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{
"table": {
"Title": "Logarithmic capacity of regular polygons",
"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 CITE{SolyninZalgaller}.",
"Parameters": {
"n": {
"type": "Z",
"title": "number of sides",
"display": "$n$",
"constraints": "$n\\geq3$"
}
},
"Tags": [
"analysis",
"polygon",
"special values"
],
"Links": {
"DLMFGamma": {
"title": "DLMF 5.5: Gamma recurrence, reflection and duplication",
"url": "https://dlmf.nist.gov/5.5"
},
"DLMFAGM": {
"title": "DLMF 19.8: arithmetic-geometric mean and elliptic integrals",
"url": "https://dlmf.nist.gov/19.8#i"
}
},
"References": {
"SolyninZalgaller": {
"bib": "A. Yu. Solynin and V. A. Zalgaller, An isoperimetric inequality for logarithmic capacity of polygons, Annals of Mathematics 159 (2004), 277\u2013303, Theorem 1.",
"doi": "10.4007/annals.2004.159.277",
"arxiv": "math/0510127",
"mr": "2052355",
"zbl": "1060.31001"
}
},
"Data properties": {
"type": "R",
"rigour": "proven",
"complete": "no",
"complete-note": "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",
"digits": "100",
"rigour details": "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.\n\nAll 28 entries were compared with the capacity/area formula in CITE{SolyninZalgaller} evaluated at 600 bits and with independent Schwarz\u2013Christoffel 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."
},
"Display properties": {
"number-header": "$\\kappa_n$"
},
"Numbers": {
"3": {
"number": "0.7304992431031591790794365589230298485194656779037308941028877172207606779428657203882681325963248532",
"comment": "Equilateral triangle."
},
"4": {
"number": "0.8346268416740731862814297327990468089939930134903470024498273701036819927095264118696911603512753241",
"comment": "Square."
},
"5": "0.8888552730139685196185131777411997768321924953217311565498368476306530953660281937564793297599220655",
"6": {
"number": "0.9203713733179424976555185645431729947262457919498943383433001997731018080805685639363385068818052860",
"comment": "Regular hexagon."
},
"7": "0.9402225423282581845064778905225410552810705288081291632814952991504732418300792706612797503058274595",
"8": "0.9535055293740722197741177590360440208529867995201213061558388754293159023332562076544820599226477968",
"9": "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"
},
"Keywords": [
"transfinite diameter",
"outer conformal radius",
"regular polygon capacity"
],
"Comments": {
"scale": "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 HREF{Area_of_the_regular_n-gon}[the table of regular-polygon areas]. Logarithmic capacity scales linearly with lengths.",
"extremal": "Among polygons with a fixed number of sides and prescribed area, the regular polygon uniquely minimizes logarithmic capacity up to rigid motion CITE{SolyninZalgaller}."
},
"Formulas": {
"gamma": "$\\displaystyle \\kappa_n=\\frac{\\Gamma(1+1/n)}{\\Gamma(1-1/n)\\Gamma(1+2/n)}=\\frac{n\\sin(\\pi/n)\\Gamma(1+1/n)}{2^{2/n}\\sqrt{\\pi}\\,\\Gamma(1/2+1/n)}$, where $\\Gamma$ is the HREF{Values_of_the_Gamma_function_at_rational_numbers}[Gamma function]. The area formula in CITE{SolyninZalgaller} gives the second expression; recurrence, reflection and duplication give the first CITE{DLMFGamma}.",
"scale": "$\\displaystyle \\operatorname{cap}(s\\overline{P_n})=s\\kappa_n\\quad(s>0),\\qquad A_n=\\frac n2\\sin\\frac{2\\pi}{n}.$ A regular polygon with side length $1$ has capacity $\\kappa_n/[2\\sin(\\pi/n)]$; a regular polygon with area $1$ has capacity $\\kappa_n/\\sqrt{A_n}$.",
"square": "$\\displaystyle \\kappa_4=\\frac{\\sqrt2}{\\pi}K(1/2)$, where HREF{Complete_elliptic_integral_of_the_first_kind_K#1/2}[$K(1/2)$] is the complete elliptic integral in the parameter convention. Equivalently, $\\kappa_4=[\\sqrt2\\operatorname{AGM}(1,1/\\sqrt2)]^{-1}$ CITE{DLMFAGM}.",
"limit": "$\\displaystyle \\cos(\\pi/n)\\leq\\kappa_n\\leq1,\\qquad\\lim_{n\\to\\infty}\\kappa_n=1.$ The bounds follow from the inscribed and circumscribed disks and monotonicity of logarithmic capacity."
