Power Series Expansion for Real Area Hyperbolic Cosecant
Theorem
The (real) area hyperbolic cosecant function has a Taylor series expansion:
| \(\ds \arcsch x\) | \(=\) | \(\ds \begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : \size x \ge 1 \\ \ds \ln \dfrac 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n} } {2 n} } & : 0 < x \le 1 \\ \ds -\ln \paren {-\dfrac 2 x} + \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n} } {2 n} } & : -1 \le x < 0 \\ \end {cases}\) |
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| \(\ds \) | \(\) | \(\ds \) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \begin {cases}
\dfrac 1 x - \dfrac 1 2 \dfrac 1 {3 x^3} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {5 x^5} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {7 x^7} + \cdots & : \size x \ge 1 \\ \ln \dfrac 2 x - \paren {\dfrac 1 2 \dfrac {x^2} 2 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^4} 4 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^6} 6 + \cdots} & : 0 < x \le 1 \\ -\map \ln {-\dfrac 2 x} + \paren {\dfrac 1 2 \dfrac {x^2} 2 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^4} 4 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^6} 6 + \cdots} & : -1 \le x < 0 \\ \end {cases}\) |
Proof
From Power Series Expansion for Real Area Hyperbolic Sine:
| \(\ds \arsinh x\) | \(=\) | \(\ds \begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1} & : \size x \le 1 \\ \ds \ln 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } } & : x \ge 1 \\ \ds -\ln \paren {-2 x} + \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } } & : x \le -1 \\ \end {cases}\) |
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| \(\ds \) | \(\) | \(\ds \) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \begin {cases}
x - \dfrac 1 2 \dfrac {x^3} 3 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^5} 5 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^7} 7 + \cdots & : \size x \le 1 \\ \ln 2 x - \paren {\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {4 x^4} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {6 x^6} + \cdots} & : x \ge 1 \\ -\ln \paren {-2 x} + \paren {\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {4 x^4} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {6 x^6} + \cdots} & : x \le -1 \\ \end {cases}\) |
From Real Area Hyperbolic Sine of Reciprocal equals Real Area Hyperbolic Cosecant:
- $\map \arsinh {\dfrac 1 x} = \arcsch x$
So:
| \(\ds \arcsch x\) | \(=\) | \(\ds \begin {cases}
\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : \size x \ge 1 \\ \ds \ln \dfrac 2 x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n} } {2 n} } & : 0 < x \le 1 \\ \ds -\ln \paren {-\dfrac 2 x} + \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n} } {2 n} } & : -1 \le x < 0 \\ \end {cases}\) |
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| \(\ds \) | \(\) | \(\ds \) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \begin {cases}
\dfrac 1 x - \dfrac 1 2 \dfrac 1 {3 x^3} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {5 x^5} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {7 x^7} + \cdots & : \size x \ge 1 \\ \ln \dfrac 2 x - \paren {\dfrac 1 2 \dfrac {x^2} 2 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^4} 4 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^6} 6 + \cdots} & : 0 < x \le 1 \\ -\map \ln {-\dfrac 2 x} + \paren {\dfrac 1 2 \dfrac {x^2} 2 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^4} 4 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^6} 6 + \cdots} & : -1 \le x < 0 \\ \end {cases}\) |
Hence the result.
$\blacksquare$