Inverse tangent integral

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

${\mathrm{Ti}}_2(x)={\int }_0^x\frac{\arctan t}{t}dt$

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:

${\mathrm{Ti}}_2(x)={\int }_0^x\frac{\arctan t}{t}dt$

The arctangent is taken to be the principal branch; that is, −Template:Pi/2 < arctan(t) < Template:Pi/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 $1-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\cdots \approx 0.915966$.

Integral relations

${\int }_0^{\pi /2}\mathrm{arcsinh}(r\sin \theta )\mathrm{d}\theta ={\mathrm{Ti}}_2(r),|r|\le 1$

${\int }_0^{\pi /2}\mathrm{arctanh}(r\sin \theta )\mathrm{d}\theta =2{\mathrm{Ti}}_2\left(\frac{1-\sqrt{1-r^2}}{r}\right),|r|\le 1\setminus \{0\}$

${\int }_0^{\pi /2}\mathrm{arctanh}(p\sin \theta )\arctan (q\sin \theta )\mathrm{d}\theta =\pi {\mathrm{Ti}}_2(\frac{1-\sqrt{1-p^2}}{p}\cdot \frac{\sqrt{1+q^2}-1}{q}),|p|\le 1\setminus \{0\},q\ne 0$

${\int }_0^{\alpha }{\int }_0^{\beta }\frac{\mathrm{d}x\mathrm{d}y}{1+x^2{y}^2}={\mathrm{Ti}}_2(\alpha \beta ),|\alpha \beta |\le 1$

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:

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

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

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