363:(mathL)  β =  Vons(PART)/c
 
2406:(mathL)		EIods(POIo,t=0,1st stage)  =  K_1st + f_sphereCapSurf(EIods(POIo,t))
 
2474:(mathL)/*	generative form  /%
	EIods(POIo,t,2nd stage) =  K_1st + f_sphereCapSurf(EIods(POIo,t,1st stage))
 
2650:(mathL)/*	generative form   /%
	EIods(POIo,t,2nd stage) =  K_1st + f_sphereCapSurf(EIods(POIo,t,1st stage))
 
2889:(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"  
/%
	Rocs(POIo) = Rpcs(POIo(t),t=0)
 
2958:(mathL)/*	differentiable form  EIods(POIo,t,2nd stage)  /%
	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))]}
 
2969:(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}
 
2979:(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"  
/%
	ETods(POIo,t=0,2nd stage)
	=		E0ods(POIo,t)					*{1 + 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}
 
3100:(mathL)/*	generative form   /%
	EIods(POIo,t,3rd stage) =  K_1st + f_sphereCapSurf(EIods(POIo,t,2nd stage))
 
3287:(mathL)/*	differentiable form   /%
	EIods(POIo,t,3rd 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)
				+						35/8	*β^6*Rocs(POIo)^5	*sin(Aθpc(POIo(t),t))^6	*Rpcs(POIo(t),t)^(-7)
				-	λ(Vons(PART))	*1		 						                       	*Rpcs(POIo(t),t)^(-2)
				-	λ(Vons(PART))	*1		*β^2*Rocs(POIo)^1	*sin(Aθpc(POIo(t),t))^2	*Rpcs(POIo(t),t)^(-3)
				-	λ(Vons(PART))	*5/4	*β^4*Rocs(POIo)^2	*sin(Aθpc(POIo(t),t))^4	*Rpcs(POIo(t),t)^(-4)
			 )
		+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}
 
3341:(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 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + 35/8*β^6*sin(Aθpc(POIo(t),t=0))^6}
		-	E0pds(POIp)*λ(Vons(PART))	*{1 + 	 β^2*sin(Aθpc(POIo(t),t=0))^2 + 5/4	*β^4*sin(Aθpc(POIo(t),t=0))^4}
 
3371:(mathL)/*	generative form   /%
	EIods(POIo,t,4th stage) =  K_1st + f_sphereCapSurf(EIods(POIo,t,3rd stage))
 
3582:(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))}}}}
 
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 + 	 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}
 
3653:(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}
 
3659:(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
		ETpds(POIp)						= 1  3/2  21/8  35/8   455/64
		ETpds(POIp)*λ(Vons(PART))	= 1  1	  5/4	  5/3