/* Note that f_sphereCapSurf(EIods(POIo,t=0)) is the round-off error for each stage, and the term containing it is dropped before progressing to the for symbols & HIGHLY restrictive conditions : see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" Also note : /% K_1st = + 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2) f_sphereCapSurf{x} = β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): x] ] /% >>>>>>>>>>>> generative form 2425:(mathL)/* generative form /% EIods(POIo,t,1st stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,0th stage)) 2454:(mathL)/* generative form /% EIods(POIo,t,2nd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,1st stage)) 3081:(mathL)/* generative form /% EIods(POIo,t,3rd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,2nd stage)) 3368:(mathL)/* generative form /% EIods(POIo,t,4th stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,3rd stage)) >>>>>>>>>>>> differentiable form 2441:(mathL)/* differentiable form /% EIods(POIo,t,1st stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *sin(Aθpc(POIo(t),t))^2 *Rpcs(POIo(t),t)^(-5) - λ(Vons(PART)) *1 ) + f_sphereCapSurf{EIods(POIo,t)} 3027:(mathL)/* differentiable form /% EIods(POIo,t,2nd stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *sin(Aθpc(POIo(t),t))^2 *Rpcs(POIo(t),t)^(-5) + 21/8 *β^4*Rocs(POIo)^4 *sin(Aθpc(POIo(t),t))^4 *Rpcs(POIo(t),t)^(-6) - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo) *sin(Aθpc(POIo(t),t))^2 *Rpcs(POIo(t),t)^(-3) ) + f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]} 3284:(mathL)/* differentiable form /% EIods(POIo,t,3rd stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^6 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^4 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}} 3579:(mathL)/* differentiable form /% EIods(POIo,t,4th stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^6 + 455/64*β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(-8)*sin(Aθpc(POIo(t),t))^8 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^4 - λ(Vons(PART)) *5/3 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^6 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}} >>>>>>>>>>>> HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) 3016:(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% EIods(POIo,t=0,2nd stage) = E0ods(POIo,t) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4} - E0ods(POIo,t)*λ(Vons(PART))*{1 + β^2*sin(Aθpc(POIo(t),t=0))^2} 3242:(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% EIods(POIo,t=0,3rd stage) = + E0pds(POIp) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 3/2*β^4*sin(Aθpc(POIo(t),t=0))^4 + 21/8 *β^6*sin(Aθpc(POIo(t),t=0))^6} - E0pds(POIp)*λ(Vons(PART)) *{1 + β^2*sin(Aθpc(POIo(t),t=0))^2 - β^4*sin(Aθpc(POIo(t),t=0))^4} 3639:(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% EIods(POIo,t,4th stage) = + E0pds(POIp) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + 35/8*β^6*sin(Aθpc(POIo(t),t=0))^6 + 455/64*β^8*sin(Aθpc(POIo(t),t=0))^8} - E0pds(POIp)*λ(Vons(PART)) *{1 + 1* β^2*sin(Aθpc(POIo(t),t=0))^2 + 5/4 *β^4*sin(Aθpc(POIo(t),t=0))^4 + 5/3 *β^6*sin(Aθpc(POIo(t),t=0))^6} ????:(mathL)/* theoretical target - binomial series /% EIods(POIo,t,4th stage) = + E0pds(POIp) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 3/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + -1/16*β^6*sin(Aθpc(POIo(t),t=0))^6 + 3/128*β^8*sin(Aθpc(POIo(t),t=0))^8} - E0pds(POIp)*λ(Vons(PART)) *{1 + 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 3/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + -1/16*β^6*sin(Aθpc(POIo(t),t=0))^6} (endMath) ????:(mathL) binomial series = 1 3/2 3/8 -1/16 3/128 -3/256 7/1024 Lucas = 1 3/2 15/8 35/16 12Oct2019 Howell ETpds(POIp) = 1 3/2 21/8 35/8 455/64 ETpds(POIp)*λ(Vons(PART)) = 1 1 5/4 5/3 (endMath)