inverse tangent integral

The inverse tangent integral is defined by:

:$\backslash operatorname\{Ti\}\_2(x)\; =\; \backslash int\_0^x\; \backslash frac\{\backslash arctan\; t\}\{t\}\; \backslash ,\; dt$

The arctangent is taken to be the principal branch; that is, −{{pi}}/2 < arctan(t) < {{pi}}/2 for all real t.{{harvnb|Lewin|1981|pp=38–39|loc=Section 2.1}}

Its power series representation is

:$\backslash operatorname\{Ti\}\_2(x)\; =\; x\; -\; \backslash frac\{x^3\}\{3^2\}\; +\; \backslash frac\{x^5\}\{5^2\}\; -\; \backslash frac\{x^7\}\{7^2\}\; +\; \backslash cdots$

which is absolutely convergent for $|x|\; \backslash le\; 1.${{harvnb|Lewin|1981|pp=38–39|loc=Section 2.1}}

The inverse tangent integral is closely related to the dilogarithm $\backslash operatorname\{Li\}\_2(z)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{z^n\}\{n^2\}$ and can be expressed simply in terms of it:

:$\backslash operatorname\{Ti\}\_2(z)\; =\; \backslash frac\{1\}\{2i\}\; \backslash left(\; \backslash operatorname\{Li\}\_2(iz)\; -\; \backslash operatorname\{Li\}\_2(-iz)\; \backslash right)$

That is,

:$\backslash operatorname\{Ti\}\_2(x)\; =\; \backslash operatorname\{Im\}(\backslash operatorname\{Li\}\_2(ix))$

for all real x.

The inverse tangent integral is an odd function:

:$\backslash operatorname\{Ti\}\_2(-x)\; =\; -\backslash operatorname\{Ti\}\_2(x)$

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

:$\backslash operatorname\{Ti\}\_2(x)\; -\; \backslash operatorname\{Ti\}\_2\; \backslash left(\backslash frac\{1\}\{x\}\; \backslash right)\; =\; \backslash frac\{\backslash pi\}\{2\}\; \backslash log\; x$

valid for all x > 0 (or, more generally, for Re(x) > 0).

This can be proven by differentiating and using the identity $\backslash arctan(t)\; +\; \backslash arctan(1/t)\; =\; \backslash pi/2$.

The special value Ti₂(1) is Catalan's constant $1\; -\; \backslash frac\{1\}\{3^2\}\; +\; \backslash frac\{1\}\{5^2\}\; -\; \backslash frac\{1\}\{7^2\}\; +\; \backslash cdots\; \backslash approx\; 0.915966$.{{harvnb|Lewin|1981|pp=39–40|loc=Section 2.2}}

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

:$\backslash operatorname\{Ti\}\_\{n\}(x)\; =\; \backslash sum\backslash limits\_\{k=0\}^\{\backslash infty\}\backslash frac\{(-1)^\{k\}x^\{2k+1\}\}\{\backslash left(2k+1\backslash right)^\{n\}\}=x\; -\; \backslash frac\{x^3\}\{3^n\}\; +\; \backslash frac\{x^5\}\{5^n\}\; -\; \backslash frac\{x^7\}\{7^n\}\; +\; \backslash cdots$

is defined analogously. This satisfies the recurrence relation:{{harvnb|Lewin|1981|p=190|loc=Section 7.1.2}}

:$\backslash operatorname\{Ti\}\_\{n\}(x)\; =\; \backslash int\_0^x\; \backslash frac\{\backslash operatorname\{Ti\}\_\{n-1\}(t)\}\{t\}\; \backslash ,\; dt$

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

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

:$\backslash operatorname\{Ti\}\_2(x)\; =\; -i\; \backslash chi\_2(ix)$

Note that $\backslash chi\_2(x)$ can be expressed as $\backslash int\_0^x\; \backslash frac\{\backslash operatorname\{artanh\}\; t\}\{t\}\; \backslash ,\; dt$, 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 $\backslash Phi(z,s,a)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{z^n\}\{(n+a)^s\}:${{MathWorld |id=InverseTangentIntegral |title=Inverse Tangent Integral}}

:$\backslash operatorname\{Ti\}\_2(x)\; =\; \backslash frac\{1\}\{4\}\; x\; \backslash Phi(-x^2,\; 2,\; 1/2)$

The notation Ti₂ and Ti_n is due to Lewin. Spence (1809){{Cite book | last= Spence | first= William | date= 1809 | title= 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 | location= London | url= https://babel.hathitrust.org/cgi/pt?id=uc1.c3118082}} studied the function, using the notation $\backslash overset\{n\}\{\backslash operatorname\{C\}\}(x)$. The function was also studied by Ramanujan.{{Cite journal | first= S. | last= Ramanujan | author-link= Srinivasa Ramanujan | journal= Journal of the Indian Mathematical Society | volume= 7 | date= 1915 | pages= 93–96 | title= On the integral $\backslash int\_0^x\; \backslash frac\{\backslash tan^\{-1\}\; t\}\{t\}\; \backslash ,\; dt$}} Appears in: {{Cite book | title= Collected Papers of Srinivasa Ramanujan | editor-first1= G. H. | editor-last1= Hardy |editor-link= G. H. Hardy | editor-first2= P. V. | editor-last2=Seshu Aiyar | editor-first3= B. M. | editor-last3= Wilson | editor-link3= Bertram Martin Wilson | date = 1927 | pages= 40–43 }}

{{Reflist}}

• {{Cite book | last= Lewin | first= L. | title= Dilogarithms and Associated Functions | location= London | publisher= Macdonald | year= 1958 | mr= 0105524 | zbl= 0083.35904 }}
• {{Cite book | last= Lewin | first= L. | title= Polylogarithms and Associated Functions | location= New York | publisher= North-Holland | year= 1981 | isbn= 978-0-444-00550-2 }}

Category:Special functions