title: Inverse tangent integral
text: # Inverse tangent integral
  
  The inverse tangent integral is a special function, defined by:
  
      $\operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt$
  
  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:
  
      $\operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt$
  
  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
  
      $\operatorname {Ti} _{2}(x)=x-{\frac {x^{3}}{3^{2}}}+{\frac {x^{5}}{5^{2}}}-{\frac {x^{7}}{7^{2}}}+\cdots$
  
  which is absolutely convergent for $|x|\leq 1.$
  
  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:
  
      $\operatorname {Ti} _{2}(z)={\frac {1}{2i}}\left(\operatorname {Li} _{2}(iz)-\operatorname {Li} _{2}(-iz)\right)$
  
  That is,
  
      $\operatorname {Ti} _{2}(x)=\operatorname {Im} (\operatorname {Li} _{2}(ix))$
  
  for all real x.
  
  ## Properties
  
  The inverse tangent integral is an odd function:
  
      $\operatorname {Ti} _{2}(-x)=-\operatorname {Ti} _{2}(x)$
  
  The values of $Ti_2(x)$ and $Ti_2(1/x)$ are related by the identity
  
      $\operatorname {Ti} _{2}(x)-\operatorname {Ti} _{2}\left({\frac {1}{x}}\right)={\frac {\pi }{2}}\log x$
  
  valid for all x > 0 (or, more generally, for Re(x) > 0). This can be proven by differentiating and using the identity $\arctan(t)+\arctan(1/t)=\pi /2$.
  
  The special value $Ti_2(1)$ 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
  
      $\operatorname {Ti} _{n}(x)=x-{\frac {x^{3}}{3^{n}}}+{\frac {x^{5}}{5^{n}}}-{\frac {x^{7}}{7^{n}}}+\cdots$
  
  is defined analogously. This satisfies the recurrence relation:
  
      $\operatorname {Ti} _{n}(x)=\int _{0}^{x}{\frac {\operatorname {Ti} _{n-1}(t)}{t}}\,dt$
  
  ## 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:
  
      $\operatorname {Ti} _{2}(x)=-i\chi _{2}(ix)$
  
  Note that $\chi _{2}(x)$ can be expressed as $∫_0^x$ artanh⁡t/t dt{\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}}}:}
  
      $\operatorname {Ti} _{2}(x)={\frac {1}{4}}x\Phi (-x^{2},2,1/2)$
  
  ## History
  
  The notation $Ti_2$ and $Ti_n$ is due to Lewin. Spence (1809) studied the function, using the notation ${\overset {n}{\operatorname {C} }}(x)$. The function was also studied by Ramanujan.
  
  