants alpha(k,n), see A163927, and gamma(k,n) = G(k,n), see A163930.

%C For information about the asymptotic expansion of the E(x,m,n) see A163932.

%C Values of E(x,m,n) can be evaluated with the Maple program.

%H G. C. Greubel, <a href="/A163931/b163931.txt">Table of n, a(n) for n = 0..5000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251.

%H J. W. Meijer and N. H. G. Baken, <a href="http://dx.doi.org/10.1016/0167-7152(87)90041-1">The Exponential Integral Distribution</a>, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

%H M.S. Milgram, <a href="http://dx.doi.org/10.1090/S0025-5718-1985-0777276-4">The generalized integro-exponential function</a>, Math. of Computation, Vol. 44, pp. 443-458, 1985.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">The Exponential Integral</a>.

%F E(x=1,m=2,n=1) = gamma^2/2 + Pi^2/12 + Sum_{k=1..infinity} ((-1)^k/(k^2*k!).

%F E(x=0,n,m) = (1/(n-1))^m for n=>2.

%F int(E(t,m,n), t=0..x) = 1/n^m - E(x,n,n+1).

%F dE(x,m,n+1)/dx = - E(x,m,n).

%F E(x,m,n+1) = (1/n)*(E(x,m-1,n+1)-x*E(x,m,n)).

%F E(x,m,n) = (-1)^m*((-x)^(n-1)/(n-1)!*sum(alpha(kz,n)*(G(m-2*kz,n) + Sum_{kz=0..floor(m/2)} (G(m-2*kz-i,n)*log(x)^i/i!,i=1..m-2*kz))) + Sum_{kx=0..(n-2)} ((-x)^kx/((kx-n+1)^m*kx!)) + Sum_{ky=n..infinity}((-x)^ky/((ky-n+1)^m*ky!))).

%e E(1,2,1) = 0.09784319721667017932553778904528008276958226953026576557442124245....

%p E:= proc(x,m,n) local nmax, kmax, EI, k1, k2, n1, n2; option remember: nmax:=20; kmax:=20; k1:=0: for n1 from 0 to nmax do alpha(k1,n1):=1 od: for k1 from 1 to kmax do for n1 from 1 to nmax do alpha(k1,n1) := (1/k1)*sum(sum(p^(-2*(k1-i1)),p=0..n1-1)*alpha(i1, n1),i1=0..k1-1) od; od: for n2 from 0 to kmax do G(0,n2):=1 od: for n2 from 1 to nmax do for k2 from 1 to kmax do G(k2,n2):=(1/k2)*(((gamma-sum(p^(-1),p=1..n2-1))*G(k2-1,n2)+ sum((Zeta(k2-i2)-sum(p^(-(k2-i2)), p=1..n2-1))*G(i2,n2),i2=0..k2-2))) od; od: EI:= evalf((-1)^m*((-x)^(n-1)/(n-1)!*sum(alpha(kz,n)*(G(m-2*kz,n)+sum(G(m-2*kz-i,n)*ln(x)^i/i!,i=1..m-2*kz)), kz=0..floor(m/2)) + sum((-x)^kx/((kx-n+1)^m*kx!),kx=0..n-2) + sum((-x)^ky/((ky-n+1)^m*ky!),ky=n..infinity))); return(EI): end:

%t Join[{0}, RealDigits[ N[ EulerGamma^2/2 + Pi^2/12 - HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -1], 104]][[1]]] (* _Jean-François Alcover_, Nov 07 2012, from 1st formula *)

%o (PARI) t=1; Euler^2/2 + Pi^2/12 + sumalt(k=1, t*=k; (-1)^k/(k^2*t)) \\ _Charles R Greathouse IV_, Nov 07 2016

%Y Cf. A163927 (alpha(k,n)), A163930 (gamma(k,n) = G(k,n)), A163932.

%Y Cf. A068985 (E(x=1,m=0,n) = exp(-1)) and A099285 (E(x=1,m=1,n=1)).

%Y Cf. A001563 (n*n!), A002775 (n^2*n!), A091363 (n^3*n!) and A091364 (n^4*n!).

%K cons,easy,nonn

%O 0,2

%A _Johannes W. Meijer_ & Nico Baken (n.h.g.baken(AT)tudelft.nl), Aug 13 2009, Aug 17 2009

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