Logarithmic capacity of regular polygons

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Numbers

$n$

$\kappa_n$

0.7304992431031591790794365589230298485194656779037308941028877172207606779428657203882681325963248532

comment: Equilateral triangle.

0.8346268416740731862814297327990468089939930134903470024498273701036819927095264118696911603512753241

comment: Square.

0.8888552730139685196185131777411997768321924953217311565498368476306530953660281937564793297599220655

0.9203713733179424976555185645431729947262457919498943383433001997731018080805685639363385068818052860

comment: Regular hexagon.

0.9402225423282581845064778905225410552810705288081291632814952991504732418300792706612797503058274595

0.9535055293740722197741177590360440208529867995201213061558388754293159023332562076544820599226477968

0.9628192586646530066306035310522177079567361203890762686951808449921441229967180916080191356782427910

10:

0.9695971616706296241008753846822811337655008760201694521911344046504877333815936880015310334478062367

11:

0.9746809080551337015329599026117054905091695315639821222817002480493460891450769145555248589701531686

12:

0.9785904997806327688179169796716777923762887640879620488486912283646244040155350708704671777406234958

13:

0.9816610217835278138794065231407808959558520019192397271006737343719132711230073551607417890204995199

14:

0.9841161777206661109744589037204686309887036114418259144290440979461855280344866504136591296604277765

15:

0.9861098826947237591909892931506095293791817050354060097140050624409691039333356342821846917572847215

16:

0.9877508215994487738247709035478422744137149342047392189515546877945799850430681712903169385280201101

17:

0.9891174955848752659263649762044119205001368344132900329798927813031591218169115630580599465635936951

18:

0.9902677417201082341063948272393938707563161768034162620577033991811190932227458984893934923724793757

19:

0.9912449280292169588298151611938085952125696860976279165675445429167665523214490080500993111487410574

20:

0.9920820866512279940467789574531276895339956421858722248222131244298472597526570569847452319591108787

21:

0.9928047332003234469044225129127138361816441553197237518222059044826351169611837164987600385025470828

22:

0.9934328285330678289923867922389426432997314225985510145284442645180594299833118827850007127863497624

23:

0.9939821684448785098655476154654796098883503929146000995412129902323540251860633121148569366352864385

24:

0.9944653842124845044561711419148928638807784702050932938827497383327835020948330841743143671043523536

25:

0.9948926736666199446222229165869003168962392564166241815535490638925143423208761029886438058255113823

26:

0.9952723426232609266638088072925061673235972955421754925508807751306836187297040603495841870250975744

27:

0.9956112108596021897471234308486177301599079090125291647838143238950119170125087924239874710722541208

28:

0.9959149200112485703888714740153134892453104994995755578840545101099175326792930425477524990507666979

29:

0.9961881695560749028037590851089251131155855045278000693867992089334930499797267979979710521867008843

30:

0.9964348994536133396896567518337769275901239600028377437615314323726469292255136728011458594045676543

Definition

For the closed regular $n$-gon $\overline{P_n}$ of circumradius $1$, the logarithmic capacity $\kappa_n$ is the positive coefficient of $w$ in its conformal map from $|w|>1$ onto $\mathbb C\setminus\overline{P_n}$, fixing infinity [1].

Parameters

$n$

— number of sides ($n\geq3$)

Formulas

(1)

$\displaystyle \kappa_n=\frac{\Gamma(1+1/n)}{\Gamma(1-1/n)\Gamma(1+2/n)}=\frac{n\sin(\pi/n)\Gamma(1+1/n)}{2^{2/n}\sqrt{\pi}\,\Gamma(1/2+1/n)}$, where $\Gamma$ is the Gamma function. The area formula in [1] gives the second expression; recurrence, reflection and duplication give the first [2].

(5)

$\displaystyle \operatorname{cap}(s\overline{P_n})=s\kappa_n\quad(s>0),\qquad A_n=\frac n2\sin\frac{2\pi}{n}.$ A regular polygon with side length $1$ has capacity $\kappa_n/[2\sin(\pi/n)]$; a regular polygon with area $1$ has capacity $\kappa_n/\sqrt{A_n}$.

(3)

$\displaystyle \kappa_4=\frac{\sqrt2}{\pi}K(1/2)$, where $K(1/2)$ is the complete elliptic integral in the parameter convention. Equivalently, $\kappa_4=[\sqrt2\operatorname{AGM}(1,1/\sqrt2)]^{-1}$, where AGM denotes the arithmetic-geometric mean [3].

(4)

$\displaystyle \cos(\pi/n)\leq\kappa_n\leq1,\qquad\lim_{n\to\infty}\kappa_n=1.$ The bounds follow from the inscribed and circumscribed disks and monotonicity of logarithmic capacity.

Comments

(5)

The domain $P_n$ is the interior of the convex hull of $\{e^{2\pi i j/n}:0\leq j<n\}$, with centre $0$ and a vertex at $1$. Its area $A_n$ is the circumradius-one value in the table of regular-polygon areas. Logarithmic capacity scales linearly with lengths.

(6)

Among polygons with a fixed number of sides and prescribed area, the regular polygon uniquely minimizes logarithmic capacity up to rigid motion [1].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
R = RealBallField(461)
def capacity(n):
    a = QQ(1) / QQ(n)
    return R(1+a).gamma() / (R(1-a).gamma()*R(1+2*a).gamma())
values = {n: capacity(n) for n in range(3, 31)}

References

[1]

A. Yu. Solynin and V. A. Zalgaller, An isoperimetric inequality for logarithmic capacity of polygons, Annals of Mathematics 159 (2004), 277–303, Theorem 1. (arXiv) (doi) (zbMATH) (MR)

Links

[2]

DLMF 5.5: Gamma recurrence, reflection and duplication

[3]

DLMF 19.8: arithmetic-geometric mean and elliptic integrals

Similar tables

Area of the regular n-gon — The circumradius-one branch gives the areas $A_n$ for the same polygons.

Complete elliptic integral of the first kind $K(m)$ — The lemniscatic value $K(1/2)$ determines the square capacity.

Values of the Gamma function at rational numbers — Gamma values at rational arguments determine every polygon capacity.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every side count $3\leq n\leq30$ at circumradius $1$. This contiguous discrete family covers small polygonal symmetries, including nonconstructible polygons; $30$ is a noncanonical editorial cutoff that limits depth as the shapes approach the unit disk)

How they were obtained:

Computed with Arb real balls at 461 working bits, using exact rational Gamma arguments and 128 guard bits. Every entry retained at least 457 bits of relative accuracy. Each 100-significant-digit decimal was checked by exact rational comparison to contain the full computed ball within one unit of its last place.

All 28 entries were compared with the capacity/area formula in [1] evaluated at 600 bits and with independent Schwarz–Christoffel boundary-length quadrature at 130 and 170 decimal digits. The quadrature used no Gamma evaluations and passed known-integral controls first. The square was also checked with a separately iterated AGM and a certified bracket. These independent calculations check the implementation; the error guarantee comes from the Arb enclosures. Reproducible code, the argument-selection justification and numerical evidence are attached.