The inverse tangent integral is a special function, defined by: Equivalently, it can be defined by a power series, or in terms of the dilogarithm, a closely related special function.
The inverse tangent integral is defined by: The arctangent is taken to be the principal branch; that is, −π/2 < arctan(t) < π/2 for all real t.[1] Its power series representation is which is absolutely convergent for
[1] The inverse tangent integral is closely related to the dilogarithm
Li
{\textstyle \operatorname {Li} _{2}(z)=\sum _{n=1}^{\infty }{\frac {z^{n}}{n^{2}}}}
and can be expressed simply in terms of it: That is, for all real x.
[1] The inverse tangent integral is an odd function:[1] The values of Ti2(x) and Ti2(1/x) are related by the identity valid for all x > 0 (or, more generally, for Re(x) > 0).
This can be proven by differentiating and using the identity
[2][3] The special value Ti2(1) is Catalan's constant
[3] Similar to the polylogarithm
Li
{\textstyle \operatorname {Li} _{n}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{n}}}}
, the function is defined analogously.
This satisfies the recurrence relation:[4] By this series representation it can be seen that the special values
( 1 ) = β ( n )
represents the Dirichlet beta function.
The inverse tangent integral is related to the Legendre chi function
{\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
[5] The notation Ti2 and Tin is due to Lewin.
Spence (1809)[6] studied the function, using the notation
The function was also studied by Ramanujan.