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Inverse tangent integral

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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

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.

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.}

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.

Properties

The inverse tangent integral is an odd function:

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} .

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} .

Generalizations

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:

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

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:

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}}}:}

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

The notation Ti2 and Tin is due to Lewin. Spence (1809) studied the function, using the notation C n ( x ) {\displaystyle {\overset {n}{\operatorname {C} }}(x)} . The function was also studied by Ramanujan.

References

  1. ^ Lewin 1981, pp. 38–39, Section 2.1
  2. ^ 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.
  3. ^ 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.
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