Values of the arithmetic-geometric mean

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Numbers

$a$

$b$

$\text{agm}(a,b)$

$\sqrt2$:

1.198140234735592207439922492280323878227212663215651558263674952946405214143915670835885556489793389

1.456791031046906869186432383265081974973863943221305590794172383267926454580250900257473712818448444

1.863616783244896542355689031024270595157532856826853722220441226978325794579357223412777796614727710

2.243028580287602570127802192829066540508973142406609975918823701387404804228999572241115059094195022

2.604008190530940288696427448725034330927065255637649487748413254455679280568518526956121116854328855

2.951328742390572760219270635177090504120654922946806134466697995198185549266872303981975340287141917

3.287921981826418106223098714072612344643650010569253540046217265736608471272099543542450019040641738

3.615756177597362748733192906502120611643324561464556329010802999458975229689874716887911805712572257

3.936235503649555477978926175500747897827897640737163942458201226349598978791472980786266544198726912

10:

4.250407094932274861728164318373134866798467864190192859670147662223755312740903784525285460787617179

11:

4.559078717091160669597575844661316136795084548014541710542427667751288633913257693101651331654131486

12:

4.862890376598251977741808909636041298369186248916010562049178110999654725956168152953347061084770534

13:

5.162360281696470422035921077342339787076734739061128968529271051127546278985209694179638759086855265

14:

5.457915625839004704472692470141994295833656411332850364902539033883678209185291010563280629355044786

15:

5.749913939609722414938736462396561539239740421701891649834623458756035031405475865714890677001391060

16:

6.038658340417193651451838056751828375285039544140860144834880855132210563234114876933556470316225721

sqrt(3):

1.340933408990525735227339312519361901602872987191217883135600277575615405018736227980871202038603784

sqrt(5):

1.556086778541937427975619106435818475364867272775271970614859784257178841152580415078209492365340376

sqrt(6):

1.643945443135381154956134178599033804502658013450571327015822644328614669331413343709529634439601787

sqrt(7):

1.723328335951541379469884045371297413268308107360689232279036253395920224032973670029271341638766519

sqrt(8):

1.796122982928329497635119641583339163679491232044118809647290141033559067919515803435307777433394574

sqrt(10):

1.926732604773774420758583099338140427466787708574274354600292655904593106771750156362471615149291493

sqrt(11):

1.986157740820355049770889427321746921119234510263253193329537584756095047720600549284154514624255670

sqrt(12):

2.042419760637451473867200450647654649887107606943397041594016990132347735408649391024914045620498297

sqrt(13):

2.095933702997821789181167011460489520611291622836739224656233689842893167707710977872441223743713635

sqrt(14):

2.147032798015794840857659526557912822979862818110060488340851499641581594093529507118392192876935286

sqrt(15):

2.195989224672134794956186113497215678550240553774379157283724683491526363810331272242792961851169098

$1/\sqrt2$

0.8472130847939790866064991234821916364814459103269421850605793726597340048341347597232002939946112299

$1/\sqrt2$

param-latex:

$1/\sqrt2$

Definition

The arithmetic-geometric mean $\text{agm}(a,b)$ of two non-negative real numbers $a$ and $b$ is defined as the simultaneous limit $\lim a_n = \lim g_n$ of the sequences $(a_n)$ and $(g_n)$ given by $a_0 = a$, $g_0 = b$, $a_{n+1} = (a_n + g_n)/2$, $g_{n+1} = \sqrt{a_n g_n}$. This table lists values of $\text{agm}(a,b)$ at selected positive initial pairs.

Parameters

$a$

— real number ($a \geq 0$)

$b$

— real number ($b \geq 0$)

Formulas

(1)

$\text{agm}(a,b) = \frac{(a+b)\pi}{4K((a-b)/(a+b))}$, where $K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}$ is the complete elliptic integral of the first kind [3].

(2)

$\text{agm}(a,b) = \text{agm}(b,a)$

(3)

$\text{agm}(a,a) = a$

(4)

$\text{agm}(0,a) = 0$

(5)

$\operatorname{agm}(1/\sqrt2,1)=2\pi^{3/2}/\Gamma(1/4)^2$ and $\operatorname{agm}(1,\sqrt2)=2\sqrt2\pi^{3/2}/\Gamma(1/4)^2$.

Comments

(6)

Due to the symmetry (2) and trivial cases (3), (4), we only list values for $a < b$.

(7)

The positive AGM is invariant under one arithmetic/geometric iteration and is related to a complete elliptic integral [6]. The retained lemniscatic values are supplemented by a unit-normalized integer-input family $(1,n)$ and a quadratic-input family $(1,\sqrt d)$. These are bounded exploratory references for identifying means and elliptic integrals with simple coefficients, not claims of a published application for each pair. Perfect squares are omitted from the second family because their input pairs already occur in the first; no Cartesian box of both initial values is taken.

Programs

(P1)

Python

from balanced_values import BalancedValues
from numberdb._write import to_text
reference = BalancedValues()
for parameters in reference.enumerate():
    print(parameters, to_text(reference.value(parameters, reference.digits), reference.digits))

Links

[1]

MathWorld: Arithmetic-Geometric Mean

[2]

Wikipedia: Arithmetic-geometric mean

[3]

Wikipedia: Complete elliptic integral of the first kind

[4]

DLMF 19.8(i): Gauss's Arithmetic-Geometric Mean

[5]

DLMF 5.12: Beta Function

[6]

DLMF 19.8: AGM iteration and elliptic integrals

Data properties

Entries are of type: real number

Table is complete: no (it retains both lemniscatic pairs and adds $(1,n)$ for $1\leq n\leq16$ and $(1,\sqrt d)$ for nonsquare $2\leq d\leq15$; these practical simple-coefficient bounds are exploratory reference choices, and the diagonal anchor is exact)

How well the digits are known: proven

How they were obtained:

All 29 stored entries are recomputed from exact rational or enclosed square-root inputs using Arb at 100 decimal digits plus 160 guard bits. Existing numeric strings and row annotations are retained unchanged; their written intervals strictly contain the fresh enclosures. Exact anchors and rational polynomial integrals use exact arithmetic.

Every inexact result is finite, with at least 488 relative accuracy bits. Independent mpmath computations at 150 and 200 decimal digits check every value; the AGM check uses its elliptic-integral expression, and the incomplete-gamma check uses integer/half-integer closed forms. Known nonzero and exact controls, shifted-last-place rejection controls, enumeration checks and applicable identities pass. The attached balanced-evidence.json records the checks, and balanced_checks.py repeats them. The current generate.py and balanced-selection.json reproduce the full retained-plus-added selection; older auxiliary attachments are preserved as historical sources, not claimed as the default enumeration.