The quarter-Gamma ratio G* = Γ(1/4)/Γ(3/4) as an area, an arc, a moment, a motion, and a lattice sum
cpaci
This notebook collects, in executable form, the classical identities that meet at the number
*
G
=
Γ
1
4

Γ
3
4

=2.9586751191886…
. Every statement below is an elementary consequence of classical results: the beta integral, Gamma reflection, Legendre's relation, the arithmetic-geometric mean, the theta transformation, and the two-squares theorem. Nothing here is claimed as new. The notation G* is a convenience for keeping one recurring ratio visible; it attaches no claim of discovery or priority, and no physical interpretation.
The notebook has two purposes. First, to let a reader check each identity symbolically where the system can, and numerically to 40 digits otherwise. A numerical agreement is a check of transcription, not a proof; the proofs are one or two lines each and are indicated in the text. Second, to ask the community five concrete questions about prior literature, collected in the last section.
How to use it: evaluate the notebook from top to bottom (Evaluation > Evaluate Notebook). The cells are independent of any package. The plots take a few seconds. Outputs are not stored in the file.

Setup

The helper
agree
reports whether the system decides an identity symbolically and, in every case, the difference of the two sides at 40 digits. Both arguments must be exact expressions.
ClearAll[gStar, i0, i2, varpi, agree, sl, lemniscate, speed, r2, iP, jP, wP, pN, sN, deltaN, gS, wS];
gStar = Gamma[1/4]/Gamma[3/4];
agree[lhs_, rhs_] := Module[{sym},
sym = TimeConstrained[FullSimplify[lhs == rhs], 20, $Failed];
<|"decided symbolically" -> TrueQ[sym],
"difference at 40 digits" -> N[lhs, 40] - N[rhs, 40]|>]
N[gStar, 50]
Reflection fixes the product of the two quarter-Gamma values; the ratio is the quantity studied here.
Γ
1
4
Γ
3
4
=π
2
,
*
G
:=
Γ
1
4

Γ
3
4

agree[Gamma[1/4] Gamma[3/4], Pi Sqrt[2]]

Two lemniscatic integrals

Everything rests on two integrals. The substitution
v=
4
u
turns each into a beta integral.
I
0
=
1
∫
0
u
1-
4
u
=
π
*
G
4
,
I
2
=
1
∫
0
2
u
u
1-
4
u
=
π
*
G
Both integrals run from 0 to 1. Their product does not contain G*, and their ratio contains nothing else:
I
0
I
2
=
π
4
,
I
0
I
2
=
2
(
*
G
)
4
i0 = Integrate[1/Sqrt[1 - u^4], {u, 0, 1}];
i2 = Integrate[u^2/Sqrt[1 - u^4], {u, 0, 1}];
varpi = 2 i0; (* the lemniscate constant *)
{i0, i2}
{agree[i0, Sqrt[Pi] gStar/4], agree[i2, Sqrt[Pi]/gStar],
agree[i0 i2, Pi/4], agree[i0/i2, gStar^2/4]}
The product is Legendre's relation at the self-complementary modulus
k=
1
2
, since
K=
2
I
0
and
I
2
=
2E-K
2
. Wolfram Language uses the parameter m = k^2, so the modulus 1/Sqrt[2] is the parameter 1/2.
{agree[EllipticK[1/2], Sqrt[2] i0], agree[(2 EllipticE[1/2] - EllipticK[1/2])/Sqrt[2], i2],
agree[2 EllipticK[1/2] EllipticE[1/2] - EllipticK[1/2]^2, Pi/2]}

