Power Series Expansion for Real Area Hyperbolic Sine
Theorem
The (real) area hyperbolic sine function has a Taylor series expansion:
\(\ds \arsinh x\) | \(=\) | \(\ds \begin {cases} \ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1} & : \size x < 1 \\ \ds \ln 2 x + \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } } & : x \ge 1 \\ \ds -\ln \paren {-2 x} - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } } & : x \le -1 \\ \end {cases}\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \begin {cases} x - \dfrac {x^3} {2 \times 3} + \dfrac {1 \times 3 x^5} {2 \times 4 \times 5} - \dfrac {1 \times 3 \times 5 x^7} {2 \times 4 \times 6 \times 7} + \cdots & : \size x < 1 \\ \ln 2 x + \dfrac 1 {2 \times 2 x^2} - \dfrac {1 \times 3} {2 \times 4 \times 4 x^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6 x^6} + \cdots & : x \ge 1 \\ -\ln \paren {-2 x} - \paren {\dfrac 1 {2 \times 2 x^2} - \dfrac {1 \times 3} {2 \times 4 \times 4 x^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6 x^6} + \cdots} & : x \le -1 \\ \end {cases}\) |
Proof
Lemma 1
\(\ds \dfrac 1 {\sqrt {x^2 + 1} }\) | \(=\) | \(\ds 1 - \frac 1 2 x^2 + \frac {1 \times 3} {2 \times 4} x^4 - \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^6 + \cdots\) | ||||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} x^{2 n}\) |
for $x \in \R: -1 < x < 1$.
$\Box$
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
\(\ds \int_0^x \frac 1 {\sqrt {t^2 + 1} } \rd t\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \int_0^x \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} t^{2 n} \rd t\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \arsinh x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}\) | Derivative of Inverse Hyperbolic Sine |
$\Box$
We will now prove that the series converges for $-1 < x < 1$.
\(\ds \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}\) | \(\sim\) | \(\ds \frac {\paren {-1}^n \paren {2 n}^{2 n} e^{-2 n} \sqrt {4 \pi n} } {2^{2 n} n^{2 n} e^{-2 n} 2 \pi n} \frac {x^{2 n + 1} } {2 n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {\sqrt {\pi n} } \frac {x^{2 n + 1} } {2 n + 1}\) |
Then:
\(\ds \size {\frac 1 {\sqrt {\pi n} } \frac {x^{2 n + 1} } {2 n + 1} }\) | \(<\) | \(\ds \size {\frac {x^{2 n + 1} } {n^{3/2} } }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \frac 1 {n^{3/2} }\) |
Hence by Convergence of P-Series:
- $\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^{3/2} }$
is convergent.
So by the Comparison Test, the Taylor series is convergent for $-1 \le x \le 1$.
$\Box$
Another lemma:
Lemma $2$
- $\map \ln {1 + \sqrt {1 + x^2} } = \ln 2 + \dfrac 1 2 \cdot \dfrac {x^2} 2 - \dfrac {1 \times 3} {2 \times 4} \cdot \dfrac {x^4} 4 + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \cdot \dfrac {x^6} 6 - \cdots$
This holds for $x \in \R: \size x < 1$.
$\Box$
Let $x \ge 1$.
Let $z = \dfrac 1 x$.
Then we have:
- $0 < \dfrac 1 z \le 1$
Now we consider:
\(\ds \map \arsinh {\dfrac 1 z} + \map \ln {2 z}\) | \(=\) | \(\ds \map \ln {2 z} + \map \ln {\dfrac 1 z + \sqrt {\dfrac 1 {z^2} + 1} }\) | Definition of Real Area Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {2 z \paren {\dfrac 1 z + \dfrac {\sqrt {z^2 + 1} } z} }\) | Sum of Logarithms and simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {2 \paren {1 + \sqrt {z^2 + 1} } }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln 2 + \map \ln {1 + \sqrt {z^2 + 1} }\) | Sum of Logarithms | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arsinh {\dfrac 1 z}\) | \(=\) | \(\ds -\map \ln {2 z} + \ln 2 + \map \ln {1 + \sqrt {z^2 + 1} }\) | rearranging | ||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {\dfrac 1 {2 z} } + \ln 2 + \map \ln {1 + \sqrt {z^2 + 1} }\) | Logarithm of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {\dfrac 1 z} + \map \ln {1 + \sqrt {z^2 + 1} }\) | Sum of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {\dfrac 1 z} + \ln 2 + \dfrac 1 2 \cdot \dfrac {z^2} 2 - \dfrac {1 \times 3} {2 \times 4} \cdot \dfrac {z^4} 4 + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \cdot \dfrac {z^6} 6 - \cdots\) | Lemma $2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {\dfrac 2 z} + \dfrac 1 2 \cdot \dfrac {z^2} 2 - \dfrac {1 \times 3} {2 \times 4} \cdot \dfrac {z^4} 4 + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \cdot \dfrac {z^6} 6 - \cdots\) | Sum of Logarithms | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \arsinh x\) | \(=\) | \(\ds \map \ln {2 x} + \dfrac 1 {2 \times 2 x^2} - \dfrac {1 \times 3} {2 \times 4 \times 4 x^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6 x^6} - \cdots\) | substituting $x \gets \dfrac 1 z$ |
Now let $x \le -1$.
We have that Inverse Hyperbolic Sine is Odd Function.
That is:
- $\arsinh x = -\map \arsinh {-x}$
Thus for $x \le -1$:
\(\ds \arsinh x\) | \(=\) | \(\ds -\map \arsinh {-x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\map \ln {-2 x} - \paren {\dfrac 1 {2 \times 2 \paren {-x}^2} - \dfrac {1 \times 3} {2 \times 4 \times 4 \paren {-x}^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6 \paren {-x}^6} - \cdots}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\map \ln {-2 x} - \paren {\dfrac 1 {2 \times 2 x^2} - \dfrac {1 \times 3} {2 \times 4 \times 4 x^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6 x^6} - \cdots}\) |
Hence the result.
$\blacksquare$
Also presented as
This can also be presented as:
\(\ds \arsinh x\) | \(=\) | \(\ds \begin {cases} \ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1} & : \size x < 1 \\ \ds \pm \paren {\ln \size {2 x} + \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } } } & : \size x \ge 1 \end {cases}\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \begin {cases} x - \dfrac 1 2 \dfrac {x^3} 3 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^5} 5 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^7} 7 + \cdots & : \size x < 1 \\ \pm \paren {\ln 2 x + \dfrac 1 {2 \times 2 x^2} - \dfrac {1 \times 3} {2 \times 4 \times 4 x^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6 x^6} + \cdots} & : \size x \ge 1 \end {cases}\) |
where $\pm$ is $+$ for $x \ge 1$ and $-$ for $x \le -1$.
Also see
- Power Series Expansion for Real Area Hyperbolic Cosine
- Power Series Expansion for Real Area Hyperbolic Tangent
- Power Series Expansion for Real Area Hyperbolic Cotangent
- Power Series Expansion for Real Area Hyperbolic Secant
- Power Series Expansion for Real Area Hyperbolic Cosecant
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Hyperbolic Functions: $20.39$
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
- Ma Joad (https://math.stackexchange.com/users/516814/ma-joad), Power Series Expansion for Real Inverse Hyperbolic Sine, URL (version: 2023-02-26): https://math.stackexchange.com/q/4647185
for a citation that looks like: