Hyperbolic Secant in terms of Secant

Theorem

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

Then:

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

where:

$\sec$ denotes the secant function

$\sech$ denotes the hyperbolic secant

$i$ is the imaginary unit: $i^2 = -1$.

Proof

\(\ds \sec \paren {i z}\)

\(=\)

\(\ds \frac 1 {\cos \paren {i z} }\)

Definition of Complex Secant Function

\(\ds \)

\(=\)

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

Hyperbolic Cosine in terms of Cosine

\(\ds \)

\(=\)

\(\ds \sech z\)

Definition of Hyperbolic Secant

$\blacksquare$

Also see

• Hyperbolic Sine in terms of Sine
• Hyperbolic Cosine in terms of Cosine
• Hyperbolic Tangent in terms of Tangent
• Hyperbolic Cotangent in terms of Cotangent
• Hyperbolic Cosecant in terms of Cosecant

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.78$: Relationship between Hyperbolic and Trigonometric Functions