Theorem: Power Series Expansion for Real Area Hyperbolic Cosecant

The (real) area hyperbolic cosecant function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \arcsch x
      | r = <nowiki>\begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : \size x \ge 1 \\ 
\ds \ln \dfrac 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n} } {2 n} } & : 0 < x \le 1 \\ 
\ds -\ln \paren {-\dfrac 2 x} + \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n} } {2 n} } & : -1 \le x < 0 \\ 
\end {cases} </nowiki>
      | c = 
}}
{{eqn | o =
}}
{{eqn | r = <nowiki>\begin {cases}
\dfrac 1 x - \dfrac 1 2 \dfrac 1 {3 x^3} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {5 x^5} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {7 x^7} + \cdots & : \size x \ge 1 \\ 
\ln \dfrac 2 x - \paren {\dfrac 1 2 \dfrac {x^2} 2 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^4} 4 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^6} 6 + \cdots} & : 0 < x \le 1 \\ 
-\map \ln {-\dfrac 2 x} + \paren {\dfrac 1 2 \dfrac {x^2} 2 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^4} 4 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^6} 6 + \cdots} & : -1 \le x < 0 \\ 
\end {cases}</nowiki>
      | c = 
}}
{{end-eqn}}

Proof:
From Power Series Expansion for Real Area Hyperbolic Sine:
{{begin-eqn}}
{{eqn | l = \arsinh x
      | r = <nowiki>\begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1} & : \size x \le 1 \\ 
\ds \ln 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } } & : x \ge 1 \\ 
\ds -\ln \paren {-2 x} + \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } } & : x \le -1 \\ 
\end {cases}</nowiki>
      | c = 
}}
{{eqn | o =
}}
{{eqn | r = <nowiki>\begin {cases}
x - \dfrac 1 2 \dfrac {x^3} 3 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^5} 5 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^7} 7 + \cdots & : \size x \le 1 \\ 
\ln 2 x - \paren {\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {4 x^4} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {6 x^6} + \cdots} & : x \ge 1 \\ 
-\ln \paren {-2 x} + \paren {\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {4 x^4} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {6 x^6} + \cdots} & : x \le -1 \\ 
\end {cases} </nowiki>
      | c = 
}}
{{end-eqn}}
From Real Area Hyperbolic Sine of Reciprocal equals Real Area Hyperbolic Cosecant:
:$\map \arsinh {\dfrac 1 x} = \arcsch x$
So:
{{begin-eqn}}
{{eqn | l = \arcsch x
      | r = <nowiki>\begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : \size x \ge 1 \\ 
\ds \ln \dfrac 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n} } {2 n} } & : 0 < x \le 1 \\ 
\ds -\ln \paren {-\dfrac 2 x} + \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n} } {2 n} } & : -1 \le x < 0 \\ 
\end {cases} </nowiki>
      | c = 
}}
{{eqn | o =
}}
{{eqn | r = <nowiki>\begin {cases}
\dfrac 1 x - \dfrac 1 2 \dfrac 1 {3 x^3} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {5 x^5} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {7 x^7} + \cdots & : \size x \ge 1 \\ 
\ln \dfrac 2 x - \paren {\dfrac 1 2 \dfrac {x^2} 2 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^4} 4 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^6} 6 + \cdots} & : 0 < x \le 1 \\ 
-\map \ln {-\dfrac 2 x} + \paren {\dfrac 1 2 \dfrac {x^2} 2 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^4} 4 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^6} 6 + \cdots} & : -1 \le x < 0 \\ 
\end {cases} </nowiki>
      | c = 
}}
{{end-eqn}}
Hence the result.
{{qed}}