Editing Disk (mathematics) (section)

===Average distance to an arbitrary internal point===
[[File:Cjcdiscin.svg|thumb|The average distance from a disk to an internal point]]To find {{math|''b''(''q'')}} we need to look separately at the cases in which the location is internal or external, i.e. in which {{math|''q'' ≶ 1}}, and we find that in both cases the result can only be expressed in terms of [[elliptic integrals|complete elliptic integrals]].

If we consider an internal location, our aim (looking at the diagram) is to compute the expected value of {{math|''r''}} under a distribution whose density is {{math|{{sfrac|1|π}}}} for {{math|0 &le; ''r'' &le; ''s''(θ)}}, integrating in polar coordinates centered on the fixed location for which the area of a cell is {{math|''r'' d''r'' dθ}}&nbsp;; hence
<math display="block">b(q) = \frac{1}{\pi} \int_0^{2\pi} \textrm{d}\theta \int_0^{s(\theta)} r^2 \textrm{d}r  = \frac{1}{3\pi} \int_0^{2\pi} s(\theta)^3 \textrm{d}\theta.</math>

Here {{math|''s''(θ)}} can be found in terms of {{math|''q''}} and {{math|θ}} using the [[Law of cosines]]. The steps needed to evaluate the integral, together with several references, will be found in the paper by Lew et al.;<ref name=lew/> the result is that
<math display="block">b(q) = \frac{4}{9\pi}\biggl\{ 4(q^2-1)K(q^2) + (q^2+7)E(q^2)\biggr\} </math>
where {{math|''K''}} and {{math|''E''}} are complete elliptic integrals of the first and second kinds.<ref>[[Abramowitz and Stegun]], 17.3.</ref> {{math|''b''(0) {{=}} {{sfrac|2|3}}}}; {{math|''b''(1) {{=}} {{sfrac|32|9π}} ≈ 1.13177}}.