Power Series Expansion for Hyperbolic Secant Function
Theorem
The hyperbolic secant function has a Taylor series expansion:
| \(\ds \sech x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {E_{2 n} x^{2 n} } {\paren {2 n}!}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 - \frac {x^2} 2 + \frac {5 x^4} {24} - \frac {61 x^6} {720} + \cdots\) |
where $E_{2 n}$ denotes the Euler numbers.
This converges for $\size x < \dfrac \pi 2$.
Proof
By definition of the Euler numbers:
- $\ds \sech x = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}$
From Odd Euler Numbers Vanish:
- $E_{2 k + 1} = 0$
for $k \in \Z$.
Hence the result.
$\blacksquare$
Also presented as
The can also be presented in the form:
| \(\ds \sech x\) | \(=\) | \(\ds 1 + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n {E_n}^* x^{2 n} } {\paren {2 n}!}\) |
where ${E_n}^*$ denotes the alternative form of the Euler numbers.
Also see
- Power Series Expansion for Hyperbolic Sine Function
- Power Series Expansion for Hyperbolic Cosine Function
- Power Series Expansion for Hyperbolic Tangent Function
- Power Series Expansion for Hyperbolic Cotangent Function
- Power Series Expansion for Hyperbolic Cosecant Function
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Hyperbolic Functions: $20.37$
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.37.$