Hyperbolic Functions | Sahithyan's Notes

For any complex number $z$ ,

\sinh z = \frac{e^z - e^{-z}}{2}, \qquad \cosh z = \frac{e^z + e^{-z}}{2}

$\sinh z$ and $\cosh z$ are entire functions.

Other Hyperbolic Functions

\tanh z = \frac{\sinh z}{\cosh z}, \qquad \coth z = \frac{\cosh z}{\sinh z}

\operatorname{sech} z = \frac{1}{\cosh z}, \qquad \operatorname{csch} z = \frac{1}{\sinh z}

$\sinh z$ and $\cosh z$ are analytic on $\mathbb{C}$ , since $e^z$ and $e^{-z}$ are.

The other 4 functions are quotients, so each is analytic wherever its denominator is non-zero.

Function	Analytic on
$\tanh z$ , $\operatorname{sech} z$	$\mathbb{C} \setminus \{i(\frac{\pi}{2} + n\pi)\}$
$\coth z$ , $\operatorname{csch} z$	$\mathbb{C} \setminus \{n\pi i\}$

Here $n \in \mathbb{Z}$ .

$\sinh z = 0$ iff $z = n\pi i$ , and $\cosh z = 0$ iff $z = i(\frac{\pi}{2} + n\pi)$ . All zeros are purely imaginary.

$\dfrac{\text{d}}{\text{d}z}\,\cosh z = \sinh z$
$\dfrac{\text{d}}{\text{d}z}\,\sinh z = \cosh z$
$\dfrac{\text{d}}{\text{d}z}\,\tanh z = \operatorname{sech}^2 z$
$\dfrac{\text{d}}{\text{d}z}\,\coth z = -\operatorname{csch}^2 z$
$\dfrac{\text{d}}{\text{d}z}\,\operatorname{sech} z = -\operatorname{sech} z \tanh z$
$\dfrac{\text{d}}{\text{d}z}\,\operatorname{csch} z = -\operatorname{csch} z \coth z$

Valid whenever the expressions involved are defined.

For $z = x + iy$ :

Both series are Maclaurin series. They converge for all $z \in \mathbb{C}$ , since $\sinh z$ and $\cosh z$ are entire.

\sinh z = \sum_{n=0}^{\infty} \frac{z^{2n+1}}{(2n+1)!} = z + \frac{z^3}{3!} + \frac{z^5}{5!} + \cdots

\cosh z = \sum_{n=0}^{\infty} \frac{z^{2n}}{(2n)!} = 1 + \frac{z^2}{2!} + \frac{z^4}{4!} + \cdots