Hyperbolic

← Functions

Definitions

• $$\begin{array}{rl}\sinh z& =\frac{\exp (z)-\exp (-z)}{2}\\ \cosh z& =\frac{\exp (z)+\exp (-z)}{2}\\ \tanh z& =\frac{\sinh z}{\cosh z}\\ \coth z& =\frac{\cosh z}{\sinh z}\\ \mathrm{sech}z& =\frac{1}{\cosh z}\\ \mathrm{csch}z& =\frac{1}{\sinh z}\end{array}$$

Fundamental Properties

$$\begin{array}{rl}\sinh (z+2\pi i)& =\sinh z\\ \cosh (z+2\pi i)& =\cosh z\\ \tanh (z+\pi i)& =\tanh z\end{array}$$

Identities

For complex numbers $a,b,z$ the following holds: $$\begin{array}{rl}{(\cosh z)}^2-{(\sinh z)}^2& =1\end{array}$$ $$\begin{array}{rl}\sinh (a+b)& =\sinh a\cosh b+\cosh a\sinh b\\ \cosh (a+b)& =\cosh a\cosh b+\cosh a\cosh b\\ \sinh a+\sinh b& =2\sinh \left(\frac{a+b}{2}\right)\cosh \left(\frac{a-b}{2}\right)\\ \cosh a+\cosh b& =2\cosh \left(\frac{a+b}{2}\right)\cosh \left(\frac{a-b}{2}\right)\end{array}$$ $$\begin{array}{rlrl}\sinh (iz)& =i\sin z& \sin (iz)& =i\sinh z\\ \cosh (iz)& =\cos z& \cos (iz)& =\cosh z\\ \tanh (iz)& =i\tan z& \tan (iz)& =i\tanh (z)\\ \coth (iz)& =-i\cot z& \cot (iz)& =-i\coth (z)\\ \mathrm{sech}(iz)& =\sec z& \sec (iz)& =\mathrm{sech}(z)\\ \mathrm{csch}(iz)& =-i\csc z& \csc (iz)& =-i\mathrm{csch}(z)\end{array}$$