Complex hyperbolic functions

The complex hyperbolic functions $\sinh z$ and $\cosh z$ extend the real hyperbolic functions analytically to all of $\mathbb{C}$, defined via the Complex exponential:

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

Both are entire.

Derivatives and identities

$\frac{d}{dz}\sinh z=\cosh z,\frac{d}{dz}\cosh z=\sinh z.$

(Note: derivatives don’t pick up signs the way sin/cos do.)

${\cosh }^{2}z-{\sinh }^{2}z=1.$

(Hyperbolic identity, with a minus sign — the geometric difference from the trig version.)

Connection to sin and cos

The fundamental identities, by direct substitution into the $e^z$ definitions:

$\sinh (iz)=i\sin z,\cosh (iz)=\cos z.$

Equivalently:

$\sin (iz)=i\sinh z,\cos (iz)=\cosh z.$

So the hyperbolic and trigonometric functions are the same family rotated $90°$ in the complex plane. The hyperbolic functions evaluated on the imaginary axis are the trig functions on the real axis (up to factors of $i$), and vice versa.

This is the reason complex sine and cosine are unbounded along the imaginary axis: they look like $\cosh y$ and $i\sinh y$ there, which grow exponentially.

Zeros

$\sinh z=0$ iff $e^z=e^{-z}$ iff $e^{2z}=1$ iff $z=n\pi i$, $n\in \mathbb{Z}$. So sinh has zeros on the imaginary axis at multiples of $\pi i$. Real zero only at $z=0$.

$\cosh z=0$ iff $z=(n+1/2)\pi i$ — also on the imaginary axis. No real zeros.

tanh, coth, sech, csch

$\tanh z=\sinh z/\cosh z$, with poles where $\cosh z=0$, i.e., $z=(n+1/2)\pi i$. The other three follow similarly.

In context

• Real-imaginary decomposition of $\sin z$ and $\cos z$ uses hyperbolic functions: $\sin (x+iy)=\sin x\cosh y+i\cos x\sinh y$. See Complex sine and cosine.
• Transmission-line equations involve $\cosh (\gamma \ell )$ and $\sinh (\gamma \ell )$ where $\gamma$ is the complex propagation constant — the same hyperbolic algebra with complex argument.
• Solutions of differential equations $y^{\prime \prime }=k^{2}y$ are $\sinh (kx),\cosh (kx)$ (real $k$), and the complex generalization handles oscillatory + exponential mixtures uniformly.
• Like sin/cos, sinh/cosh are entire-but-unbounded, illustrating Liouville’s theorem.