Inverse tangent integral

Short description: Special function related to the dilogarithm

The inverse tangent integral is a special function, defined by:

[math]\displaystyle{ \operatorname{Ti}_2(x) = \int_0^x \frac{\arctan t}{t} \, dt }[/math]

Equivalently, it can be defined by a power series, or in terms of the dilogarithm, a closely related special function.

Definition

The inverse tangent integral is defined by:

[math]\displaystyle{ \operatorname{Ti}_2(x) = \int_0^x \frac{\arctan t}{t} \, dt }[/math]

The arctangent is taken to be the principal branch; that is, −π/2 < arctan(t) < π/2 for all real t.

Its power series representation is

[math]\displaystyle{ \operatorname{Ti}_2(x) = x - \frac{x^3}{3^2} + \frac{x^5}{5^2} - \frac{x^7}{7^2} + \cdots }[/math]

which is absolutely convergent for [math]\displaystyle{ |x| \le 1. }[/math]

The inverse tangent integral is closely related to the dilogarithm [math]\displaystyle{ \operatorname{Li}_2(z) = \sum_{n=1}^\infty \frac{z^n}{n^2} }[/math] and can be expressed simply in terms of it:

[math]\displaystyle{ \operatorname{Ti}_2(z) = \frac{1}{2i} \left( \operatorname{Li}_2(iz) - \operatorname{Li}_2(-iz) \right) }[/math]

That is,

[math]\displaystyle{ \operatorname{Ti}_2(x) = \operatorname{Im}(\operatorname{Li}_2(ix)) }[/math]

for all real x.

Properties

The inverse tangent integral is an odd function:

[math]\displaystyle{ \operatorname{Ti}_2(-x) = -\operatorname{Ti}_2(x) }[/math]

The values of Ti₂(x) and Ti₂(1/x) are related by the identity

[math]\displaystyle{ \operatorname{Ti}_2(x) - \operatorname{Ti}_2 \left(\frac{1}{x} \right) = \frac{\pi}{2} \log x }[/math]

valid for all x > 0 (or, more generally, for Re(x) > 0). This can be proven by differentiating and using the identity [math]\displaystyle{ \arctan(t) + \arctan(1/t) = \pi/2 }[/math].

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

Generalizations

Similar to the polylogarithm [math]\displaystyle{ \operatorname{Li}_n(z) = \sum_{k=1}^\infty \frac{z^k}{k^n} }[/math], the function

[math]\displaystyle{ \operatorname{Ti}_n(x) = x - \frac{x^3}{3^n} + \frac{x^5}{5^n} - \frac{x^7}{7^n} + \cdots }[/math]

is defined analogously. This satisfies the recurrence relation:

[math]\displaystyle{ \operatorname{Ti}_n(x) = \int_0^x \frac{\operatorname{Ti}_{n-1}(t)}{t} \, dt }[/math]

Relation to other special functions

The inverse tangent integral is related to the Legendre chi function [math]\displaystyle{ \chi_2(x) = x + \frac{x^3}{3^2} + \frac{x^5}{5^2} + \cdots }[/math] by:

[math]\displaystyle{ \operatorname{Ti}_2(x) = -i \chi_2(ix) }[/math]

Note that [math]\displaystyle{ \chi_2(x) }[/math] can be expressed as [math]\displaystyle{ \int_0^x \frac{\operatorname{artanh} t}{t} \, dt }[/math], 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 [math]\displaystyle{ \Phi(z,s,a) = \sum_{n=0}^\infty \frac{z^n}{(n+a)^s}: }[/math]

[math]\displaystyle{ \operatorname{Ti}_2(x) = \frac{1}{4} x \Phi(-x^2, 2, 1/2) }[/math]

History

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

References

• Lewin, L. (1958). Dilogarithms and Associated Functions. London: Macdonald. 
• Lewin, L. (1981). Polylogarithms and Associated Functions. New York: North-Holland. ISBN 978-0-444-00550-2.