Grothendiecks Constant

Author

Eric W. Weisstein
January 20, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Postulated Values

(π/(2 log((Sqrt[3])+1)))

N[(π/(2Log[1+(Sqrt[3])])),20]

(8/(π ((Subscript[x, 0])+1)))

V4.2

x Integrate[(((Cos[θ])^2)/(Sqrt[1-(x^2)((Sin[θ])^2)])),{θ,0,π/2},Assumptions->0<x<1]//FullSimplify

V5.0

x Integrate[(((Cos[θ])^2)/(Sqrt[1-(x^2)((Sin[θ])^2)])),{θ,0,π/2},Assumptions->0<x<1]//FullSimplify

FullSimplify[%]

V5.2

x Integrate[(((Cos[θ])^2)/(Sqrt[1-(x^2)((Sin[θ])^2)])),{θ,0,π/2},Assumptions->0<x<1]//FullSimplify

V6.1

x Integrate[(((Cos[θ])^2)/(Sqrt[1-(x^2)((Sin[θ])^2)])),{θ,0,π/2},Assumptions->0<x<1]//FullSimplify

FullSimplify[(((Sqrt[1-(x^2)]) (EllipticE[((x^2)/(-1+(x^2)))]-EllipticK[((x^2)/(-1+(x^2)))]))/x),0<x<1]

expr1=(Sqrt[1-(x^2)]) (EllipticE[((x^2)/(-1+(x^2)))]-EllipticK[((x^2)/(-1+(x^2)))]);
expr2=EllipticE[(x^2)]-(1-(x^2))EllipticK[(x^2)];

FullSimplify[expr1-expr2,0<x<1]

Plot[expr1-expr2,{x,0,1}]

Table[expr1-expr2,{x,.01,.99,.01}]//Chop//Union

ψ[x_]:=((EllipticE[(x^2)]-(1-(x^2))EllipticK[(x^2)])/x)

x0=First[x/.FindInstance[ψ[x]==(1/8)π(x+1),x]]

N[(8/(π(x0+1))),20]

V7

FullSimplify[x Integrate[(((Cos[θ])^2)/(Sqrt[1-(x^2)((Sin[θ])^2)])),{θ,0,π/2},Assumptions->0<x<1],0<x<1]

(1/(2 K(i)-E(i)))

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

N[%,20]