An area and an arc

Take the Bernoulli lemniscate
2
r
=
2
a
cos2θ
and the quartic squircle
4
x
+
4
y
=
4
a
with the same scale a. On a branch of the lemniscate the arc element is
ℓ=
au
1-
4
u
with u = r/a, so the length is
4a
I
0
. The squircle area is a second beta integral.
L
lem
=a
π
*
G
,
A
4
=
2
a
π
2
*
G
,
A
4
=
a
2
L
lem
The last relation is the total-measure case of the squircle and lemniscate relation of Fiedorowicz and Ramalingam (arXiv:2411.19864), who prove the stronger statement for corresponding sectors and arcs. The length below is computed from an independent rational parametrization, not from the integral that defines
I
0
.
lemniscate[t_] := {Cos[t], Sin[t] Cos[t]}/(1 + Sin[t]^2);
speed[t_] = Sqrt[Simplify[D[lemniscate[t], t] . D[lemniscate[t], t]]];
lengthLem = NIntegrate[speed[t], {t, 0, 2 Pi}, WorkingPrecision -> 30];
{lengthLem, N[Sqrt[Pi] gStar, 30]}
areaSquircle = 4 Integrate[(1 - x^4)^(1/4), {x, 0, 1}];
{agree[areaSquircle, Sqrt[Pi/2] gStar], agree[areaSquircle, 4 i0/Sqrt[2]]}
Show[
ContourPlot[{x^2 + y^2 == 1, x^4 + y^4 == 1}, {x, -1.1, 1.1}, {y, -1.1, 1.1},
ContourStyle -> {Directive[Black, Dashed], Directive[Black, Thin]}],
ParametricPlot[lemniscate[t], {t, 0, 2 Pi}, PlotStyle -> Directive[Black, Thick]],
AspectRatio -> 1, Frame -> True,
PlotLabel -> "circle (dashed), squircle (thin), lemniscate (thick); common scale a = 1"]

A measurement: span over RMS radius

Sample the lemniscate uniformly in arc length and measure the distance r of each sample from the crossing. With D = 2a the tip-to-tip span,
2
r
rms
=
1
L
lem
∮
2
r
ℓ=
2
a
I
2
I
0
,
D
r
rms
=
*
G
The sampling measure is part of the statement. Uniform polar angle or uniform interior area give different constants, computed below in closed form. For a thin wire of uniform density, the polar moment about the crossing is
M
2
r
rms
.
r2[t_] = Simplify[lemniscate[t] . lemniscate[t]];
rrms = Sqrt[NIntegrate[r2[t] speed[t], {t, 0, 2 Pi}, WorkingPrecision -> 30]/lengthLem];
{2/rrms, N[gStar, 30], agree[2/Sqrt[i2/i0], gStar]}
angleMean = Integrate[Cos[2 th], {th, -Pi/4, Pi/4}]/(Pi/2);
areaMean = Integrate[r^3, {th, -Pi/4, Pi/4}, {r, 0, Sqrt[Cos[2 th]]}]/
Integrate[r, {th, -Pi/4, Pi/4}, {r, 0, Sqrt[Cos[2 th]]}];
Grid[{{"sampling rule", "r_rms/a", "D/r_rms", "numerical D/r_rms"},
{"uniform arc length", 2/gStar, gStar, N[gStar, 12]},
{"uniform polar angle", Sqrt[angleMean], 2/Sqrt[angleMean], N[2/Sqrt[angleMean], 12]},
{"uniform interior area", Sqrt[areaMean], 2/Sqrt[areaMean], N[2/Sqrt[areaMean], 12]}},
Frame -> All, Alignment -> Left]
Uniform arc length induces the radial density
ρ(u)=
1
I
0
1-
4
u
on 0 < u < 1. Its second moment is
4
2
(
*
G
)
but its fourth moment is exactly 1/3: one RMS value does not identify a distribution.
rho[u_] := 1/(i0 Sqrt[1 - u^4]);
{agree[Integrate[u^2 rho[u], {u, 0, 1}], 4/gStar^2], agree[Integrate[u^4 rho[u], {u, 0, 1}], 1/3]}

A selection statement and its negative control

Vary the exponent. For p > 0 define three integrals on (0,1), with weights 1, u^2 and u^(p/2) against
1
1-
p
u
. The half-exponent weight always pairs with the unweighted integral; the quadratic weight, which the RMS question fixed, agrees with it only at p = 4.
I(p)W(p)=
π
p
,J(p)=W(p)⟺p=4
iP[p_] := Beta[1/p, 1/2]/p; jP[p_] := Beta[3/p, 1/2]/p; wP[p_] := Beta[1/p + 1/2, 1/2]/p;
FullSimplify[FunctionExpand[iP[p] wP[p]], p > 0]
Plot[{jP[p], wP[p]}, {p, 1, 12}, PlotStyle -> {Black, Directive[Black, Dashed]},
PlotLegends -> {"J(p): weight u^2", "W(p): weight u^(p/2)"}, AxesLabel -> {"p", None},
Epilog -> {PointSize[Medium], Point[{4, N[i2]}]}]
The negative control: a normalized crossing proves nothing by itself. Comparing 4 I(p)/c with 2 Sqrt[I(p)/J(p)] only asks for I(p) J(p) = c^2/4. The product decreases strictly from infinity to 1/3, so every comparison scale c > 2/Sqrt[3] selects exactly one exponent. The choice c = Sqrt[Pi] returns p = 4; other choices return other exponents.
MapThread[{#1, p /. FindRoot[iP[p] jP[p] == #1^2/4, {p, #2}, WorkingPrecision -> 25]} &,
{{Pi/Sqrt[2], 2, Sqrt[Pi], Sqrt[2]}, {21/10, 27/10, 41/10, 11}}] // TableForm

