{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 17 "limitfourteen.mws" }} }{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 48 "The special case v(theta)=-1 , with constraints," }} }{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 36 "Notation diff(v(thtea),theta $n)=v[n]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 35 "the following substitution is made." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 435 "P(t,theta) = P[i nfinity](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[in finity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[in finity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/ 4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta, 2)))*t^2*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(th eta))+1/2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))*ln(t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"PG6$%\"tG%&thetaG,,-&F%6#%)in finityG6#F(\"\"\"*&-%\"vGF.F/-%#lnG6#F'F/!\"\"*&#F/\"\"%F/**)-%$expG6# F*\"\"#F/)-%%diffG6$-&%\"QGF,F.-%\"$G6$F(F?F?F/)F'F?F/)F3F?F/F/F/*&-&% \"VG6#F/F.F/FJF/F/*(,(*(-F=6#,$*&F?F/F*F/F/F/-&%$psiG6#FFF.F/FAF/F/*&# F/F9F/*$F@F/F/F6*&#F/F9F/-FB6$F1FGF/F6F/FJF/F3F/F//-FFF&,(FDF/*&FXF/)F ',$*&F?F/F1F/F/F/F/*&#F/F?F/*(FAF/FaoF/F3F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "gr tw();" }}{PARA 6 "" 1 "" {TEXT -1 25 "Scalar invariant library." }} {PARA 6 "" 1 "" {TEXT -1 28 "Last modified 25 March 1997." }}{PARA 6 " " 1 "" {TEXT -1 25 "`Differential Invariants`" }}{PARA 6 "" 1 "" {TEXT -1 29 "`Last modified Jan. 20, 1995`" }}{PARA 6 "" 1 "" {TEXT -1 41 "`Basis/tetrad related object definitions`" }}{PARA 6 "" 1 "" {TEXT -1 31 "`Last modified 23 January 2001`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~V ersion~1.79~(R4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%06~February~2001 G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgrave,~De nis~Pollney~and~Kayll~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyr ight~1994-2001~by~the~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%in Latest~version~available~from:~http://grtensor.phy.queensu.ca/G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%4c:/Grtii(6)/MetricsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qload(gowdy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%&gowdyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~gowdy~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"x G6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~G%\"aG-%'vectorG6#7& %\"tG%&thetaG%#x1G%#x2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elem entG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",,**-%$ex pG6#,$*&#F(F'F(-%&gammaG6$%\"tG%&thetaGF(!\"\"F(F4#F6F'%#~dGF()F4%#2~G F(F6**F+F(F4F7F8F()F5F:F(F(**F4F(-F,6#-%\"PGF3F(F8F()%#x1GF:F(F(*2F'F( F4F(F>F(-%\"QGF3F(F8F()FC%\"~GF(%#d~GF()%#x2GFHF(F(*(,&*(F4F(F>F()FEF' F(F(*&F4F(F>F6F(F(F8F()FKF:F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%, ConstraintsG7&/-%%diffG6$-%&gammaG6$%\"tG%&thetaGF-,**(F-\"\"\")-%$exp G6#-%\"PGF,\"\"#F1)-F(6$-%\"QGF,F-F8F1!\"\"*&F-F1)-F(6$F6F-F8F1F>*(F-F 1F2F1)-F(6$F*&F-F1)-F(6$F6F.F8F1F>/-F(6$F*F.,&**F8F1F-F1FIF1F AF1F>*,F8F1F-F1F2F1F:F1FEF1F>/-F(6$F6-%\"$G6$F-F8,(-F(6$F6-FU6$F.F8F1* &FAF1F-F>F>*&-F46#,$*&F8F1F6F1F1F1,&*$F9F1F1*$FDF1F>F1F1/-F(6$FF>*(F8F1FAF1F:F1F>*(F8F1FIF1FEF1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalc(WeylSq);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%*CPU~Time~G$\"#%*!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "gralter(_,13,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Co mponent simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 64 "Applying routine `Apply c onstraints repeatedly` to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine factor to object WeylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#K!