Special function related to the dilogarithm
The inverse tangent integral is a special function , defined by:
Ti 2 ( x ) = ∫ 0 x arctan t t d t {\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\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 [ edit ] The inverse tangent integral is defined by:
Ti 2 ( x ) = ∫ 0 x arctan t t d t {\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt} 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
Ti 2 ( x ) = x − x 3 3 2 + x 5 5 2 − x 7 7 2 + ⋯ {\displaystyle \operatorname {Ti} _{2}(x)=x-{\frac {x^{3}}{3^{2}}}+{\frac {x^{5}}{5^{2}}}-{\frac {x^{7}}{7^{2}}}+\cdots } which is absolutely convergent for | x | ≤ 1. {\displaystyle |x|\leq 1.} [1]
The inverse tangent integral is closely related to the dilogarithm Li 2 ( z ) = ∑ n = 1 ∞ z n n 2 {\textstyle \operatorname {Li} _{2}(z)=\sum _{n=1}^{\infty }{\frac {z^{n}}{n^{2}}}} and can be expressed simply in terms of it:
Ti 2 ( z ) = 1 2 i ( Li 2 ( i z ) − Li 2 ( − i z ) ) {\displaystyle \operatorname {Ti} _{2}(z)={\frac {1}{2i}}\left(\operatorname {Li} _{2}(iz)-\operatorname {Li} _{2}(-iz)\right)} That is,
Ti 2 ( x ) = Im ( Li 2 ( i x ) ) {\displaystyle \operatorname {Ti} _{2}(x)=\operatorname {Im} (\operatorname {Li} _{2}(ix))} for all real x .[1]
Properties [ edit ] The inverse tangent integral is an odd function :[1]
Ti 2 ( − x ) = − Ti 2 ( x ) {\displaystyle \operatorname {Ti} _{2}(-x)=-\operatorname {Ti} _{2}(x)} The values of Ti2 (x ) and Ti2 (1/x ) are related by the identity
Ti 2 ( x ) − Ti 2 ( 1 x ) = π 2 log x {\displaystyle \operatorname {Ti} _{2}(x)-\operatorname {Ti} _{2}\left({\frac {1}{x}}\right)={\frac {\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 ) = π / 2 {\displaystyle \arctan(t)+\arctan(1/t)=\pi /2} .[2] [3]
The special value Ti2 (1) is Catalan's constant 1 − 1 3 2 + 1 5 2 − 1 7 2 + ⋯ ≈ 0.915966 {\textstyle 1-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+\cdots \approx 0.915966} .[3]
Generalizations [ edit ] Similar to the polylogarithm Li n ( z ) = ∑ k = 1 ∞ z k k n {\textstyle \operatorname {Li} _{n}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{n}}}} , the function
Ti n ( x ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 ( 2 k + 1 ) n = x − x 3 3 n + x 5 5 n − x 7 7 n + ⋯ {\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 } is defined analogously. This satisfies the recurrence relation:[4]
Ti n ( x ) = ∫ 0 x Ti n − 1 ( t ) t d t {\displaystyle \operatorname {Ti} _{n}(x)=\int _{0}^{x}{\frac {\operatorname {Ti} _{n-1}(t)}{t}}\,dt} By this series representation it can be seen that the special values Ti n ( 1 ) = β ( n ) {\displaystyle \operatorname {Ti} _{n}(1)=\beta (n)} , where β ( s ) {\displaystyle \beta (s)} represents the Dirichlet beta function .
Relation to other special functions [ edit ] The inverse tangent integral is related to the Legendre chi function χ 2 ( x ) = x + x 3 3 2 + x 5 5 2 + ⋯ {\textstyle \chi _{2}(x)=x+{\frac {x^{3}}{3^{2}}}+{\frac {x^{5}}{5^{2}}}+\cdots } by:[1]
Ti 2 ( x ) = − i χ 2 ( i x ) {\displaystyle \operatorname {Ti} _{2}(x)=-i\chi _{2}(ix)} Note that χ 2 ( x ) {\displaystyle \chi _{2}(x)} can be expressed as ∫ 0 x artanh t t d t {\textstyle \int _{0}^{x}{\frac {\operatorname {artanh} t}{t}}\,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 Φ ( z , s , a ) = ∑ n = 0 ∞ z n ( n + a ) s : {\textstyle \Phi (z,s,a)=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+a)^{s}}}:} [5]
Ti 2 ( x ) = 1 4 x Φ ( − x 2 , 2 , 1 / 2 ) {\displaystyle \operatorname {Ti} _{2}(x)={\frac {1}{4}}x\Phi (-x^{2},2,1/2)} History [ edit ] The notation Ti2 and Tin is due to Lewin. Spence (1809)[6] studied the function, using the notation C n ( x ) {\displaystyle {\overset {n}{\operatorname {C} }}(x)} . The function was also studied by Ramanujan .[2]
References [ edit ] ^ a b c d e Lewin 1981 , pp. 38–39, Section 2.1 ^ a b Ramanujan, S. (1915). "On the integral ∫ 0 x tan − 1 t t d t {\displaystyle \int _{0}^{x}{\frac {\tan ^{-1}t}{t}}\,dt} ". 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.