Csch
The hyperbolic cosecant function is defined by $$\mathrm{csch}(z)=\dfrac{1}{\sinh(z)},$$ where $\sinh$ denotes the hyperbolic sine.
- Complex Csch.jpg
Domain coloring of analytic continuation of $\mathrm{csch}$.
Contents
Properties
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{csch}(z)=-\mathrm{csch}(z)\mathrm{coth}(z),$$ where $\mathrm{csch}$ denotes the hyperbolic cosecant and $\mathrm{coth}$ denotes the hyperbolic cotangent.
Proof
From the definition, $$\mathrm{csch}(z) = \dfrac{1}{\mathrm{sinh}(z)}.$$ Using the quotient rule, the derivative of sinh, and the definition of $\mathrm{coth}$, we compute $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{csch}(z) &= \dfrac{0-\mathrm{cosh}(z)}{\mathrm{sinh}^2(z)} \\ &= -\mathrm{csch}(z)\mathrm{coth}(z), \end{array}$$ as was to be shown.
