Inverse tangent integral explained

The inverse tangent integral is a special function, defined by:

\operatorname{Ti} _{2(x)} = \int ^{x} _{0} (\arctan t)/(t) dt Equivalently, it can be defined by a power series, or in terms of the dilogarithm, a closely related special function.

Definition

The inverse tangent integral is defined by:

\operatorname{Ti} _{2(x)} = \int ^{x} _{0} (\arctan t)/(t) dt The arctangent is taken to be the principal branch; that is, −/2 < arctan(t) < /2 for all real t.

Its power series representation is

\operatorname{Ti} _{2(x)} = x - (x ^{3})/(3 ^{2}) + (x ^{5})/(5 ^{2}) - (x ^{7})/(7 ^{2}) + … which is absolutely convergent for |x| \le 1 .

The inverse tangent integral is closely related to the dilogarithm $\backslash operatorname\_2(z)\; =\; \backslash sum\_^\backslash infty\; \backslash frac$ and can be expressed simply in terms of it:

\operatorname{Ti} _{2(z)} = (1)/(2 i) \left( \operatorname{Li} _{2(iz)} - \operatorname{Li} _{2(-iz)} \right) That is,\operatorname{Ti} _{2(x)} = \operatorname{Im}(\operatorname{Li} _{2(ix))} for all real x.

Properties

The inverse tangent integral is an odd function:

\operatorname{Ti} _{2(-x)} = - \operatorname{Ti} _{2(x)}

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

\operatorname{Ti} _{2(x)} - \operatorname{Ti} _{2} \left( (1)/(x) \right) = (\pi)/(2) 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 $1\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots\; \backslash approx\; 0.915966$.

Generalizations

Similar to the polylogarithm $\backslash operatorname\_n(z)\; =\; \backslash sum\_^\backslash infty\; \backslash frac$, the function

\operatorname{Ti} _{n} (x) = \sum\limits ^{inf ty} _{k =0} (( -1 ) ^{k} x ^{2k+1})/(\left( 2 k +1 \right) ^{n}) = x - (x ^{3})/(3 ^{n}) + (x ^{5})/(5 ^{n}) - (x ^{7})/(7 ^{n}) + … is defined analogously. This satisfies the recurrence relation:\operatorname{Ti} _{n} (x) = \int ^{x} _{0} (\operatorname{Ti)/(_{n-1}) (t)}{t} dt

By this series representation it can be seen that the special values

\operatorname{Ti} _{n} ( 1 ) = \beta(n) , where \beta(s) represents the Dirichlet beta function.

Relation to other special functions

The inverse tangent integral is related to the Legendre chi function $\backslash chi\_2(x)\; =\; x\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$ by:

\operatorname{Ti} _{2(x)} = - i \chi _{2(ix)} Note that \chi _{2(x)} can be expressed as $\backslash int\_0^x\; \backslash frac\; \backslash ,\; dt$, 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 $\backslash Phi(z,s,a)\; =\; \backslash sum\_^\backslash infty\; \backslash frac:$

\operatorname{Ti} _{2(x)} = (1)/(4) x \Phi( - x ^{2,} 2 , 1 / 2 )

History

The notation Ti₂ and Ti_n is due to Lewin. Spence (1809)^[1] studied the function, using the notation

\overset{n}{\operatorname{C}}(x) . The function was also studied by Ramanujan.^[2]

References

• Book: Lewin, L. . Dilogarithms and Associated Functions . London . Macdonald . 1958 . 0105524 . 0083.35904 .
• Book: Lewin, L. . Polylogarithms and Associated Functions . New York . North-Holland . 1981 . 978-0-444-00550-2 .

Notes and References

1. Book: 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 .
2. S. . Ramanujan . Srinivasa Ramanujan . Journal of the Indian Mathematical Society . 7 . 1915 . 93–96 . On the integral \int ^{x} _{0} (\tan ^{-1} t)/(t) dt . Appears in: Book: Collected Papers of Srinivasa Ramanujan . G. H. . Hardy . G. H. Hardy . P. V. . Seshu Aiyar . B. M. . Wilson . Bertram Martin Wilson . 1927 . 40–43 .