|
| |
|
|
A163931
|
|
Decimal expansion of the higher order exponential integral E(x,m=2,n=1) at x =1
|
|
63
| |
|
|
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
(list; constant; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| 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.
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 formulae.
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.
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.
|
|
|
REFERENCES
| M.S. Milgram, The generalized integro-exponential function, Math. of Computation, Vol. 44, pp. 443-458, 1985.
|
|
|
LINKS
| 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. 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.
Eric. W. Weisstein, The Exponential Integral, from Wolfram MathWorld.
|
|
|
FORMULA
| E(x=1,m=2,n=1) = gamma^2/2 + Pi^2/12 + sum ((-1)^k/(k^2*k!), k=1..infinity)
E(x=0,n,m) = (1/(n-1))^m for n=>2.
int(E(t,m,n), t=0..x) = 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(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))
|
|
|
EXAMPLE
| E(1,2,1) = 0.09784319721667017932553778904528
|
|
|
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:
|
|
|
CROSSREFS
| Cf. A163927 (alpha(k,n)), A163930 (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 * A011359 A154827 A199003
Adjacent sequences: A163928 A163929 A163930 * A163932 A163933 A163934
|
|
|
KEYWORD
| cons,easy,nonn
|
|
|
AUTHOR
| Johannes W. Meijer & Nico Baken (meijgia(AT)hotmail.com and n.h.g.baken(AT)tudelft.nl), Aug 13 2009, Aug 17 2009
|
| |
|
|