A motion, and a test from which G* cancels

Invert the arc integral. The inverse of
F(x)=
x
∫
0
u
1-
4
u
is the lemniscatic sine sl, which is JacobiSN with parameter -1. It solves
q''+2
3
q
=0
with first integral
2
(q')
+
4
q
=1
and real period
T
0
=4
I
0
=
π
*
G
. Neither π nor G* appears in the differential equation; the period emerges from the initial-value problem.
sl[s_] := JacobiSN[s, -1];
dsl[s_] = D[JacobiSN[s, -1], s];
Table[{D[sl[s], {s, 2}] + 2 sl[s]^3, D[sl[s], s]^2 + sl[s]^4 - 1, sl[s + N[4 i0, 30]] - sl[s]} /. s -> N[s0, 30],
{s0, {3/10, 1, 28/10}}]
Restore a dimensional coupling,
x''+λ
3
x
=0
with turning amplitude A. Then
T=
2π
*
G
A
λ
and, with uniform time weighting,
x
rms
=
2A
*
G
. Their product is free of G*:
λ
T
x
rms
=4
2
I
0
I
2
=2
2π
This is the period-moment product again, so it is not a new principle; but it is a usable model test, because no fitted value of G* can rescue a failed product. The cell below runs the test on a direct numerical solution, with no elliptic function in it.
lambda = 8/10; amp = 17/10;
{xsol, {ups}} = Reap[NDSolveValue[{x''[t] + lambda x[t]^3 == 0, x[0] == 0,
x'[0] == amp^2 Sqrt[lambda/2], WhenEvent[x[t] > 0, Sow[t]]}, x, {t, 0, 16},
PrecisionGoal -> 12, AccuracyGoal -> 12]];
period = Mean[Differences[ups]];
xrms = Sqrt[NIntegrate[xsol[t]^2, {t, ups[[-2]], ups[[-1]]}]/period];
<|"measured period" -> period, "predicted period" -> N[Sqrt[2 Pi] gStar/(amp Sqrt[lambda])],
"Sqrt[lambda] T x_rms" -> Sqrt[lambda] period xrms, "2 Sqrt[2 Pi]" -> N[2 Sqrt[2 Pi]]|>
Two cycle averages follow from the virial identity: the mean of q^4 is 1/3 and the mean of q'^2 is 2/3. Hence the closed phase curve
2
(q')
+
4
q
=1
encloses the area
2
3
T
0
=
8
3
I
0
=Β(1/4,3/2)
. This curve has mixed exponents; it is not the squircle.
{NIntegrate[sl[s]^4, {s, 0, N[4 i0]}]/N[4 i0], agree[Beta[1/4, 3/2], 8 i0/3]}
ParametricPlot[{{sl[s], dsl[s]}, {Sin[s], Cos[s]}}, {s, 0, 2 Pi},
PlotStyle -> {Directive[Black, Thick], Directive[Black, Dashed]}, AxesLabel -> {"q", "q'"},
PlotLabel -> "phase curves: quartic oscillator (solid), sine (dashed)"]

One curve that carries both integrals as lengths

The rectangular elastica has curvature
κ=
κ
0
sl(s)
with
s=
κ
0
(ℓ-
ℓ
0
)
2
. Its tangent angle satisfies cos ψ = sl'(s) and sin ψ = sl(s)^2 (both pairs solve the same linear system with the same initial value). Integrating the tangent gives, with h = 2/κ0,
X=hsl(s),Y=h
s
∫
0
2
sl(t)
t
Over a quarter of the curvature period the arc length is
h
I
0
and the advance along the axis is
h
I
2
. So the second integral is a length after all, and for one full period (arc length)(advance) =
16
2
h
I
0
I
2
=4π
2
h
. The curve returns in X and in direction but advances in Y: an open wave. The two differentials of this curve date from the 1690s (James Bernoulli); the product relation is classical (see Levien, The elastica: a mathematical history, 2008).
yAdv = NDSolveValue[{y'[s] == sl[s]^2, y[0] == 0}, y, {s, 0, 8 N[i0]}, PrecisionGoal -> 12, AccuracyGoal -> 12];
{yAdv[N[i0]] - N[i2], yAdv[4 N[i0]] - 4 N[i2], agree[(4 i0) (4 i2), 4 Pi]}
ParametricPlot[{yAdv[s], sl[s]}, {s, 0, 8 N[i0]}, PlotStyle -> Black, AspectRatio -> Automatic,
AxesLabel -> {"Y/h", "X/h"}, PlotLabel -> "rectangular elastica, two curvature periods",
Epilog -> {Thick, Line[{{0, -1.25}, {N[i2], -1.25}}], Text["h I2", {N[i2]/2, -1.4}]}]

