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 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 , the function
is defined analogously. This satisfies the recurrence relation: [4]
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: [1]
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 [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. [2]
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 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 , the function
is defined analogously. This satisfies the recurrence relation: [4]
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: [1]
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 [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. [2]