Hyperbolic Cosecant in terms of Cosecant

Theorem

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

Then:

$i \csch z = -\csc \paren {i z}$

where:

$\csc$ denotes the cosecant function

$\csch$ denotes the hyperbolic cosecant

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

Proof

\(\ds i \csch z\)

\(=\)

\(\ds \frac i {\sinh z}\)

Definition 2 of Hyperbolic Cosecant

\(\ds \)

\(=\)

\(\ds \frac {-1} {i \sinh z}\)

as $i^2 = -1$

\(\ds \)

\(=\)

\(\ds \frac {-1} {\sin \paren {i z} }\)

Hyperbolic Sine in terms of Sine

\(\ds \)

\(=\)

\(\ds -\csc \paren {i z}\)

Definition of Complex Cosecant Function

$\blacksquare$

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 Secant in terms of Secant

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.77$: Relationship between Hyperbolic and Trigonometric Functions