},
"Similar tables": {
"Area_of_the_regular_n-gon": "The unit-R branch gives the areas $A_n$ for the same polygons.",
"Complete_elliptic_integral_of_the_first_kind_K": "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."
},
"Programs": {
"sage": {
"language": "Sage",
"code": "import numberdb.sage as numberdb\nfrom sage.rings.rational_field import QQ\nfrom sage.rings.real_arb import RealBallField\nR = RealBallField(461)\ndef capacity(n):\n a = QQ(1) / QQ(n)\n return R(1+a).gamma() / (R(1-a).gamma()*R(1+2*a).gamma())\nvalues = {n: capacity(n) for n in range(3, 31)}"
}
}
},
"agm_table": {
"tid": "T51",
"slug": "Arithmetic-geometric_mean",
"params": [
"1/sqrt(2)",
"1"
],
"number": "0.8472130847939790866064991234821916364814459103269421850605793726597340048341347597232002939946112299"
},
"elliptic_table": {
"slug": "Complete_elliptic_integral_of_the_first_kind_K",
"params": [
"1/2"
],
"number": "1.854074677301371918433850347195260046217598823521766905585928045056021776838119978357271861650371897"
},
"area_table": {
"tid": "T17",
"slug": "Area_of_the_regular_n-gon",
"normalization": "unit-R",
"numbers": {
"3": "1.299038105676657970145584756129404275207103940357785471041855234588949762681600027810859640067936431",
"4": "2",
"5": "2.377641290737883930291098333448455358514246585314375556118264111075382925212983754296982027427028454",
"6": "2.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872863",
"7": "2.736410188638104330479555843359202125813170815480406351432222353157821060718777545953038024118468437",
"8": "2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145",
"9": "2.892544243589426968451895344582685448084019478925568056462397646020007763084313878697197024729174582",
"10": "2.938926261462365645843529773195363842988262188215729955361362403786392370811759787542520249313706679",
"11": "2.973524496005786701591997748752804324874738343439626120196624391218103049799863397299434797458029188",
"12": "3",
"13": "3.020700618284495546764099678365181054125283124661048350201160452608340141465116042973841952204204066",
"14": "3.037186173822906843330378329938511282269935094512219135111831124725422761572388950645488152421192714",
"15": "3.050524823068501558154894927561232096923547382996300772207147646127783323934098038031016278904646952",
"16": "3.061467458920718173827679872243190934090756499885016331470405085020368271680717537896110282738268377",
"17": "3.070554162590800064330074067561451391658261764561446602457723724403234498554058677298922687883778147",
"18": "3.078181289931018597396896532140336226867750307627445656185436479162432873318698396077711641046729927",
"19": "3.084644957444493130371940807192576860387627858144394326806323435829742643613194762330880478785637908",
"20": "3.090169943749474241022934171828190588601545899028814310677243113526302314094512248536036020946955687",
"21": "3.094929331314494276723116310582005218660051914721499770435095294365846819340569960412619287113502626",
"22": "3.099058125255726674825597068812785889393556888705935590141883366966336783709320849358143802970752049",
"23": "3.102662868305779119321278009929561169465682606995908438485443604388196189449214710425005379871497852",
"24": "3.105828541230249148186786051488579940188826815839166165768038487780683696985623963068415696330995986",
"25": "3.108623589560684853028546825080599605219592580552740877952818554105387502778872126114690383372570720",
"26": "3.111103635738250972933798441382754637641249295597978173617076342603810317059774335495534768871237814",
"27": "3.113314255022942409077677715454677883177849818268547794891017828004728488678962730939624923570936789",
"28": "3.115293075388401660044635902955126632528977962703629374367818222563897248399662467044134736513021388",
"29": "3.117071383059848974138325417690098430068137089222463672001783437361790650420538576291261880920647080",
"30": "3.118675362266390056526134266076877493248771409415857546107729608235742384961115644569431350840913595"
}
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