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If $a\gg k_{B}T$, the wall is thermally inaccessible and $⟨E⟩\sim 2k_{B}T$, as four quadratic terms predict. If $a\ll k_{B}T$, the spatial distribution is nearly uniform in the disk andso only the two kinetic quadratic terms contribute at leading order.

An element $m$ of a partially ordered set is minimal when $x\le m$ implies $x=m$.

An element $m$ of a partially ordered set is maximal when $m\le x$ implies $x=m$. A poset can have several maximal elements and need not have a greatest element.

For a parabolic trajectory with periapsis radius $r_p$, put $D=\tan (f/2)$, where $f$ is true anomaly. The Barker equation is $t-t_p=\sqrt{2r_p^3/(GM)}(D+D^3/3)$. Since $r=r_p(1+D^2)$, it is equivalent to the radius-time relation for a parabolic Kepler orbit. The sign of $D$ distinguishes approach from recession; $t_p$ is the periapsis time.

Multiplying a solution by $(z-z_0{)}^{\lambda }$ adds $\lambda$ to both exponents at the finite point $z_0$ and subtracts $\lambda$ from both exponents at infinity, under the convention that an exponent $\rho$ at infinity corresponds to behavior $z^{-\rho }$.

A Möbius change of independent variable permutes the singular points while carrying their characteristic exponent pairs with them.

For a second-order equation with three regular singular points, the sum of all six characteristic exponents is one.

For an order-$n$ Fuchsian differential equation with $r$ finite singular points and infinity, the sum of all characteristic exponents is $n(n-1)(r-1)/2$. The three-singularity second-order case has $n=2$ and $r=2$.