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "grmap(_,subs,P(t,theta) = P[infinity](theta)-v(theta )*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(thet a,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q]( theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](thet a),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2*ln(t),Q(t,thet a) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta))+1/2*diff(Q[infini ty](theta),`$`(theta,2))*t^(2*v(theta))*ln(t),`x`);" }}{PARA 6 "" 1 " " {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 42 "Keep all derivatives of Q[infinity](theta)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "grmap(_,subs,diff(Q[infinity ](theta),theta$5)=Q[5],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying \+ routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "g rmap(_,subs,diff(Q[infinity](theta),theta$4)=0,`x`);" }}{PARA 6 "" 1 " " {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "grmap(_,subs,diff(Q[infinity](theta),theta$3)= 0,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylS q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "grmap(_,subs,diff(Q[in finity](theta),theta)=Q[1],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Apply ing routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "grmap(_,subs,diff(Q[infinity](theta),theta$2)=0,`x`);" }{MPLTEXT 0 21 0 "" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to Weyl Sq" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 32 "Keep all derivatives o f v(theta)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,subs, diff(v(theta),theta$4)=v[4],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Appl ying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,subs,diff(v(theta),theta$3)=v[3],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "grmap(_,subs,diff(v(theta),theta$2)=0,`x`);" }} {PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "grmap(_,subs,diff(v(theta),t heta$1)=0,`x`):" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs \+ to WeylSq" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 37 "Keep all deriva tives of psi[Q](theta)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "g rmap(_,subs,diff(psi[Q](theta),`$`(theta,3))=psi[Q3],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "grmap(_,subs,diff(psi[Q](theta),`$`(theta ,2))=0,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to \+ WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "grmap(_,subs,diff(psi[Q](theta),`$`(theta ,4))=0,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to \+ WeylSq" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 17 "Apply constraints" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,subs,diff(psi[Q](theta),theta)=0,`x`):" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "grmap(_,subs,psi[Q](theta)=0 ,`x`):" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "grmap(_,subs,v(theta)=-1,`x`):" }}{PARA 6 "" 1 " " {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 30 "denom(grcomponent(WeylSq,[]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"&%Q;\"\"\")%\"tG\"\"*F&)-%$expG6#,$*&#F&\"\" #F&-%&gammaG6$F(%&thetaGF&!\"\"F1F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "factor(subs(t=0,numer(grcomponent(WeylSq,[]))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "core:= simplify(limit(factor(grcomponent(WeylSq,[])/(t*exp(gamma(t,theta)))), t=0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Ww:=factor(t*exp( gamma(t,theta))*core);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#WwG,$*(% \"tG\"\"\"-%$expG6#-%&gammaG6$F'%&thetaGF(,.*&\"#;F()-&%\"VG6#F(6#F/\" \"#F(!\"\"*&\"\"$F()-%%diffG6$-&%\"PG6#%)infinityGF8F/\"\"%F(F:*&FFF() -F?6$FA-%\"$G6$F/F9F9F(F:*(\"\"'F()F>F9F(FIF(F(*(\"\")F(FIF(F4F(F(*(\" #7F(-F*6#,$*&F9F(FAF(F(F()&%\"QGF7F9F(F(F(F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "latex(%);" }}{PARA 6 "" 1 "" {TEXT -1 67 "-t\{e^ \{\\gamma \\left( t,\\theta \\right) \}\} \\left( -16\\, \\left( V_\{ \{1\}\}" }}{PARA 6 "" 1 "" {TEXT -1 70 " \\left( \\theta \\right) \\r ight) ^\{2\}-3\\, \\left( \{\\frac \{d\}\{d\\theta\}\}P_" }}{PARA 6 " " 1 "" {TEXT -1 70 "\{\{\\infty \}\} \\left( \\theta \\right) \\right ) ^\{4\}-4\\, \\left( \{\\frac \{d^\{" }}{PARA 6 "" 1 "" {TEXT -1 70 " 2\}\}\{d\{\\theta\}^\{2\}\}\}P_\{\{\\infty \}\} \\left( \\theta \\righ t) \\right) ^\{2\}+6" }}{PARA 6 "" 1 "" {TEXT -1 66 "\\, \\left( \{\\ frac \{d\}\{d\\theta\}\}P_\{\{\\infty \}\} \\left( \\theta \\right) " }}{PARA 6 "" 1 "" {TEXT -1 70 " \\right) ^\{2\}\{\\frac \{d^\{2\}\}\{d \{\\theta\}^\{2\}\}\}P_\{\{\\infty \}\} \\left( \\theta" }}{PARA 6 "" 1 "" {TEXT -1 64 " \\right) +8\\, \\left( \{\\frac \{d^\{2\}\}\{d\{\\t heta\}^\{2\}\}\}P_\{\{\\infty \}\}" }}{PARA 6 "" 1 "" {TEXT -1 70 " \\ left( \\theta \\right) \\right) V_\{\{1\}\} \\left( \\theta \\right) \+ +12\\,\{e^" }}{PARA 6 "" 1 "" {TEXT -1 64 "\{2\\,P_\{\{\\infty \}\} \\ left( \\theta \\right) \}\}\{Q_\{\{1\}\}\}^\{2\} \\right) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 387 "V[1](theta):=(exp(P[infinity](thet a))^2*diff(Q[infinity](theta),`$`(theta,2))^2/4)*(3/2)-exp(2*P[infinit y](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*dif f(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2))+e xp(P[infinity](theta))^2*(psi[Q](theta)+diff(Q[infinity](theta),`$`(th eta,2))/4)^2+(diff(P[infinity](theta),theta$2)+diff(v(theta),theta$2)) /4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"VG6#\"\"\"6#%&thetaG,,*&# \"\"$\"\")F(*&)-%$expG6#-&%\"PG6#%)infinityGF)\"\"#F()-%%diffG6$-&%\"Q GF8F)-%\"$G6$F*F:F:F(F(F(*(-F36#,$*&F:F(F5F(F(F(-&%$psiG6#FAF)F(F " 0 "" {MPLTEXT 1 0 69 "subs(psi[Q](th eta)=0,diff(Q[infinity](theta),theta$2)=0,V[1](theta));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"%F&-%%diffG6$-&%\"PG6#%)infinityG 6#%&thetaG-%\"$G6$F1\"\"#F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "latex(1/4*diff(P[infinity](theta),`$`(theta,2)));" }}{PARA 6 "" 1 "" {TEXT -1 62 "1/4\\,\{\\frac \{d^\{2\}\}\{d\{\\theta\}^\{2\}\}\}P_ \{\{\\infty \}\} \\left( \\theta" }}{PARA 6 "" 1 "" {TEXT -1 9 " \\rig ht) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "factor(subs(Q[1]=d iff(Q[infinity](theta),theta),V[1](theta)=1/4*diff(P[infinity](theta), `$`(theta,2)),Ww));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"$\"\"\" %\"tGF&-%$expG6#-%&gammaG6$F'%&thetaGF&,**$)-%%diffG6$-&%\"PG6#%)infin ityG6#F.-%\"$G6$F.\"\"#F>F&F&*$)-F36$F5F.\"\"%F&F&*(F>F&)FAF>F&F2F&!\" \"*(FCF&-F)6#,$*&F>F&F5F&F&F&)-F36$-&%\"QGF8F:F.F>F&FFF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "latex(%);" }}{PARA 6 "" 1 "" {TEXT -1 70 "3\\,t\{e^\{\\gamma \\left( t,\\theta \\right) \}\} \\left( \\l eft( \{\\frac \{d^\{2\}" }}{PARA 6 "" 1 "" {TEXT -1 67 "\}\{d\{\\theta \}^\{2\}\}\}P_\{\{\\infty \}\} \\left( \\theta \\right) \\right) ^\{2 \}+" }}{PARA 6 "" 1 "" {TEXT -1 64 " \\left( \{\\frac \{d\}\{d\\theta \}\}P_\{\{\\infty \}\} \\left( \\theta \\right) " }}{PARA 6 "" 1 "" {TEXT -1 66 " \\right) ^\{4\}-2\\, \\left( \{\\frac \{d\}\{d\\theta\} \}P_\{\{\\infty \}\} \\left( " }}{PARA 6 "" 1 "" {TEXT -1 70 "\\theta \+ \\right) \\right) ^\{2\}\{\\frac \{d^\{2\}\}\{d\{\\theta\}^\{2\}\}\}P _\{\{\\infty \}" }}{PARA 6 "" 1 "" {TEXT -1 62 "\} \\left( \\theta \\r ight) -4\\,\{e^\{2\\,P_\{\{\\infty \}\} \\left( \\theta" }}{PARA 6 "" 1 "" {TEXT -1 66 " \\right) \}\} \\left( \{\\frac \{d\}\{d\\theta\}\}Q _\{\{\\infty \}\} \\left( \\theta" }}{PARA 6 "" 1 "" {TEXT -1 31 " \\r ight) \\right) ^\{2\} \\right) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "dsolve (diff(P[infinity](theta),`$`(theta,2))^2+diff(P[infinity](theta),theta )^4-4*exp(2*P[infinity](theta))*diff(Q[infinity](theta),theta)^2-2*dif f(P[infinity](theta),`$`(theta,2))*diff(P[infinity](theta),theta)^2=0, P[infinity](theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-&%\"PG6#%)in finityG6#%&thetaG,$-%#lnG6#,**(\"\"#\"\"\"F*F2-&%\"QGF'F)F2!\"\"*&%$_C 1GF2F*F2F2*&F1F2-%$IntG6$*&F*F2-%%diffG6$F3F*F2F*F2F2%$_C2GF6F6/F$,$-F -6#,**(F1F2F*F2F3F2F2F7F2*&F1F2F:F2F6FAF6F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "latex(P[infinity](theta) = -ln(-2*theta*Q[infinit y](theta)+_C1*theta+2*Int(theta*diff(Q[infinity](theta),theta),theta)- _C2));" }}{PARA 6 "" 1 "" {TEXT -1 66 "P_\{\{\\infty \}\} \\left( \\th eta \\right) =-\\ln \\left( -2\\,\\theta\\,Q_\{\{" }}{PARA 6 "" 1 "" {TEXT -1 70 "\\infty \}\} \\left( \\theta \\right) +\{\\it \\_C1\}\\, \\theta+2\\,\\int \\!\\theta\\," }}{PARA 6 "" 1 "" {TEXT -1 67 "\{\\fr ac \{d\}\{d\\theta\}\}Q_\{\{\\infty \}\} \\left( \\theta \\right) \{d \\theta\}-\{" }}{PARA 6 "" 1 "" {TEXT -1 18 "\\it \\_C2\} \\right) " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 13 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }