Gauss's Constant

Author

Eric W. Weisstein
October 6, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/GausssConstant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GausssConstant.html.

©2023 Wolfram Research, Inc. except for portions noted otherwise

Numeric

A014549

N[((2 EllipticK[-1])/π),20]

Integrals

(2/π) (∫_0^1)(1/(Sqrt[1-(x^4)]))dx

(2/π)Integrate[(1/(Sqrt[1-(x^4)])),{x,0,1}]

(2/π) (∫_0^(π/2))(1/(Sqrt[(sin^2)(θ)+1]))dθ

(2/π)Integrate[(1/(Sqrt[1+((Sin[θ])^2)])),{θ,0,π/2}]

Special functions

(L/π)

N[(L/π)/.L->2 EllipticK[-1]]

(1/(ArithmeticGeometricMean[1, (Sqrt[2])]))

N[(1/(ArithmeticGeometricMean[1,(Sqrt[2])]))]

((2 (EllipticK[-1]))/π)

N[((2 EllipticK[-1])/π)]

(((Sqrt[2]) (EllipticK[(1/2)]))/π)

N[(((Sqrt[2]) EllipticK[(1/2)])/π)]

((4 (Sqrt[2]) ((Gamma[(5/4)])^2))/(π^(3/2)))

N[((4 (Sqrt[2]) ((Gamma[(5/4)])^2))/(π^(3/2)))]

(((2 π))^(-3/2)) ((Gamma[(1/4)])^2)

N[(((2 π))^(-3/2)) ((Gamma[(1/4)])^2)]

((EllipticTheta[4, 0, (e^(-π))])^2)

N[((EllipticTheta[4,0,(E^(-π))])^2)]

(3/(2 (CarlsonRD[0, 2, 1])))

(3/(2CarlsonRD[0,2,1]))//N

((2 (CarlsonRF[0, 1, 2]))/π)

(2/π)CarlsonRF[0,1,2]//N

(CarlsonRK[1, 2])

N[CarlsonRK[1,2]]

Sums

((((∑_(n=-∞)^∞)(((-1))^n) exp(-π (n^2))))^2)

((Sum[(((-1))^n)Exp[-π (n^2)],{n,-∞,∞}])^2)//FunctionExpand

(2^(5/4)) exp(-(π/3)) ((((∑_(n=-∞)^∞)(((-1))^n) exp(-2 π (3 n+1) n)))^2)

(2^(5/4)) Exp[-π/3] ((Sum[(((-1))^n)Exp[-2π(3n+1)n],{n,-∞,∞}])^2)

Continued Fraction

ContinuedFraction[((2 EllipticK[-1])/π),100]