Periodicity of Hyperbolic Secant

Theorem

Let $k \in \Z$.

Then:

$\map \sech {x + 2 k \pi i} = \sech x$

Proof

\(\ds \map \sech {x + 2 k \pi i}\)

\(=\)

\(\ds \frac 1 {\map \cosh {x + 2 k \pi i} }\)

Definition of Hyperbolic Secant

\(\ds \)

\(=\)

\(\ds \frac 1 {\cosh x}\)

Periodicity of Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \sech\)

Definition of Hyperbolic Secant

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.90$: Periodicity of Hyperbolic Functions