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Inverse Tangent Integral

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The inverse tangent integral Ti_2(x) is defined in terms of the dilogarithm Li_2(x) by

Li_2(ix)=1/4Li_2(-x^2)+iTi_2(x)
(1)

(Lewin 1958, p. 33). It has the series

Ti_2(x)=sum_(k=1)^infty(-1)^(k-1)(x^(2k-1))/((2k-1)^2)
(2)

and gives in closed form the sum

sum_(n=1)^infty(sin[(4n-2)x])/((2n-1)^2)=Ti_2(tanx)-xln(tanx)
(3)

that was considered by Ramanujan (Lewin 1958, p. 39). The inverse tangent integral can be expressed in terms of the dilogarithm as

Ti_2(x)=1/(2i)[Li_2(ix)-Li_2(-ix)],
(4)

in terms of Legendre's chi-function as

Ti_2(x)=-ichi_2(ix),
(5)

in terms of the Lerch transcendent by

Ti_2(x)=1/4xPhi(-x^2,2,1/2),
(6)

and as the integral

Ti_2(x)=int_0^x(tan^(-1)(x^'))/(x^')dx^'.
(7)

Ti_2(x) has derivative

(dTi_2(x))/(dx)=(tan^(-1)x)/x.
(8)

It satisfies the identities

Ti_2(x)-Ti_2(1/x)=1/2pisgn(x)ln|x| 1/2Ti_2((2x)/(1-x^2))=Ti_2(x)+Ti_2(-x,1)-Ti_2(x,1),
(9)

where

Ti_2(x,a)=int_0^x(tan^(-1)x^')/(a+x^')dx^'
(10)

is the generalized inverse tangent function.

Ti_2(x) has the special value

Ti_2(1)=K,
(11)

where K is Catalan's constant, and the functional relationships

3Ti_2(1)-2Ti_2(1/2)-Ti_2(1/3)-1/2Ti_2(3/4)=1/2piln2,
(12)

the two equivalent identities

3Ti_2(2-sqrt(3))=2Ti_2(1)-1/4piln(2-sqrt(3))
(13)
Ti_2(tan(1/(12)pi))=2/3Ti_2(tan(1/4pi))+1/(12)piln(tan(1/(12)pi)),
(14)

and

3Ti_2(2+sqrt(3))=2Ti_2(1)+5/4piln(2+sqrt(3))
(15)

(Lewin 1958, p. 39). The triplication formula is given by

1/3Ti_2((3x-x^3)/(1-3x^2))=Ti_2(x)+Ti_2((1-xsqrt(3))/(sqrt(3)+x)) -Ti_2((1+xsqrt(3))/(sqrt(3)-x))+1/6piln[((sqrt(3)+x)(1+xsqrt(3)))/((1-xsqrt(3))(sqrt(3)-x))],
(16)

which leads to

Ti_2(tan(1/(24)pi))-Ti_2(tan(5/(24)pi))+2/3Ti_2(tan(1/8pi)) +1/6piln[(tan(5/(24)pi))/(tan(1/8pi))]=0
(17)

and the algebraic form

Ti_2((sqrt(3)-sqrt(2))/(sqrt(2)+1))-Ti_2((sqrt(3)-sqrt(2))/(sqrt(2)-1))+2/3Ti_2(sqrt(2)-1) =1/6piln[(sqrt(2)-1)/((sqrt(3)-sqrt(2))(sqrt(2)+1))]
(18)

(Lewin 1958, p. 41).

See also

Dilogarithm, L-Algebraic Number, Legendre's Chi-Function, Lerch Transcendent, Rogers L-Function

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References

Finch, S. R. "Inverse Tangent Integral." §1.7.6 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 57, 2003.Lewin, L. "The Inverse Tangent Integral" and "The Generalized Inverse Tangent Integral." Chs. 2-3 in Dilogarithms and Associated Functions. London: Macdonald, pp. 33-90, 1958.Lewin, L. Polylogarithms and Associated Functions. Amsterdam, Netherlands: North-Holland, p. 45, 1981.Nielsen, N. "Der Eulersche Dilogarithmus und seine Verallgemeinerungen." Nova Acta Leopoldina, Abh.der Kaiserlich Leopoldinisch-Carolinischen Deutschen Akad. der Naturforsch. 90, 121-212, 1909.

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Inverse Tangent Integral

Cite this as:

Weisstein, Eric W. "Inverse Tangent Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseTangentIntegral.html

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