# 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
    $\operatorname {Ti} _{n}(x)=\sum \limits _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{\left(2k+1\right)^{n}}}=x-{\frac {x^{3}}{3^{n}}}+{\frac {x^{5}}{5^{n}}}-{\frac {x^{7}}{7^{n}}}+\cdots$
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 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.