Inverse tangent integral

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

${\mathrm{Ti}}_2(x)=\underset{0}{\overset{x}{\int }}\frac{\arctan t}{t}dt$

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:

${\mathrm{Ti}}_2(x)=\underset{0}{\overset{x}{\int }}\frac{\arctan t}{t}dt$

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

${\mathrm{Ti}}_2(x)=x-\frac{x^3}{3^2}+\frac{x^5}{5^2}-\frac{x^7}{7^2}+\cdots$

which is absolutely convergent for $|x|\le 1.$ The inverse tangent integral is closely related to the dilogarithm ${\mathrm{Li}}_2(z)={\sum }_{n=1}^{\mathrm{\infty }}\frac{z^n}{n^2}$ and can be expressed simply in terms of it:

${\mathrm{Ti}}_2(z)=\frac{1}{2i}({\mathrm{Li}}_2(iz)-{\mathrm{Li}}_2(-iz))$

That is,

${\mathrm{Ti}}_2(x)=\mathrm{Im}({\mathrm{Li}}_2(ix))$

for all real x.

Properties

The inverse tangent integral is an odd function:

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

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

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

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

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

is defined analogously. This satisfies the recurrence relation:

${\mathrm{Ti}}_n(x)=\underset{0}{\overset{x}{\int }}\frac{{\mathrm{Ti}}_{n-1}(t)}{t}dt$

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

Relation to other special functions

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

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

Note that ${\chi }_2(x)$ 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 $\mathrm{\Phi }(z,s,a)={\sum }_{n=0}^{\mathrm{\infty }}\frac{z^n}{(n+a{)}^s}:$

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

History

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

References

• 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.