Comparison sine, cosine and cotangent functions

For a real number $k\in \mathbb{R}$ the comparison sine is the smooth function ${\mathrm{sn}}_k:\mathbb{R}\to \mathbb{R}$ given by ${\mathrm{sn}}_k(t):=\{\begin{array}{ll}{\displaystyle \frac{\sin (\sqrt{k}t)}{\sqrt{k}}},& k>0,\\ t,& k=0,\\ {\displaystyle \frac{\sinh (\sqrt{-k}t)}{\sqrt{-k}}},& k<0.\end{array}$ The comparison cosine is its derivative ${\mathrm{cs}}_k:={\mathrm{sn}}_k^{\mathrm{\prime }}$, so that ${\mathrm{cs}}_k(t)=\{\begin{array}{ll}\cos (\sqrt{k}t),& k>0,\\ 1,& k=0,\\ \cosh (\sqrt{-k}t),& k<0.\end{array}$ The comparison cotangent is the quotient ${\mathrm{ct}}_k:={\mathrm{cs}}_k\mathrm{/}{\mathrm{sn}}_k$, defined at every real $t$ with ${\mathrm{sn}}_k(t)\mathrm{\ne }0$. Thus its domain is $\mathbb{R}\setminus \{m\pi \mathrm{/}\sqrt{k}:m\in \mathbb{Z}\}$ when $k>0$, and $\mathbb{R}\setminus \{0\}$ when $k\le 0$; these are unions of the open intervals on which ${\mathrm{sn}}_k$ has constant sign.

The following data are part of the definition and are used throughout the pair: ${\mathrm{sn}}_k(0)=0$, ${\mathrm{sn}}_k^{\mathrm{\prime }}(0)={\mathrm{cs}}_k(0)=1$, ${\mathrm{cs}}_k(0)=1$, and ${\mathrm{sn}}_k$ is odd while ${\mathrm{cs}}_k$ is even. The positive domain of ${\mathrm{sn}}_k$ is $t\in (0,\pi \mathrm{/}\sqrt{k})$ when $k>0$ and $t\in (0,+\mathrm{\infty })$ when $k\le 0$; there ${\mathrm{sn}}_k(t)>0$. For $k\le 0$, ${\mathrm{ct}}_k(t)>0$ for every $t>0$. For $k>0$, ${\mathrm{ct}}_k$ is positive on $(0,\pi \mathrm{/}(2\sqrt{k}))$, zero at $\pi \mathrm{/}(2\sqrt{k})$, and negative on $(\pi \mathrm{/}(2\sqrt{k}),\pi \mathrm{/}\sqrt{k})$. The terminal value is ${\mathrm{sn}}_k(\pi \mathrm{/}\sqrt{k})=0$: the spherical endpoint is excluded from the domain of ${\mathrm{ct}}_k$, and no value of ${\mathrm{ct}}_k$ at that endpoint is asserted. For $k\le 0$ no positive zero of ${\mathrm{sn}}_k$ exists.

In dimension $n\ge 2$ the same functions describe the radial behaviour of the model spaces: ${\mathrm{sn}}_k$ is the profile of the model normal Jacobi fields and ${\mathrm{sn}}_k^{n-1}$ the model radial density used on this page. Only the displayed piecewise formulas, their stated values at $0$ and the stated sign domains are part of this definition; the differential identities satisfied by these functions are proved separately.