Hyperbolic sine ($\sinh$)

The hyperbolic sine is the odd functionIts derivative is $\cosh x\ge 1$, so it increases strictly from $-\infty$ to $\infty$ and is a bijection of $\mathbb{R}$ onto itself. For example, $\sinh 0=0$, $\sinh 1\approx 1.175201$, and $\sinh (-1)\approx -1.175201$. It is the vertical coordinate on the unit hyperbola parametrized by $(\cosh x,\sinh x)$.

The Taylor seriesshows that $\sinh x\approx x$ near zero. Its tangent line there is $y=x$; unlike the ordinary sine, the hyperbolic sine grows without bound. Its antiderivative is $\cosh x+C$. In the catenary $y=a\cosh (x/a)$, the hyperbolic sine gives both the slope, through the derivative $y^{\prime }=\sinh (x/a)$, and the arc length $s(x)=a\sinh (x/a)$ from the lowest point for $x\ge 0$.

For real $x,y$, the exponential function definitions give $\sinh (x+iy)=\sinh x\cos y+i\cosh x\sin y$. Taking the squared complex modulus and using ${\cosh }^{2}x-{\sinh }^{2}x=1$ gives the identity. In particular $|\sinh (x+iy)|\ge |\sinh x|$, uniformly in $y$. This controls the vertical sides of rectangular contours in residue theorem arguments with hyperbolic functions.