Inverse tangent integral
# 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, −π/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 $|x|\leq 1.$

The inverse tangent integral is closely related to the dilogarithm $Li_2⁡(z)=∑_{n=1}^∞$ $z^n/n^2$ {\textstyle \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 $Ti_2(x)$ and $Ti_2(1/x)$ 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 $Ti_2(1)$ is Catalan's constant $1−1/3^2+1/5^2−1/7^2+⋯≈0.915966{\textstyle$ 1-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+\cdots \approx 0.915966}.

## Generalizations

Similar to the polylogarithm $Li_n⁡(z)=∑_{k=1}^∞$ $z^k/k^n$ {\textstyle \operatorname {Li} _{n}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{n}}}}, the function

    $\operatorname {Ti} _{n}(x)=x-{\frac {x^{3}}{3^{n}}}+{\frac {x^{5}}{5^{n}}}-{\frac {x^{7}}{7^{n}}}+\cdots$

is defined analogously. This satisfies the recurrence relation:

    $\operatorname {Ti} _{n}(x)=\int _{0}^{x}{\frac {\operatorname {Ti} _{n-1}(t)}{t}}\,dt$

## Relation to other special functions

The inverse tangent integral is related to the Legendre chi function $χ_2(x)=x+x^3/3^2+x^5/5^2+⋯{\textstyle$ \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 $∫_0^x$ artanh⁡t/t dt{\textstyle \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 Φ(z, s, $a)=∑_{n=0}^∞$ $z^n/((n+a)^s)$ :{\textstyle \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 $Ti_2$ and $Ti_n$ is due to Lewin. Spence (1809) studied the function, using the notation ${\overset {n}{\operatorname {C} }}(x)$. The function was also studied by Ramanujan.

