Cosecant in terms of Hyperbolic Cosecant
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Theorem
Let $z \in \C$ be a complex number.
Then:
- $i \csc = -\csch \paren {i z}$
where:
- $\csc$ denotes the cosecant function
- $\csch$ denotes the hyperbolic cosecant
- $i$ is the imaginary unit: $i^2 = -1$.
Proof
\(\ds i \csc x\) | \(=\) | \(\ds \frac i {\sin z}\) | Definition of Complex Cosecant Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {-i \sin z}\) | $i^2 = -1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {-\sinh \paren {i z} }\) | Sine in terms of Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds -\csch \paren {i z}\) | Definition 2 of Hyperbolic Cosecant |
$\blacksquare$
Also see
- Sine in terms of Hyperbolic Sine
- Cosine in terms of Hyperbolic Cosine
- Tangent in terms of Hyperbolic Tangent
- Cotangent in terms of Hyperbolic Cotangent
- Secant in terms of Hyperbolic Secant
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.83$: Relationship between Hyperbolic and Trigonometric Functions