A mean, a lattice sum, and an L-function

Gauss's arithmetic-geometric mean M gives the same value, and it is also a length: the rectangle with sides
2a
I
0
and
2a
I
2
has area
π
2
a
and height a M(1, Sqrt[2]).
*
G
=
2
π
M(1,√2)
,2
I
2
=M(1,√2)=
π
ϖ
{agree[2 Sqrt[Pi]/ArithmeticGeometricMean[1, Sqrt[2]], gStar],
agree[2 i2, ArithmeticGeometricMean[1, Sqrt[2]]], agree[2 i2, Pi/varpi]}
At the self-complementary modulus the nome is e^(-π), so the same ratio is a Gaussian sum over the square lattice. It equals the area of the squircle divided by the area of the circle of the same width; no tiling is asserted, the elliptic integral connects them.
*
G
2π
=
2
θ
3
(0,
-π
e
)
=
∑
(m,n)
-π(
2
m
+
2
n
)
e
z1 = Sum[Exp[-Pi (mm^2 + nn^2)], {mm, -12, 12}, {nn, -12, 12}];
{agree[Sqrt[2 Pi] EllipticTheta[3, 0, Exp[-Pi]]^2, gStar], N[z1, 40] - N[gStar/Sqrt[2 Pi], 40]}
The theta transformation
Z(σ)=
1
σ
Z(1/σ)
has the fixed point σ = 1. Differentiating there gives 2 Z'(1) = -Z(1), so for the Gaussian law on the lattice with weight
-π(
2
m
+
2
n
)
e
the mean squared radius is exactly 1/(2π). The normalizing sum contains G*; this observable does not.
N[Sum[(mm^2 + nn^2) Exp[-Pi (mm^2 + nn^2)], {mm, -12, 12}, {nn, -12, 12}]/z1, 40] - N[1/(2 Pi), 40]
Grouping lattice points by squared radius brings in the two-squares theorem and the character
χ
4
. The same character writes the ratio as a product,
*
G
=∏
χ
4
(j)
Γ(j/4)
over j = 1, 3, and its L-function (the Dirichlet beta function) records the ratio through its derivative at zero:
log
*
G
=β'(0)+log2
And @@ Table[SquaresR[2, k] == 4 DivisorSum[k, KroneckerSymbol[-4, #] &], {k, 1, 500}]
betaPrime0 = D[4^(-s) (Zeta[s, 1/4] - Zeta[s, 3/4]), s] /. s -> 0;
{N[betaPrime0, 40] - N[Log[gStar/2], 40], FullSimplify[betaPrime0 - Log[gStar/2]]}

The period lattice

Over the complex numbers sl(iz) = i sl(z). The numbers ϖ and iϖ are anti-periods, sl(z + ϖ) = -sl(z), and the period lattice is generated by (1 + i)ϖ and (1 - i)ϖ (Cox and Hyde, The Galois theory of the lemniscate, J. Number Theory 135 (2014), equation (7)). The real period 2ϖ = 4 I0 is a lattice vector but not a primitive one. As numbers only, the modulus of a primitive period is Sqrt[2] ϖ = 2 K(1/Sqrt[2]) = the area of the unit squircle. Equal numbers do not identify the objects.
With[{z0 = N[31/100 + 17 I/100, 30], w = N[varpi, 30]},
{sl[I z0] - I sl[z0], sl[z0 + (1 + I) w] - sl[z0], sl[z0 + (1 - I) w] - sl[z0],
sl[z0 + w] + sl[z0], sl[z0 + I w] + sl[z0]}]
Show[ComplexPlot[sl[z], {z, -32/10 - 32 I/10, 32/10 + 32 I/10}, PlotLabel -> "sl(z): zeros and poles on a square lattice"],
Graphics[{White, EdgeForm[Black], Table[Disk[N[varpi] {mm + nn, mm - nn}, 0.07], {mm, -1, 1}, {nn, -1, 1}],
Black, Arrowheads[Small], Arrow[{{0, 0}, N[varpi] {1, 1}}], Arrow[{{0, 0}, N[varpi] {-1, 1}}]}]]
The white discs are lattice points m(1 + i)ϖ + n(1 - i)ϖ; the arrows are the generators (1 + i)ϖ and i(1 + i)ϖ, whose ratio is i.

