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, âÂÂ/2 < arctan(t) < /2 for all real t.
Its power series representation is
which is absolutely convergent for
The inverse tangent integral is closely related to the dilogarithm and can be expressed simply in terms of it:
That is,
for all real x.
The inverse tangent integral is an odd function:
The values of Ti<sub>2</sub>(x) and Ti<sub>2</sub>(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 .
The special value Ti<sub>2</sub>(1) is Catalan's constant .
Similar to the polylogarithm , the function
is defined analogously. This satisfies the recurrence relation:
By this series representation it can be seen that the special values , where represents the Dirichlet beta function.
The inverse tangent integral is related to the Legendre chi function by:
Note that can be expressed as , 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
The notation Ti<sub>2</sub> and Ti<sub>n</sub> is due to Lewin. Spence (1809) studied the function, using the notation . The function was also studied by Ramanujan.