Hyperbolic Cosecant Function is Odd

Theorem

Let $\csch: \C \to \C$ be the hyperbolic cosecant function on the set of complex numbers.

Then $\csch$ is odd:

$\map \csch {-x} = -\csch x$

Proof

\(\ds \map \csch {-x}\)

\(=\)

\(\ds \frac 1 {\map \sinh {-x} }\)

Definition 2 of Hyperbolic Cosecant

\(\ds \)

\(=\)

\(\ds \frac 1 {-\sinh x}\)

Hyperbolic Sine Function is Odd

\(\ds \)

\(=\)

\(\ds -\csch x\)

$\blacksquare$

Also see

• Hyperbolic Sine Function is Odd
• Hyperbolic Cosine Function is Even
• Hyperbolic Tangent Function is Odd
• Hyperbolic Cotangent Function is Odd
• Hyperbolic Secant Function is Even

• Cosecant Function is Odd

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.17$: Functions of Negative Arguments