Three computations with error bounds

agmSteps = Rest[NestList[{(#[[1]] + #[[2]])/2, Sqrt[#[[1]] #[[2]]]} &, {1, Sqrt[2]}, 4]];
TableForm[Table[With[{lo = 2 Sqrt[Pi]/agmSteps[[n, 1]], hi = 2 Sqrt[Pi]/agmSteps[[n, 2]]},
{n, N[lo, 25], N[hi, 25], TrueQ[lo <= gStar <= hi]}], {n, 1, 4}],
TableHeadings -> {None, {"AGM steps", "lower", "upper", "encloses G*"}}]
sN[n_] := 1 + 2 Sum[Exp[-Pi k^2], {k, 1, n}];
deltaN[n_] := 2 Exp[-Pi (n + 1)^2]/(1 - Exp[-Pi (2 n + 3)]);
TableForm[Table[With[{lo = Sqrt[2 Pi] sN[n]^2},
With[{hi = lo + Sqrt[2 Pi] (2 sN[n] deltaN[n] + deltaN[n]^2)},
{n, N[lo, 25], N[hi - lo, 5], TrueQ[lo < gStar <= hi]}]], {n, 1, 4}],
TableHeadings -> {None, {"theta terms", "lower", "width", "encloses G*"}}]
pN[n_] := Product[(4 k + 3)/(4 k + 1), {k, 0, n - 1}]/Sqrt[n];
TableForm[Table[With[{lo = pN[n] Exp[-1/(64 n^2)], hi = pN[n] Exp[-1/(64 n^2) + 5/(2048 n^4)]},
{n, N[lo, 20], N[hi, 20], TrueQ[lo < gStar < hi]}], {n, {1, 2, 5, 10, 50}}],
TableHeadings -> {None, {"product factors", "lower", "upper", "encloses G*"}}]
{NIntegrate[Exp[-3 t] (1 - Sech[t/4])/(2 t), {t, 0, Infinity}, WorkingPrecision -> 25],
N[Log[pN[3]/gStar], 25]}

Two normalizations side by side

rows = {{"lemniscate length / a", Sqrt[Pi] gS, 2 wS},
{"squircle area / a^2", Sqrt[Pi/2] gS, Sqrt[2] wS},
{"oscillator period T0", Sqrt[Pi] gS, 2 wS},
{"K(1/Sqrt[2])", Sqrt[2 Pi] gS/4, wS/Sqrt[2]},
{"pendulum at 90 degrees, T Sqrt[g/l]", Sqrt[2 Pi] gS, 2 Sqrt[2] wS},
{"elastica curvature period, kappa0 Lambda", 2 Sqrt[Pi] gS, 4 wS},
{"Z(1), squircle/circle area ratio", gS/Sqrt[2 Pi], Sqrt[2] wS/Pi},
{"I2", Sqrt[Pi]/gS, Pi/(2 wS)},
{"M(1, Sqrt[2])", 2 Sqrt[Pi]/gS, Pi/wS},
{"beta'(0)", Log[gS/2], Log[wS/Sqrt[Pi]]},
{"E/K at the self-complementary modulus", 1/2 + 2/gS^2, 1/2 + Pi/(2 wS^2)},
{"I0/I2", gS^2/4, wS^2/Pi},
{"span / RMS radius", gS, 2 wS/Sqrt[Pi]}};
Grid[Prepend[{#[[1]], #[[2]], #[[3]],
N[#[[2]] /. gS -> gStar, 30] - N[#[[3]] /. wS -> varpi, 30]} & /@ rows,
{"quantity", "with G*", "with varpi", "difference"}] /. {gS -> "G*", wS -> "varpi"},
Frame -> All, Alignment -> Left]
Reading of the table, a judgement about notation and not a theorem: in every length, period and area row the lemniscate constant is the shorter form, and G* arrives with a factor Sqrt[π] or Sqrt[2π]. In the quotient and moment rows G* carries no factor of π. The middle rows are mixed. So G* is the natural normalization for quotient and moment questions, and ϖ for period, length and area questions. The symbol has no obligation to replace the older one.

