The inverse tangent integral is a

special function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defin ...

, defined by: :

\operatorname_2(x) = \int_0^x \frac \, dt

Equivalently, it can be defined by a

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...

, or in terms of the

dilogarithm In mathematics, Spence's function, or dilogarithm, denoted as , is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself: :\operatorname_2(z) = -\int_0^z\, du \text ...

, a closely related special function.

Definition

The inverse tangent integral is defined by: :

\operatorname_2(x) = \int_0^x \frac \, dt

The

arctangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). S ...

is taken to be the

principal branch In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane. Examples Trigonometric inverses Principal branches are use ...

; that is, −/2 < arctan(''t'') < /2 for all real ''t''. Its

representation is :

\operatorname_2(x) = x - \frac + \frac - \frac + \cdots

which is

absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is sa ...

for

, x,  \le 1.

The inverse tangent integral is closely related to the

\operatorname_2(z) = \sum_^\infty \frac

and can be expressed simply in terms of it: :

\operatorname_2(z) = \frac \left( \operatorname_2(iz) - \operatorname_2(-iz) \right)

That is, :

\operatorname_2(x) = \operatorname(\operatorname_2(ix))

for all real ''x''.

Properties

The inverse tangent integral is an

odd function In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power s ...

: :

\operatorname_2(-x) = -\operatorname_2(x)

The values of Ti₂(''x'') and Ti₂(1/''x'') are related by the identity :

\operatorname_2(x) - \operatorname_2 \left(\frac \right) = \frac \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 Ti₂(1) is

Catalan's constant In mathematics, Catalan's constant , is defined by : G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots, where is the Dirichlet beta function. Its numerical value is approximately : It is not known whether is irra ...

1 - \frac + \frac - \frac + \cdots \approx 0.915966

Generalizations

Similar to the

polylogarithm In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natu ...

\operatorname_n(z) = \sum_^\infty \frac

, the function :

\operatorname_n(x) = x - \frac + \frac - \frac + \cdots

is defined analogously. This satisfies the recurrence relation: :

\operatorname_n(x) = \int_0^x \frac \, dt

Relation to other special functions

The inverse tangent integral is related to the

Legendre chi function In mathematics, the Legendre chi function is a special function whose Taylor series is also a Dirichlet series, given by \chi_\nu(z) = \sum_^\infty \frac. As such, it resembles the Dirichlet series for the polylogarithm, and, indeed, is trivially ...

\chi_2(x) = x + \frac + \frac + \cdots

by: :

\operatorname_2(x) = -i \chi_2(ix)

Note that

\chi_2(x)

can be expressed as

\int_0^x \frac \,  dt

, similar to the inverse tangent integral but with the

inverse hyperbolic tangent In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The ...

instead. The inverse tangent integral can also be written in terms of the

Lerch transcendent In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who publi ...

\Phi(z,s,a) = \sum_^\infty \frac:

\operatorname_2(x) = \frac x \Phi(-x^2, 2, 1/2)

History

The notation Ti₂ and Ti_''n'' is due to Lewin. Spence (1809) studied the function, using the notation

\overset(x)

. The function was also studied by Ramanujan. Appears in:

References

* * {{Cite book , last= Lewin , first= L. , title= Polylogarithms and Associated Functions , location= New York , publisher= North-Holland , year= 1981 , isbn= 978-0-444-00550-2 Special functions