Inverse tangent integral

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.

Definition

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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]

Properties

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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]

Generalizations

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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.

Relation to other special functions

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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]

 

History

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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]

References

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  1. ^ a b c d e Lewin 1981, pp. 38–39, Section 2.1
  2. ^ a b Ramanujan, S. (1915). "On the integral  ". Journal of the Indian Mathematical Society. 7: 93–96. Appears in: Hardy, G. H.; Seshu Aiyar, P. V.; Wilson, B. M., eds. (1927). Collected Papers of Srinivasa Ramanujan. pp. 40–43.
  3. ^ a b Lewin 1981, pp. 39–40, Section 2.2
  4. ^ Lewin 1981, p. 190, Section 7.1.2
  5. ^ Weisstein, Eric W. "Inverse Tangent Integral". MathWorld.
  6. ^ Spence, William (1809). An essay on the theory of the various orders of logarithmic transcendents; with an inquiry into their applications to the integral calculus and the summation of series. London.