What is known about the number

The same checks in Wolfram|Alpha

Readers without a Wolfram Language kernel can paste these plain queries into Wolfram|Alpha. The cell after the list sends the first one from inside the notebook (it needs a network connection).
◼
  • Gamma(1/4)/Gamma(3/4)
    • ◼
  • integrate 1/sqrt(1-x^4) from 0 to 1
    • ◼
  • (integrate 1/sqrt(1-x^4) from 0 to 1) * (integrate x^2/sqrt(1-x^4) from 0 to 1)
    • ◼
  • 2 sqrt(pi) / agm(1, sqrt(2))
    • ◼
  • sqrt(2 pi) * EllipticTheta[3, 0, e^(-pi)]^2
    • ◼
  • area enclosed by x^4 + y^4 = 1
    • ◼
  • arc length of the lemniscate r^2 = cos(2 theta)
    • WolframAlpha["Gamma(1/4)/Gamma(3/4)", "Result"]

    Five questions for the community

    The literature search behind this notebook was not exhaustive. None of the following is claimed as new; each is a question about where it already appears. References, counterexamples and corrections are all welcome.
    1
    .
    The measurement. Is the characterization (tip-to-tip span)/(RMS distance from the crossing, uniform in arc length) = Γ(1/4)/Γ(3/4) for the Bernoulli lemniscate, or the equivalent polar moment of a uniform lemniscate wire, stated anywhere?
    2
    .
    The lattice mean. For the Gaussian law on the square lattice with weight exp(-π(m^2 + n^2)), the mean squared radius is exactly 1/(2π), by the theta transformation at its fixed point. Where is this recorded (theta functions, discrete Gaussians, lattice sums)?
    3
    .
    The product enclosure. Is the two-term bound for Γ(N + 3/4)/(Sqrt[N] Γ(N + 1/4)) displayed above a stated corollary of the general error-bound theory for ratios of Gamma functions (Frenzen 1987; Buric and Elezovic 2011; DLMF 5.11(iii))?
    4
    .
    The dual weight. The relation I(p) W(p) = π/p is close to published generalized Legendre relations and to Wallis-type products on clovers. Which reference states it in this form, and is the observation that the quadratic and half-exponent weights agree only at p = 4 made anywhere?
    5
    .
    The model test. For the purely quartic oscillator, is the G*-free product Sqrt[λ] T x_rms = 2 Sqrt[2π] in the nonlinear-oscillation literature?
    A sixth, broader question: on which class of problems, if any, is the ratio a better normalization than the lemniscate constant, beyond the quotient and moment rows of the table above?

    Sources

    ◼
  • NIST Digital Library of Mathematical Functions: 5.5 (reflection), 5.12 (beta integral), 5.11(iii) (Gamma ratios), 19.7 (Legendre's relation), 19.8 (AGM), 19.20 (lemniscatic integrals), 20.7 (theta transformation), 22.8 and 22.15 (Jacobian functions), 25.11 (Hurwitz zeta), 27.13 (sums of squares). https://dlmf.nist.gov
    • ◼
  • Z. Fiedorowicz and M. V. Ramalingam, An Elementary Proof of a Remarkable Relation Between the Squircle and Lemniscate, arXiv:2411.19864; and Generalizations of the Squircle-Lemniscate Relation and Keplerian Dynamics, arXiv:2601.17358.
    • ◼
  • D. A. Cox and T. Hyde, The Galois theory of the lemniscate, Journal of Number Theory 135 (2014), 43-59, arXiv:1208.2653.
    • ◼
  • D. A. Cox, The Arithmetic-Geometric Mean of Gauss, L'Enseignement Mathématique (2) 30 (1984), 275-330.
    • ◼
  • J. Todd, The lemniscate constants, Communications of the ACM 18 (1975), 14-19.
    • ◼
  • R. Levien, The elastica: a mathematical history, Technical Report UCB/EECS-2008-103, University of California at Berkeley (2008).
    • ◼
  • D. Kershaw, Some extensions of W. Gautschi's inequalities for the gamma function, Mathematics of Computation 41 (1983), 607-611. C. L. Frenzen, Error bounds for asymptotic expansions of the ratio of two gamma functions, SIAM J. Math. Anal. 18 (1987), 890-896.
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  • Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics 187 (1996), 1319-1348.
    • A longer account, with proofs, figures, 363 reproducible numerical checks and a register of 120 questions, is the manuscript Between the Circle and the Lemniscate: a mathematical conversation in ten voices (cpaci, 2026).