Inverse tangent integral

Definition

The inverse tangent integral is defined by: :\operatorname{Ti}_2(x) = \int_0^x \frac{\arctan t}{t} , dt The arctangent is taken to be the principal branch; that is, −/2

Its power series representation is :\operatorname{Ti}_2(x) = x - \frac{x^3}{3^2} + \frac{x^5}{5^2} - \frac{x^7}{7^2} + \cdots which is absolutely convergent for |x| \le 1.

The inverse tangent integral is closely related to the dilogarithm \operatorname{Li}2(z) = \sum{n=1}^\infty \frac{z^n}{n^2} and can be expressed simply in terms of it: :\operatorname{Ti}_2(z) = \frac{1}{2i} \left( \operatorname{Li}_2(iz) - \operatorname{Li}_2(-iz) \right) That is, :\operatorname{Ti}_2(x) = \operatorname{Im}(\operatorname{Li}_2(ix)) for all real x.

Properties

The inverse tangent integral is an odd function: :\operatorname{Ti}_2(-x) = -\operatorname{Ti}_2(x)

The values of Ti2(x) and Ti2(1/x) are related by the identity :\operatorname{Ti}_2(x) - \operatorname{Ti}_2 \left(\frac{1}{x} \right) = \frac{\pi}{2} \log x valid for all x 0 (or, more generally, for Re(x) 0). This can be proven by differentiating and using the identity \arctan(t) + \arctan(1/t) = \pi/2.

The special value Ti2(1) is Catalan's constant 1 - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots \approx 0.915966.

Generalizations

Similar to the polylogarithm \operatorname{Li}n(z) = \sum{k=1}^\infty \frac{z^k}{k^n}, the function :\operatorname{Ti}{n}(x) = \sum\limits{k=0}^{\infty}\frac{(-1)^{k}x^{2k+1}}{\left(2k+1\right)^{n}}=x - \frac{x^3}{3^n} + \frac{x^5}{5^n} - \frac{x^7}{7^n} + \cdots is defined analogously. This satisfies the recurrence relation: :\operatorname{Ti}{n}(x) = \int_0^x \frac{\operatorname{Ti}{n-1}(t)}{t} , dt

By this series representation it can be seen that the special values \operatorname{Ti}_{n}(1)=\beta(n), where \beta(s) represents the Dirichlet beta function.

Relation to other special functions

The inverse tangent integral is related to the Legendre chi function \chi_2(x) = x + \frac{x^3}{3^2} + \frac{x^5}{5^2} + \cdots by: :\operatorname{Ti}_2(x) = -i \chi_2(ix) Note that \chi_2(x) can be expressed as \int_0^x \frac{\operatorname{artanh} t}{t} , dt, similar to the inverse tangent integral but with the inverse hyperbolic tangent instead.

The inverse tangent integral can also be written in terms of the Lerch transcendent \Phi(z,s,a) = \sum_{n=0}^\infty \frac{z^n}{(n+a)^s}: :\operatorname{Ti}_2(x) = \frac{1}{4} x \Phi(-x^2, 2, 1/2)

History

The notation Ti2 and Tin is due to Lewin. Spence (1809) studied the function, using the notation \overset{n}{\operatorname{C}}(x). The function was also studied by Ramanujan.

References

References

1. {{harvnb. Lewin. 1981
2. {{harvnb. Lewin. 1981
3. {{harvnb. Lewin. 1981
4. "Inverse Tangent Integral".
5. Spence, William. (1809). "An essay on the theory of the various orders of logarithmic transcendents; with an inquiry into their applications to the integral calculus and the summation of series".
6. Ramanujan, S.. (1927). "On the integral $\backslash int\_0^x\; \backslash frac\{\backslash tan^\{-1\}\; t\}\{t\}\; ,\; dt$}} Appears in: {{Cite book". Journal of the Indian Mathematical Society.