Inverse tangent integral

The inverse tangent integral is a special function, defined by:
 * $$\operatorname{Ti}_2(x) = \int_0^x \frac{\arctan t}{t} \, dt$$

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:
 * $$\operatorname{Ti}_2(x) = \int_0^x \frac{\arctan t}{t} \, dt$$

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

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.