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 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: $20.37$