Inverse tangent integral

Properties

The inverse tangent integral is an odd function:

${\displaystyle {\mathrm{Ti}}_2(-x)=-{\mathrm{Ti}}_2(x)}$

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

${\displaystyle {\mathrm{Ti}}_2(x)-{\mathrm{Ti}}_2\left(\frac{1}{x}\right)=\frac{\pi }{2}\log x}$

valid for all x > 0 (or, more generally, for Re(x) > 0). This can be proven by differentiating and using the identity ${\displaystyle \arctan (t)+\arctan (1/t)=\pi /2}$.

The special value Ti₂(1) is Catalan's constant $1-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\cdots \approx 0.915966$.

Generalizations

Similar to the polylogarithm ${\mathrm{Li}}_n(z)=\sum _{k=1}^{\mathrm{\infty }}\frac{z^k}{k^n}$, the function

${\displaystyle {\mathrm{Ti}}_n(x)=\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k{x}^{2k+1}}{{\left(2k+1\right)}^n}=x-\frac{x^3}{3^n}+\frac{x^5}{5^n}-\frac{x^7}{7^n}+\cdots }$

is defined analogously. This satisfies the recurrence relation:

${\displaystyle {\mathrm{Ti}}_n(x)={\int }_0^x\frac{{\mathrm{Ti}}_{n-1}(t)}{t}dt}$

By this series representation it can be seen that the special values ${\displaystyle {\mathrm{Ti}}_n(1)=\beta (n)}$, where ${\displaystyle \beta (s)}$ represents the Dirichlet beta function.

Relation to other special functions

The inverse tangent integral is related to the Legendre chi function ${\chi }_2(x)=x+\frac{x^3}{3^2}+\frac{x^5}{5^2}+\cdots$ by:

${\displaystyle {\mathrm{Ti}}_2(x)=-i{\chi }_2(ix)}$

Note that ${\displaystyle {\chi }_2(x)}$ can be expressed as ${\int }_0^x\frac{\mathrm{artanh}t}{t}dt$, 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 $\mathrm{\Phi }(z,s,a)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{(n+a{)}^s}:$

${\displaystyle {\mathrm{Ti}}_2(x)=\frac{1}{4}x\mathrm{\Phi }(-x^2,2,1/2)}$

History

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