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

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, 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, 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

Offset

0,2

Comments

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

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.

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 A090998.

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

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251.

J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

M. S. Milgram, The generalized integro-exponential function, Math. of Computation, Vol. 44, pp. 443-458, 1985.

Eric Weisstein's World of Mathematics, The Exponential Integral.

Formula

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

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

Integral_{t=0..x} E(t,m,n) = 1/n^m - E(x,n,n+1).

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

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

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

Example

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

Maple

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:

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A163927 (alpha(k,n)), A090998 (gamma(k,n) = G(k,n)), A163932.

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

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

Sequence in context: A086278 A081855 A019887 * A277774 A011359 A154827

Adjacent sequences: A163928 A163929 A163930 * A163932 A163933 A163934

Keyword

cons,easy,nonn

Author

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

Status

approved