Hyperbolic Secant in terms of Secant
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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