Special function related to the dilogarithm
The inverse tangent integral is a special function, defined by:
![{\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b92224e2a4a10024a3c0d40b90f0b3a6dcd78807)
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:
![{\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b92224e2a4a10024a3c0d40b90f0b3a6dcd78807)
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
![{\displaystyle \operatorname {Ti} _{2}(x)=x-{\frac {x^{3}}{3^{2}}}+{\frac {x^{5}}{5^{2}}}-{\frac {x^{7}}{7^{2}}}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4cb2ab07dd987c96ad2ec75b0fc7681e8f2c6b8)
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:
![{\displaystyle \operatorname {Ti} _{2}(z)={\frac {1}{2i}}\left(\operatorname {Li} _{2}(iz)-\operatorname {Li} _{2}(-iz)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff48c678aceb4e8f548e6ed7359f7f8242970f5)
That is,
![{\displaystyle \operatorname {Ti} _{2}(x)=\operatorname {Im} (\operatorname {Li} _{2}(ix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52525101ad02a59e86f15ba44d4400fd6646ad70)
for all real x.[1]
Properties
The inverse tangent integral is an odd function:[1]
![{\displaystyle \operatorname {Ti} _{2}(-x)=-\operatorname {Ti} _{2}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58386497e90baf448f9b3a80880ab810d0d46284)
The values of Ti2(x) and Ti2(1/x) are related by the identity
![{\displaystyle \operatorname {Ti} _{2}(x)-\operatorname {Ti} _{2}\left({\frac {1}{x}}\right)={\frac {\pi }{2}}\log x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc3db2515d59c889c594864873739c951a0d645f)
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
Similar to the polylogarithm
, the function
![{\displaystyle \operatorname {Ti} _{n}(x)=\sum \limits _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{\left(2k+1\right)^{n}}}=x-{\frac {x^{3}}{3^{n}}}+{\frac {x^{5}}{5^{n}}}-{\frac {x^{7}}{7^{n}}}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/95ff2b62aa579068e044e27c2e71d883b092eb41)
is defined analogously. This satisfies the recurrence relation:[4]
![{\displaystyle \operatorname {Ti} _{n}(x)=\int _{0}^{x}{\frac {\operatorname {Ti} _{n-1}(t)}{t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/952b599b70a516ebee2f5c4c585e237323f5974e)
By this series representation it can be seen that the special values
, where
represents the Dirichlet beta function.
Relation to other special functions
The inverse tangent integral is related to the Legendre chi function
by:[1]
![{\displaystyle \operatorname {Ti} _{2}(x)=-i\chi _{2}(ix)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75b531f19b44ee7df018fe035ab53e3fb0b8569b)
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]
![{\displaystyle \operatorname {Ti} _{2}(x)={\frac {1}{4}}x\Phi (-x^{2},2,1/2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb1b78b4976d993f3f9ad9960c6ccc51d752e43)
History
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
- ^ a b c d e Lewin 1981, pp. 38–39, Section 2.1
- ^ 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. - ^ a b Lewin 1981, pp. 39–40, Section 2.2
- ^ Lewin 1981, p. 190, Section 7.1.2
- ^ Weisstein, Eric W. "Inverse Tangent Integral". MathWorld.
- ^ 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.
- Lewin, L. (1958). Dilogarithms and Associated Functions. London: Macdonald. MR 0105524. Zbl 0083.35904.
- Lewin, L. (1981). Polylogarithms and Associated Functions. New York: North-Holland. ISBN 978-0-444-00550-2.