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Abel's Integral

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Abel's integral is the definite integral

I=int_0^infty(tdt)/((e^(pit)-e^(-pit))(t^2+1))
(1)
=1/2int_(-infty)^infty(tdt)/((e^(pit)-e^(-pit))(t^2+1))
(2)
=1/2int_0^infty(tcsch(pit))/(t^2+1)dt
(3)
=1/2int_0^infty(csch(pi/x))/(x(x^2-1))dx
(4)
=1/(pi^2)int_0^1(lnxdx)/((x^2-1)[((lnx)/pi)^2+1])
(5)
=sum_(n=1,3,5,...)Ci(pin)
(6)
=1/2ln2-1/4
(7)
=0.09657359...
(8)

(OEIS A102047), where Ci(x) is a cosine integral.

See also

Definite Integral

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References

Abel, N. H. "Oplösning af et Par Opgaver ved Hjelp af bestemt Integraler." Mag. for naturvidenskaberne 1, 55-68, 1823. Reprinted as "Solution de quelques problémes à l'aide d'intégrales définies." Paper II in Abel, N. H. Œ (Ed. L. Sylow and S. Lie). Christiania [Oslo], Norway, pp. 11-27, 1881. Reprinted by J. Gabay, 1992.Abel, N. H. "Oplösning af nogle Opgaver ved Hjelp af bestemt Integraler." Mag. for naturvidenskaberne 1, 205-215, 1823. Reprinted as "Solution de quelques problémes à l'aide d'intégrales définies." Paper II in Abel, N. H. Œ (Ed. L. Sylow and S. Lie). Christiania [Oslo], Norway, 1881. Reprinted by J. Gabay, 1992.Sloane, N. J. A. Sequence A102047 in "The On-Line Encyclopedia of Integer Sequences."

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Abel's Integral

Cite this as:

Weisstein, Eric W. "Abel's Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AbelsIntegral.html

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