Secant in terms of Hyperbolic Secant

Theorem

Let $z \in \C$ be a complex number.

Then:

$\sec z = \map \sech {i z}$

where:

$\sec$ denotes the secant function
$\sech$ denotes the hyperbolic secant
$i$ is the imaginary unit: $i^2 = -1$.


Proof

\(\ds \sec z\) \(=\) \(\ds \frac 1 {\cos z}\) Definition of Complex Secant Function
\(\ds \) \(=\) \(\ds \frac 1 {\map \cosh {i z} }\) Cosine in terms of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \map \sech {i z}\) Definition of Hyperbolic Secant

$\blacksquare$


Also see


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.84$: Relationship between Hyperbolic and Trigonometric Functions
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