Littlewood-Salem-Izumi Constant

Author

Eric W. Weisstein
March 10, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/Littlewood-Salem-IzumiConstant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Littlewood-Salem-IzumiConstant.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Definition

Integrate[u^-α Cos[u],{u,0,3 π/2},Assumptions->0<α<1]

FunctionExpand[(((((((3 π)/2)))^(1-α)) HypergeometricPFQ[{(1/2)-(α/2)},{(1/2),(3/2)-(α/2)},-((9 (π^2))/16)])/(1-α))]

FullSimplify[%%]

FindRoot[Integrate[u^-α Cos[u],{u,0,3 π/2}]==0,{α,1/2}]

Timing[ax=α/.FindRoot[Integrate[u^-α Cos[u],{u,0,3 π/2}]==0,{α,1/2},WorkingPrecision->5000];]

lsi=FindInstance[(((((((3 π)/2)))^(1-α)) HypergeometricPFQ[{(1/2)-(α/2)},{(1/2),(3/2)-(α/2)},-((9 (π^2))/16)])/(1-α))==0&&α>0,α][[1,1,2]]

N[lsi,20]

N[%]

(ax1=N[lsi,5000]);//Timing

N[ax1,1000]

(ax2=α/.FindRoot[Integrate[u^-α Cos[u],{u,0,3 π/2}]==0,{α,1/2},WorkingPrecision->5000];)//Timing

ax1-ax2

Properties

Element[lsi,Algebraics]

Element[lsi,Rationals]