Zolotarev-Schur Constant

Author

Eric W. Weisstein
August 5, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/Zolotarev-SchurConstant.nb.

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

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

Value

FindInstance[(((EllipticK[(c^2)]-EllipticE[(c^2)]) )^3)+(1-(c^2)) EllipticK[(c^2)]-(1+(c^2)) EllipticE[(c^2)]==0&&0<c<1,c]

FullSimplify[%]

FindRoot[(((EllipticK[(c^2)]-EllipticE[(c^2)]) )^3)+(1-(c^2)) EllipticK[(c^2)]-(1+(c^2)) EllipticE[(c^2)],{c,.5,.9}]

σ=With[{c=Root[{((EllipticE[((#1)^2)])^3)-3 ((EllipticE[((#1)^2)])^2) EllipticK[((#1)^2)]+EllipticK[((#1)^2)] (-1-((EllipticK[((#1)^2)])^2)+((#1)^2))+EllipticE[((#1)^2)] (1+3 ((EllipticK[((#1)^2)])^2)+((#1)^2))&,0.9155020553896763963055240364016622896543129422884607667501`30.102999566398122}]},
(1/(c^2))(((1-((EllipticE[(c^2)])/(EllipticK[(c^2)]))))^2)]

N[σ,1000]