{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 "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 "Warning" -1 7 1 {CSTYLE "" -1 -1 " Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 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 "Maple O utput" -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 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 112 "A series of calculatio ns to help with understanding Jacobi elliptic functions, their periods , and their inverses" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "The quarter period of the elliptic functions i s usually referred to by K(k), since it depends on the modulus k." }} {PARA 0 "" 0 "" {TEXT -1 87 "Maple uses the more complicated, but comp letely unambiguous to it, symbol EllipticK(k)." }}{PARA 0 "" 0 "" {TEXT -1 100 "By putting a question mark in front of the name, you are asking Maple to open up a window which will" }}{PARA 0 "" 0 "" {TEXT -1 105 "give you syntax answers, and other sorts of answers, concernin g the function to which it gives that name." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "?EllipticK" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Let's first create a plot." }} {PARA 0 "" 0 "" {TEXT -1 80 "Since it has value pi/2 for k=0, we plot \+ its ratio to that, as a function of k, " }}{PARA 0 "" 0 "" {TEXT -1 49 "and see that it goes to infinity as k goes to 1, " }}{PARA 0 "" 0 "" {TEXT -1 60 "but rises quite slowly as k increases initially away f rom 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plot(EllipticK(k )*2/evalf(Pi),k=0..1,0..4,labels=[k,\"K(k)\"],thickness=3,title=` K(k) divided by pi/2`);" }}{PARA 13 "" 1 "" {GLPLOT2D 622 622 622 {PLOTDATA 2 "6'-%'CURVESG6$7ao7$$\"\"!F)$\"+**********!#57$$\"+;arz@!# 6$\"+5)=,+\"!\"*7$$\"+XTFwSF0$\"+*y:/+\"F37$$\"+\"z_\"4iF0$\"+Nf'4+\"F 37$$\"+S&phN)F0$\"+HDv,5F37$$\"+*=)H\\5F,$\"+X(pF+\"F37$$\"+[!3uC\"F,$ \"+)[CR+\"F37$$\"+J$RDX\"F,$\"+3#Q`+\"F37$$\"+)R'ok;F,$\"+n!Qq+\"F37$$ \"+1J:w=F,$\"+)[y*35F37$$\"+3En$4#F,$\"+#QP7,\"F37$$\"+/RE&G#F,$\"+1TX 85F37$$\"+D.&4]#F,$\"+N@@;5F37$$\"+vB_5F37$$\"+v'Hi#HF,$\" +sP]A5F37$$\"+(*ev:JF,$\"++,t5F37$$\"+(RQb@&F ,$\"+]@!43\"F37$$\"+=>Y2aF,$\"+gaG)3\"F37$$\"+yXu9cF,$\"+g\\%o4\"F37$$ \"+\\y))GeF,$\"+&z%Q16F37$$\"+i_QQgF,$\"+??Z;6F37$$\"+!y%3TiF,$\"+lt,F 6F37$$\"+O![hY'F,$\"+\"fN(R6F37$$\"+#Qx$omF,$\"+](z@:\"F37$$\"+u.I%)oF ,$\"+6+nm6F37$$\"+(pe*zqF,$\"+A,-\"=\"F37$$\"+C\\'QH(F,$\"+&yM#)>\"F37 $$\"+8S8&\\(F,$\"+***=h@\"F37$$\"+0#=bq(F,$\"+i1(oB\"F37$$\"+2s?6zF,$ \"+#3#ef7F37$$\"+IXaE\")F,$\"+AS['G\"F37$$\"+l*RRL)F,$\"+.%[hJ\"F37$$ \"+`<.Y&)F,$\"+waN^8F37$$\"+8tOc()F,$\"+y2p#R\"F37$$\"+\\Qk\\*)F,$\"+l aPQ9F37$$\"+p0;r\"*F,$\"+u)yR]\"F37$$\"+mTAq#*F,$\"+<0*)R:F37$$\"+lxGp $*F,$\"+0.b\"e\"F37$$\"+A-\"\\Z*F,$\"+T2dM;F37$$\"+!oK0e*F,$\"+X?X+Dy\"F37$$\"+KRF37$$\"+i 2/P)*F,$\"+2G%f)>F37$$\"+N1?k)*F,$\"+J&*3U?F37$$\"+30O\"*)*F,$\"+Q&[66 #F37$$\"+W/%\\!**F,$\"+a4j_@F37$$\"+\"Q?&=**F,$\"+ark+AF37$$\"+<.5K**F ,$\"+AdfdAF37$$\"+a-oX**F,$\"+vj\\FBF37$$\"+B-Z_**F,$\"+;xTpBF37$$\"+ \">g#f**F,$\"+/t)yT#F37$$\"+g,0m**F,$\"+2jIvCF37$$\"+F,%G(**F,$\"+E!)p XDF37$$\"+'4I'z**F,$\"+#=1mj#F37$$\"+k+U')**F,$\"+2'p\\w#F37$$\"+K+@$* **F,$\"+o!\\[)HF37$%*undefinedGFi`l-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+ AXESLABELSG6$%\"kGQ%K(k)6\"-%&TITLEG6#%6~K(k)~divided~by~pi/2G-%*THICK NESSG6#\"\"$-%%VIEWG6$;F($\"\"\"F);F($\"\"%F)" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "To see what it does around the origin, we next ask for a power series about the origin, noting that it does begin with the qua rter period of the " }}{PARA 0 "" 0 "" {TEXT -1 37 "trigonometric func tions, namely pi/2." }}{PARA 0 "" 0 "" {TEXT -1 111 "However, it might be that you are not as familiar with the syntax for the command \"ser ies\" as you want, so you " }}{PARA 0 "" 0 "" {TEXT -1 71 "could, firs t, ask for more information about it, as is done just below:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "?series" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "series(EllipticK(k),k,8);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#+-%\"kG,$*&\"\"#!\"\"%#PiG\"\"\"F*\"\"!,$*&\"\")F(F)F *F*F',$*(\"\"*F*\"$G\"F(F)F*F*\"\"%,$*(\"#DF*\"$7&F(F)F*F*\"\"'-%\"OG6 #F*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Since the integral that defines the quarter period has a denominator that involves the square root of " }}{PARA 0 "" 0 "" {TEXT -1 103 "1-k^2*u^2 in the denominato r, in principle we could determine the series above by inserting the s eries " }}{PARA 0 "" 0 "" {TEXT -1 65 "below---for that square root--- and then integrating term by term." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "series(1/sqrt(1-k^2*u^2),k,14);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"kG\"\"\"\"\"!,$*&\"\"#!\"\"%\"uGF)F%F),$*(\"\"$F% \"\")F*F+\"\"%F%F0,$*(\"\"&F%\"#;F*F+\"\"'F%F5,$*(\"#NF%\"$G\"F*F+F/F% F/,$*(\"#jF%\"$c#F*F+\"#5F%F>,$*(\"$J#F%\"%C5F*F+\"#7F%FC-%\"OG6#F%\"# 9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "We have been told that K(k) \+ goes to infinity as k goes to 1;" }}{PARA 0 "" 0 "" {TEXT -1 84 "the c alculation just below shows us how it does that, namely it goes in the same way" }}{PARA 0 "" 0 "" {TEXT -1 45 "as the logarithm of the squa re root of 1-k^2." }}{PARA 0 "" 0 "" {TEXT -1 90 "We see this by notin g that the infinity cancels out if we divide the two and take a limit. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "limit(EllipticK(k)/log( sqrt(1-k^2)),k=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "One can also look more directly; h owever, it's not nearly as illuminating." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "series(EllipticK(k),k=1,6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+1,&%\"kG\"\"\"F&!\"\",&-%#lnG6#*&^#!\"#F&\"\"##F&F/F&* &#F&F/F&-F*6#F$F&F'\"\"!,(#F&\"\"%F&*&#F&F/F&F)F&F'*&F7F&F3F&F&F&,(#\" \"(\"#KF'*&#\"\"&\"#;F&F)F&F&*&#FBF?F&F3F&F'F/,(*&#F>F?F&F)F&F'*&#F>\" #kF&F3F&F&#\"#<\"#'*F&\"\"$,(#\"%v<\"&)G7F'*&#\"$p\"\"%C5F&F)F&F&*&#FV \"%[?F&F3F&F'F8,(*&#\"$p#FZF&F)F&F'*&#Fhn\"%'4%F&F3F&F&#\"&*y9\"'!)G7F &FB-%\"OG6#F&\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "In principl e this ought to begin with something like the power series for log(sqr t(1-k*k))), " }}{PARA 0 "" 0 "" {TEXT -1 74 "but in fact it has these \+ nasty square roots of -2, and things like that. " }}{PARA 0 "" 0 "" {TEXT -1 68 "We can eliminate that behavior, by inserting a 4 into our expression" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "series(Ellip ticK(k)/log(4/sqrt(1-k^2)),k=1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# +),&%\"kG\"\"\"F&!\"\"F&\"\"!*&,(#F&\"\"#F&*&#F&F,F&-%#lnG6#*&^#!\"#F& F,F+F&F'*&#F&\"\"%F&-F06#F$F&F&F&,&F/F&*&#F&F,F&F8F&F'F'F&-%\"OG6#F&F, " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "The one below is also somewh at interesting, since the behavior of K(k) is really rather \"nasty\" \+ near k=1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "series(Ellipti cK(k)-log(4/sqrt(1-k^2)),k=1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+' ,&%\"kG\"\"\"F&!\"\",(#F&\"\"#F&*&#F&F*F&-%#lnG6#*&^#!\"#F&F*F)F&F'*&# F&\"\"%F&-F.6#F$F&F&F&-%\"OG6#F&F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Now, let's plot some of the elliptic sines themselves:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "plot([JacobiSN(u,0.1),Jacob iSN(u,0.6),JacobiSN(u,0.9),JacobiSN(u,0.995)],u=0..16,color=[black,red ,green,blue],labels=[u,\"sn\"],title=`sn(u;k)\\n for k=0.1, 0.6, 0.9, \+ 0.995`);" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6)-%'C URVESG6$7]s7$$\"\"!F)F(7$$\"+LBxV'[t\"F-7$$\"+mYa([$F-$\"+( zEmT$F-7$$\"+\\;z/]F-$\"+slr'z%F-7$$\"+L'Q?_'F-$\"+n$>g1'F-7$$\"+\\:MG #)F-$\"+/%\\_K(F-7$$\"+mWkM**F-$\"+^m*=P)F-7$$\"+(yD_;\"!\"*$\"+l(Q3=* F-7$$\"+Fr)pL\"FL$\"+l>V@(*F-7$$\"+@'fCU\"FL$\"+s^T&))*F-7$$\"+:@$z]\" FL$\"+;3\"z(**F-7$$\"+i$o1b\"FL$\"+h08(***F-7$$\"+4YS$f\"FL$\"+amF)*** F-7$$\"+c39O;FL$\"+>rM\")**F-7$$\"+.r()y;FL$\"+V8PY**F-7$$\"+!*\\OP=FL $\"+)=l/m*F-7$$\"+wG&e*>FL$\"+2-\\M\"*F-7$$\"+.z&*f@FL$\"+ytS]$)F-7$$ \"+IH1CBFL$\"+))f$GM(F-7$$\"+$e!y$\\#FL$\"+q)3Q4'F-7$$\"+O#)\\jEFL$\"+ Zf'\\6$F-7$$\"+q\\%=+$FL$\"+K-0r9F-7$$\"+r0'e< $FL$!+LgjOE!#67$$\"+th()\\LFL$!+OlL!*>F-7$$\"+4#\\J]$FL$!+FOkiMF-7$$\" +YAUcOFL$!+3K;`[F-7$$\"+$Qr*GQFL$!+Y+=!G'F-7$$\"+?0_,SFL$!+0BJ?vF-7$$ \"+g\"yZ<%FL$!+qu2T&)F-7$$\"++e.[VFL$!+\"[osI*F-7$$\"+?(=:V%FL$!+@uty& *F-7$$\"+S;+:XFL$!+()G/%y*F-7$$\"++JucXFL$!+Z/Wh)*F-7$$\"+gX[)f%FL$!+d 1#=#**F-7$$\"+?gASYFL$!+TM3l**F-7$$\"+![n>o%FL$!+gr:\"***F-7$$\"+CF()> ZFL$!+=d!*****F-7$$\"+pzxdZFL$!+];V%***F-7$$\"+8Ko&z%FL$!+3Cuu**F-7$$ \"+e%)eL[FL$!+e['3%**F-7$$\"+Z*)R4\\FL$!+q#\\2$)*F-7$$\"+O%4_)\\FL$!+% z'ok'*F-7$$\"+W9]l^FL$!+4Q'40*F-7$$\"+`MzX`FL$!+wFpX\")F-7$$\"+KW_)\\& FL$!+\\Q^rrF-7$$\"+8aD^cFL$!+N9VIgF-7$$\"+]y\"*GeFL$!+,0'z_%F-7$$\"+*G !e1gFL$!+8%Q@)GF-7$$\"+];%Q;'FL$!+acDZ8F-7$$\"+8I5@jFL$\"+jS*=@#Fir7$$ \"+]Ok$\\'FL$\"+_PAM>F-7$$\"+*G%=mmFL$\"+3WI*e$F-7$$\"+yP[IoFL$\"+QXCm ]F-7$$\"+pKy%*pFL$\"+(f$41kF-7$$\"+]D@mrFL$\"+s/N>wF-7$$\"+L=kPtFL$\"+ y7h4')F-7$$\"+Gu1&\\(FL$\"+T[G)H*F-7$$\"+BI\\_wFL$\"+z4me(*F-7$$\"+l)FL$\"+3b=WrF-7$$\"+(>lx\"))FL$\"+%)*\\S*eF-7$$\"+E8f $)*)FL$\"+].u\"[%F-7$$\"+Vf!\\:*FL$\"+9>H$*GF-7$$\"+f0AE$*FL$\"+<]X>7F -7$$\"+!\\=Q\\*FL$!+zB%H`%Fir7$$\"+>kTh'*FL$!+zz@8@F-7$$\"+MgdB)*FL$!+ O[)Gm$F-7$$\"+\\ct&)**FL$!+sKw:^F-7$$\"+D'yl,\"!\")$!+m$R%plF-7$$\"+'o $eM5F`dl$!+w'f/\"yF-7$$\"+M?w]5F`dl$!+EDj6()F-7$$\"+\"QSp1\"F`dl$!+G)f kQ*F-7$$\"++tdv5F`dl$!+\\=%zk*F-7$$\"+?U@%3\"F`dl$!+gl6Q)*F-7$$\"+!oK& )3\"F`dl$!+N6-1**F-7$$\"+S6&G4\"F`dl$!+,Cjb**F-7$$\"++'pr4\"F`dl$!+DC' o)**F-7$$\"+g!)[,6F`dl$!+Kel****F-7$$\"+E7S06F`dl$!+:mI&***F-7$$\"+$R9 $46F`dl$!+?j!e(**F-7$$\"+gvA86F`dl$!+(H$=T**F-7$$\"+E29<6F`dl$!+%)z[\" *)*F-7$$\"+fq'\\7\"F`dl$!+V<>Z(*F-7$$\"+#R$zK6F`dl$!+i)ePa*F-7$$\"+!*e !*\\6F`dl$!+MV?**))F-7$$\"+)Q=q;\"F`dl$!+eI>'*zF-7$$\"+:*>J=\"F`dl$!+: UGLpF-7$$\"+U9A*>\"F`dl$!+C!f4p&F-7$$\"+y@0;7F`dl$!+Tv-NUF-7$$\"+8H)GB \"F`dl$!+Z;neEF-7$$\"+L!Q$\\7F`dl$!++s\"R/\"F-7$$\"+`Jzl7F`dl$\"+(etI* fFir7$$\"+R,-$G\"F`dl$\"+a%[8I#F-7$$\"+DrC+8F`dl$\"+>AnMRF-7$$\"+g(QoJ \"F`dl$\"+mFW(R&F-7$$\"+&RIML\"F`dl$\"+o\"z8r'F-7$$\"+QxR]8F`dl$\"+%o: S'yF-7$$\"+\"3ltO\"F`dl$\"+v>M\"z)F-7$$\"+E>>%Q\"F`dl$\"+U//j%*F-7$$\" +q(=5S\"F`dl$\"+Yj_p)*F-7$$\"+,V)[S\"F`dl$\"+ST9C**F-7$$\"+K)\\(39F`dl $\"+-&yS'**F-7$$\"+i`h79F`dl$\"+*ps#*)**F-7$$\"+$*3[;9F`dl$\"+y4p****F -7$$\"+CkM?9F`dl$\"+I&=`***F-7$$\"+a>@C9F`dl$\"+%fhh(**F-7$$\"+&[x!G9F `dl$\"+PtCU**F-7$$\"+;I%>V\"F`dl$\"+5Ri$*)*F-7$$\"+aVm\\9F`dl$\"+)H))[ [*F-7$$\"+\"p&Qn9F`dl$\"+0DT\"y)F-7$$\"+neB$[\"F`dl$\"+$=J'>zF-7$$\"+V g3*\\\"F`dl$\"+rZ\")foF-7$$\"+Oc)f^\"F`dl$\"+8%>5a&F-7$$\"+H_)G`\"F`dl $\"+p#pP1%F-7$$\"+'Hg!\\:F`dl$\"+#o2,a#F-7$$\"+j`Bl:F`dl$\"+uvs&\\*Fir 7$$\"+#oFL$\"+UY\"=\"**F-7$Fbp $\"+?&4q!)*F-7$Fgp$\"+_c]e%*F-7$F\\q$\"+&Q+w#*)F-7$Faq$\"+ve&)z\")F-7$ Ffq$\"+!pcdA(F-7$F[r$\"+Dz\"=2'F-7$F`r$\"+mP!*GZF-7$Fer$\"+/n-!=$F-7$F [s$\"+^HX3:F-7$F`s$!+14gT;!#77$Fes$!+W^wS:F-7$Fjs$!+]y#p>$F-7$F_t$!+QH '=t%F-7$Fdt$!+#QmX5'F-7$Fit$!+@hcysF-7$Fcu$!+Y-k3#)F-7$Fgv$!+(oR!R*)F- 7$F[x$!+;[*GV*F-7$Fex$!+'3E2x*F-7$$\"+SaNv]FL$!+Uok**)*F-7$Fjx$!+>1!f( **F-7$$\"+YWd5_FL$!+)QQW***F-7$$\"+[ukb_FL$!+9L'*****F-7$$\"+]/s+`FL$! +SV[#***F-7$F_y$!+S$*)>(**F-7$$\"+U*e@U&FL$!+^`X2**F-7$Fdy$!+p];0)*F-7 $$\"+A**)[d&FL$!+*G!ok'*F-7$Fiy$!+&=Ta[*F-7$F^z$!+Kj#[\"*)F-7$Fcz$!+7E !G7)F-7$Fhz$!+WT+LsF-7$F][l$!+En+ohF-7$Fb[l$!+oqm5[F-7$Fg[l$!+3O&QG$F- 7$F\\\\l$!+h7v8)F-7$F_cl$\"+$*fTdsF-7$Fdcl$\"+pT%4;'F-7$Ficl$\"+)QV%*)[F-7$F^dl$\" +LF\\,LF-7$Fddl$\"+,O&yd\"F-7$Fidl$!+8i^(4$Fhim7$F^el$!+o>rQ;F-7$Fhel$ !+0O:\"H$F-7$F\\gl$!+JB%*=[F-7$F`hl$!+jWOdgF-7$Fjhl$!+O11MrF-7$F_il$!+ +(=J6)F-7$Fdil$!++DG#)))F-7$Fiil$!+SV+=%*F-7$F^jl$!+sabx(*F-7$$\"+5oj2 7F`dl$!+_sN(*)*F-7$Fcjl$!+>:V'eZ*F-7$Fg[m$!+9YW@*)F-7$F\\\\m$!+mX!*e\")F-7$Fa\\m$!+[GZB sF-7$Ff\\m$!+=&yJ4'F-7$F[]m$!+DW3\\ZF-7$F`]m$!+Vj2l**F-7$$\"+U4ep:F`d l$\"++(*4))**F-7$$\"+Al#Rd\"F`dl$\"+dy,****F-7$$\"+-@Fy:F`dl$\"+/s%y** *F-7$F_cm$\"+,ge%)**F-7$$\"+T)38f\"F`dl$\"+rWp@**F-7$Fdcm$\"+%3v*4)*F- -Ficm6&F[dm$\"*++++\"F`dlF(F(-F$6$7hpF'7$F+$\"+r&\\zs\"F-7$F1$\"+&)3$ \\O$F-7$F6$\"+zyxdYF-7$F;$\"+C6u&z&F-7$F@$\"+-j?voF-7$FE$\"+kW$pu(F-7$ FP$\"+`(G!e*)F-7$FZ$\"+mM>X$*F-7$Fho$\"+mU.C'*F-7$F]p$\"+[J-/)*F-7$Fbp $\"+[#49#**F-7$Fgp$\"+zP8')**F-7$F\\q$\"+j.?)***F-7$Faq$\"+[cIc**F-7$F fq$\"+H)fa&)*F-7$F[r$\"+b@r(o*F-7$F`r$\"+[!R#R%*F-7$Fer$\"+CToz!*F-7$F [s$\"+,QN*e)F-7$F`s$\"+SY(f-)F-7$Fes$\"+])>$=tF-7$Fjs$\"+R\\$zK'F-7$F_ t$\"+T)=07&F-7$Fdt$\"+l-6)p$F-7$Fit$\"+'*\\\">5#F-7$Fcu$\"+\")3r1YFir7 $Fgv$!+[HR.7F-7$F[x$!+22XlEF-7$Fex$!+\\!H\\-%F-7$Fjx$!+`I/baF-7$F_y$!+ =ZR_mF-7$Fdy$!+;Je#[(F-7$Fiy$!+D(yc:)F-7$Fcz$!+;kc<#*F-7$Fhz$!+$4u<^*F -7$F][l$!+%etYs*F-7$Fb[l$!+^.R\"))*F-7$Fg[l$!++H_q**F-7$F\\\\l$!++;))* ***F-7$Fa\\l$!+Q!*ex**F-7$Ff\\l$!+?RB(*)*F-7$F[]l$!+n4a^(*F-7$F`]l$!+x Ir\\&*F-7$Fe]l$!+i9&)o#*F-7$F]`l$!+lE#zI)F-7$Fb`l$!+%4s:c(F-7$Fg`l$!+H G=3mF-7$F\\al$!+=jF&f&F-7$Faal$!+PX!>T%F-7$Ffal$!+v(R>'HF-7$F[bl$!+gW2 y8F-7$F`bl$\"+$Q%))pKFir7$Febl$\"+kw*\\,#F-7$Fjbl$\"+<^qoNF-7$F_cl$\"+ Ge'Q'\\F-7$Fdcl$\"+UwvGhF-7$Ficl$\"+ja\\,rF-7$F^dl$\"+=F5mzF-7$Fddl$\" +K\")GH')F-7$F^el$\"+#o0!=%*F-7$Fhel$\"+CIJx'*F-7$F\\gl$\"+=ee^)*F-7$F `hl$\"+Z`Y[**F-7$Fjhl$\"+zan%***F-7$F_il$\"+)zm6***F-7$Fdil$\"+OU\"3$* *F-7$Fiil$\"+%p6u\")*F-7$F^jl$\"+@#z.k*F-7$Fcjl$\"+,;Ot$*F-7$Fhjl$\"+g WH.!*F-7$Fb[m$\"+%=b#))yF-7$Fg[m$\"+B([X/(F-7$F\\\\m$\"+Wwr\"*fF-7$Fa \\m$\"+BC)Hx%F-7$Ff\\m$\"+(Q_qO$F-7$F[]m$\"+2\"zfx\"F-7$F`]m$\"+R2vV'* Fhim7$Fe]m$!+q/XF-7$Fb`m$!+H[2$o&F -7$Fg`m$!+^Ex?oF-7$F\\am$!+=4vLxF-7$Faam$!+`V+x$)F-7$Ffam$!+2hmy))F-7$ F[bm$!+\")f&HG*F-7$F`bm$!+qt7x&*F-7$Febm$!+wn-v(*F-7$Fjbm$!+(=Vb!**F-7 $F_cm$!+r&3>)**F-7$Fdcm$!+(pd()***F--Ficm6&F[dmF(F`^oF(-F$6$7^oF'7$F+$ \"+,iRE'yFy(F-7$Fdcm $\"+\"46hQ)F--Ficm6&F[dmF(F(F`^o-%+AXESLABELSG6$%\"uGQ#sn6\"-%&TITLEG6 #%Dsn(u;k)|+~for~k=0.1,~0.6,~0.9,~0.995G-%%VIEWG6$;F(Fdcm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "We c an see that the elliptic sine not only has a longer period as the valu e of k increases, " }}{PARA 0 "" 0 "" {TEXT -1 48 "but, also, it becom es more and more flat on top." }}{PARA 0 "" 0 "" {TEXT -1 98 "This is \+ a behavior completely consistent with the behavior of an actual pendul um, so this is good!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "The question below is a good one to see what so rt of more \"fancy\" things one can do to a plot." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 14 "?plot[options]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Now, let's look at the elliptic cosine, instead, noticing that, " }}{PARA 0 "" 0 "" {TEXT -1 100 "as k increases, certainly its period increases in exactly the same way as that of the elliptic sine ," }}{PARA 0 "" 0 "" {TEXT -1 91 "but near its maximum value it become s more and more peaked, rather than more and more flat." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "plot([JacobiCN(u,0.1),JacobiCN(u,0 .6),JacobiCN(u,0.9),JacobiCN(u,0.995)],u=0..16,color=[black,red,green, blue],labels=[u,\"cn\"],title=`cn(u;k)\\n for k=0.1, 0.6, 0.9, 0.995`) ;;" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6)-%'CURVESG 6$7_s7$$\"\"!F)$\"+**********!#57$$\"+K3VfV!#6$\"+X#*\\!***F,7$$\"+l;' )=()F0$\"+cd,i**F,7$$\"+]#HyI\"F,$\"+,eg9**F,7$$\"+LBxVS2oF,7$$\"+mW kM**F,$\"+`m7paF,7$$\"+(yD_;\"!\"*$\"+u;'Q'RF,7$$\"+Fr)pL\"F]o$\"+2s(Q M#F,7$$\"+:@$z]\"F]o$\"+ty+VmF07$$\"+.r()y;F]o$!+)ehU.\"F,7$$\"+!*\\OP =F]o$!+L>o$e#F,7$$\"+wG&e*>F]o$!+H<`pSF,7$$\"+.z&*f@F]o$!+w6)=]&F,7$$ \"+IH1CBF]o$!+g$G%)y'F,7$$\"+$e!y$\\#F]o$!+&Gw(GzF,7$$\"+O#)\\jEF]o$!+ $GoH%))F,7$$\"+.;nKGF]o$!+'psC]*F,7$$\"+q\\%=+$F]o$!+%y37*)*F,7$$\"+r) [`/$F]o$!+(*G#e%**F,7$$\"+rF&))3$F]o$!+/3i\")**F,7$$\"+smNKJF]o$!+'3K& )***F,7$$\"+r0'e<$F]o$!+mM_'***F,7$$\"+sWO>KF]o$!+)*))fv**F,7$$\"+s$oG E$F]o$!+f&*zN**F,7$$\"+sAP1LF]o$!+9P?x)*F,7$$\"+th()\\LF]o$!+]l#**z*F, 7$$\"+4#\\J]$F]o$!+&yp8Q*F,7$$\"+YAUcOF]o$!+lkQV()F,7$$\"+$Qr*GQF]o$!+ \"3))>y(F,7$$\"+?0_,SF]o$!+QZF\"f'F,7$$\"+g\"yZ<%F]o$!+:5'4?&F,7$$\"++ e.[VF]o$!+-5:dOF,7$$\"+S;+:XF]o$!+$z.q1#F,7$$\"+![n>o%F]o$!+Oc]/UF07$$ \"+e%)eL[F]o$\"+DC\"f3\"F,7$$\"+O%4_)\\F]o$\"+B_%yc#F,7$$\"+W9]l^F]o$ \"+$ok?D%F,7$$\"+`MzX`F]o$\"+3Gm+eF,7$$\"+KW_)\\&F]o$\"+/U$*zOF,7$$\"+BI\\_wF]o$\"+ Yxp$=#F,7$$\"+&z(HAyF]o$\"+\\7,:^F07$$\"+mD5#*zF]o$!+!*)>^<\"F,7$$\"+, ?[o\")F]o$!+%R08*GF,7$$\"+O9'[M)F]o$!+C>R>XF,7$$\"+_-S)\\)F]o$!+#)*Hf# eF,7$$\"+p!R>l)F]o$!++h=(*pF,7$$\"+(>lx\"))F]o$!+#)pPy!)F,7$$\"+E8f$)* )F]o$!+iKYR*)F,7$$\"+Vf!\\:*F]o$!+\"e'Hs&*F,7$$\"+f0AE$*F]o$!+wzOD**F, 7$$\"+U+7o$*F]o$!+sZtn**F,7$$\"+D&>+T*F]o$!+aug#***F,7$$\"+2!>>X*F]o$! +P1%*****F,7$$\"+!\\=Q\\*F]o$!+I4s*)**F,7$$\"+tzrN&*F]o$!+&*p'>'**F,7$ $\"+buhx&*F]o$!+s%Hn\"**F,7$$\"+Qp^>'*F]o$!+%)34a)*F,7$$\"+>kTh'*F]o$! +Ba;u(*F,7$$\"+MgdB)*F]o$!+;M,0$*F,7$$\"+\\ct&)**F]o$!+q$yBf)F,7$$\"+D 'yl,\"!\")$!+(H%RRvF,7$$\"+'o$eM5F`fl$!+U`tWiF,7$$\"+M?w]5F`fl$!+&oM*4 \\F,7$$\"+\"QSp1\"F`fl$!+(z@)[MF,7$$\"+?U@%3\"F`fl$!+y`0#z\"F,7$$\"+g! )[,6F`fl$!+s%\\kH)!#77$$\"+E29<6F`fl$\"+\\;J=\"F`fl$\"+[u@1sF,7$$\"+U9A*>\"F`fl$\"+f:rA#)F,7$$\"+y@0;7F`fl$\"+ `\"[*e!*F,7$$\"+8H)GB\"F`fl$\"+Ln4S'*F,7$$\"+t/6T7F`fl$\"+\"fWf#)*F,7$ $\"+L!Q$\\7F`fl$\"+$ei`%**F,7$$\"+8=X`7F`fl$\"+0)z)z**F,7$$\"+$flvD\"F `fl$\"+V<^(***F,7$$\"+t$z;E\"F`fl$\"+_tA)***F,7$$\"+`Jzl7F`fl$\"+w`-#) **F,7$$\"+YmSu7F`fl$\"+L$oM*)*F,7$$\"+R,-$G\"F`fl$\"+&\\(eJ(*F,7$$\"+K Oj\"H\"F`fl$\"+a#Gw\\*F,7$$\"+DrC+8F`fl$\"+_kQ$>*F,7$$\"+g(QoJ\"F`fl$ \"+,.H=%)F,7$$\"+&RIML\"F`fl$\"+N^K8uF,7$$\"+QxR]8F`fl$\"+#3cr<'F,7$$ \"+\"3ltO\"F`fl$\"+wHulZF,7$$\"+E>>%Q\"F`fl$\"+O)zFB$F,7$$\"+q(=5S\"F` fl$\"+n&3,h\"F,7$$\"+$*3[;9F`fl$\"+9mWhyF\\hl7$$\"+;I%>V\"F`fl$!+A%=ZX \"F,7$$\"+aVm\\9F`fl$!+Gd5oJF,7$$\"+\"p&Qn9F`fl$!+UW,%y%F,7$$\"+neB$[ \"F`fl$!+.$*o0hF,7$$\"+Vg3*\\\"F`fl$!+B+>wsF,7$$\"+Oc)f^\"F`fl$!+*>)[C $)F,7$$\"+H_)G`\"F`fl$!+(Gbq8*F,7$$\"+'Hg!\\:F`fl$!+0R,s'*F,7$$\"+j`Bl :F`fl$!+&[8[&**F,7$$\"+U4ep:F`fl$!+Kkm')**F,7$$\"+Al#Rd\"F`fl$!+*ek'** **F,7$$\"+-@Fy:F`fl$!+CCy$***F,7$$\"+#okGo*F,-%'COLOURG6&%$RGBGF)F)F)-F$6$7] s7$F($\"+++++5F]o7$F.$\"+^8]!***F,7$F4$\"+n$\\?'**F,7$F9$\"+e_x9**F,7$ F>$\"+,c*)[)*F,7$FC$\"+0Ffi'*F,7$FH$\"+eMV1%*F,7$FM$\"+Sw]2))F,7$FR$\" +;B')Q!)F,7$FW$\"+\"*yA6qF,7$Ffn$\"+f(GP&eF,7$F[o$\"+OaH(f%F,7$Fao$\"+ vUa\"G$F,7$Ffo$\"+fy2P>F,7$F[p$\"+Viu[dF07$F`p$!+;1YEpF07$Fep$!+_z8b>F ,7$Fjp$!+rQ-YKF,7$F_q$!+)yC`]%F,7$Fdq$!+brQ_dF,7$Fiq$!+o$=H\"pF,7$F^r$ !+Q%Hc%zF,7$Fcr$!+,F@6))F,7$Fgs$!+z\")*3[*F,7$F[u$!+@Qd&))*F,7$$\"+JW> )Q$F]o$!+\"onf$**F,7$$\"+!p7lU$F]o$!+I4\">(**F,7$$\"+\\4$[Y$F]o$!+efF$ ***F,7$F`u$!+Il)*****F,7$$\"+ouYTNF]o$!+Q'=?***F,7$$\"+FdyzNF]o$!+&z+% p**F,7$$\"+()R5=OF]o$!+iO@K**F,7$Feu$!+'>*e!))*F,7$Fju$!+CD@v%*F,7$F_v $!+)*Qi4))F,7$Fdv$!+\"z'\\?zF,7$Fiv$!+#e$HdoF,7$F^w$!+jnB6dF,7$Fcw$!+W ^e#[%F,7$Fhw$!+$=8(>LF,7$F]x$!+FO1H@F,7$Fbx$!+?.NQpF07$Fgx$\"+9f\\zuF0 7$F\\y$\"+%3nV'>F,7$Fay$\"+%**\\k;$F,7$Ffy$\"+>([0`%F,7$F[z$\"+-LnKeF, 7$Fez$\"+y\\L0pF,7$Fi[l$\"+\")Q>ryF,7$F]]l$\"+O*Row)F,7$Fg]l$\"+[$QaW* F,7$$\"+LSL[nF]o$\"+DU*)z'*F,7$F\\^l$\"+,&e?&)*F,7$$\"+^'e:(oF]o$\"+D \")*Q\"**F,7$$\"+BNj7pF]o$\"+bMAf**F,7$$\"+'R3P&pF]o$\"+)>]y)**F,7$Fa^ l$\"+v6m****F,7$$\"+)eSw.(F]o$\"+wo+%***F,7$$\"+4z\\!3(F]o$\"+[(=+(**F ,7$$\"+H_NBrF]o$\"+GQ!y#**F,7$Ff^l$\"+x$\\v')*F,7$$\"+\">F>D(F]o$\"+0` 0%p*F,7$F[_l$\"+#)Q^_%*F,7$F`_l$\"+%)y]X))F,7$Fe_l$\"+\"\\')R0)F,7$Fj_ l$\"+6$*3MqF,7$F_`l$\"+D]q%)eF,7$Fd`l$\"+pU[&f%F,7$Fi`l$\"+nXrVKF,7$F^ al$\"+4$Rl.#F,7$Fcal$\"+kQVU\")F07$Fhal$!+ISr=^F07$F]bl$!+)HKR$=F,7$Fb bl$!+-5;$=$F,7$Fgbl$!+Y*e))\\%F,7$F[dl$!+f\\:JdF,7$F_el$!+;RnzoF,7$Fde l$!+&zZ'**F,7$$\"+%z&*G1\"F`fl$!+*GAG#**F,7$F^gl$!+Uu\"[')*F,7$Fcgl$!+Hs* GW*F,7$Fhgl$!+GRHi()F,7$F^hl$!+fNlczF,7$Fchl$!+e(3v+(F,7$Fhhl$!+/f8YeF ,7$F]il$!+PT-%f%F,7$Fbil$!+8\\shLF,7$Fgil$!+WoZ(4#F,7$F\\jl$!+hw>xvF07 $Fajl$\"+b3r#)eF07$F[[m$\"+*e`'**=F,7$F_\\m$\"+'Q&*\\>$F,7$Fi\\m$\"+\" >-v^%F,7$Fc]m$\"+h]1#y&F,7$Fh]m$\"+tZI:pF,7$F]^m$\"+C2EHzF,7$Fb^m$\"+3 @N+))F,7$Fg^m$\"+7Gof%*F,7$$\"+/&ydP\"F`fl$\"+T;n&p*F,7$F\\_m$\"+r*pg' )*F,7$$\"+P')R)Q\"F`fl$\"+N4vD**F,7$$\"+[`g#R\"F`fl$\"+j*f!o**F,7$$\"+ f?\"oR\"F`fl$\"+;j\"G***F,7$Fa_m$\"+%e8*****F,7$$\"+,V)[S\"F`fl$\"+J)Q 3***F,7$$\"+K)\\(39F`fl$\"+F'fo'**F,7$$\"+i`h79F`fl$\"+&*H1G**F,7$Ff_m $\"+S*)eu)*F,7$$\"+a>@C9F`fl$\"+?LUC(*F,7$F[`m$\"+U!o%=&*F,7$F``m$\"+I X%>&))F,7$Fe`m$\"+9[L^zF,7$Fj`m$\"+8=o))pF,7$F_am$\"+Lmx9fF,7$Fdam$\"+ uR$Go%F,7$Fiam$\"+`em\"R$F,7$F^bm$\"+97F,-Ffdm6&Fhdm$\"*++++\"F`flF(F(-F$6$7dpF\\em7$ F.$\"+fS]!***F,7$F4$\"+AD4i**F,7$F9$\"+$\"+v\"y&\\)*F,7 $FC$\"+m9)fm*F,7$FH$\"+_$foT*F,7$FM$\"+&>\\\")F,7$F fn$\"+n)pLK'F,7$Fao$\"+7e^WWF,7$F[p$\"+a3C;FF,7$Fep$\"+&ya7D\"F,7$F_q$ !+DK4(*=F07$Fiq$!+g83%p\"F,7$Fcr$!+`3j,LF,7$F[u$!+:*R37&F,7$Feu$!+R5&[ \"oF,7$F_v$!+,_a*e)F,7$Fdv$!+L*p5H*F,7$Fiv$!+MBgw(*F,7$$\"+gsx*Q%F]o$! +06va)*F,7$$\"+?(=:V%F]o$!+!pSl\"**F,7$$\"+!=gKZ%F]o$!+0s_h**F,7$F^w$! +mMQ*)**F,7$$\"++JucXF]o$!+g^!*****F,7$$\"+gX[)f%F]o$!+rZ,$***F,7$$\"+ ?gASYF]o$!++Jwo**F,7$Fcw$!+7#Gt#**F,7$Fhw$!+\\XAQ'*F,7$F]x$!+Z5Ba\"*F, 7$Fgx$!+*e+jY(F,7$Fay$!+hUh'y&F,7$F[z$!+y')oxQF,7$Fi[l$!+E8RIBF,7$Fg]l $!+gJ]swF07$Fa^l$\"+1u;\"p'F07$F[_l$\"+NiF:AF,7$Fe_l$\"+p2X`PF,7$F_`l$ \"+qh#ec&F,7$Fi`l$\"+J$fa](F,7$Fcal$\"+i\\8u*)F,7$Fhal$\"+k!y7b*F,7$F] bl$\"+\"Q!f/**F,7$$\"+\")*>k-*F]o$\"+*R#Ga**F,7$$\"+N'[#p!*F]o$\"+hk*f )**F,7$$\"+!Hx?6*F]o$\"+A!)[****F,7$Fbbl$\"+)\\_Y***F,7$$\"+)fMx>*F]o$ \"+5u_r**F,7$$\"+_KcS#*F]o$\"+I;HI**F,7$$\"+1>R$G*F]o$\"+#ph7()*F,7$Fg bl$\"+Gc)[z*F,7$F[dl$\"+8'Q:M*F,7$F_el$\"+Se,\"o)F,7$Fdel$\"+AGz,zF,7$ Fiel$\"+g'30/(F,7$F^fl$\"+wB$\\/'F,7$Fdfl$\"+:]D`]F,7$Fifl$\"+iBT*=%F, 7$F^gl$\"+^6shLF,7$Fhgl$\"+*Qmkr\"F,7$Fchl$\"+OU&GE$F07$F]il$!+\\zFu6F ,7$Fgil$!+#>Zwl#F,7$Fajl$!+g,4_VF,7$F_\\m$!+(*>M\"F`fl$!+%o(\\f'*F,7$Fb^ m$!+d8.T)*F,7$$\"+u&RYN\"F`fl$!+O%4p!**F,7$$\"+59))e8F`fl$!+ps]b**F,7$ $\"+YK7j8F`fl$!+90Y')**F,7$Fg^m$!+p\\`****F,7$$\"+#zr:P\"F`fl$!+jcu%** *F,7$Ff[o$!+A.Js**F,7$$\"+:_)*z8F`fl$!+%Q'RK**F,7$F\\_m$!+:'*Hv)*F,7$F c\\o$!+lbM6(*F,7$Fa_m$!+r'4^[*F,7$Ff_m$!+^'o\"G*)F,7$F[`m$!+YE;G#)F,7$ Fe`m$!+:r[RjF,7$F_am$!+!\\45g%F,7$Fiam$!+'3hs(GF,7$Fcbm$!+s*37P\"F,7$F adm$\"+5!=id\"F0-Ffdm6&FhdmF(Fi`oF(-F$6$7eoF\\em7$F.$\"+T^]!***F,7$F4$ \"+(z4@'**F,7$F9$\"+1'z]\"**F,7$F>$\"+p0&)\\)*F,7$FC$\"+:(Htm*F,7$FH$ \"+04*4U*F,7$FM$\"+neKl))F,7$FR$\"+\"[L@>)F,7$Ffn$\"+A!4I]'F,7$Fao$\"+ 3^_*)[F,7$F[p$\"+gy+gNF,7$Fep$\"+_Kj%f#F,7$F_q$\"+c8:E=F,7$Fiq$\"+;/C? 7F,7$Fcr$\"+Fuj+vF07$F[u$\"+8B;MNF07$Feu$\"+dPITSF\\hl7$F_v$!+zF&*)3$F 07$Fiv$!+Qw/ipF07$Fcw$!+#yBi9\"F,7$F]x$!+bbtf;F,7$Fgx$!+,')GeCF,7$Fay$ !+**)HkM$F,7$F[z$!+)yr_n%F,7$Fi[l$!+q[o7hF,7$Fg]l$!+z%)3IyF,7$Fa^l$!+K &>LD*F,7$Faan$!+$z/&G&*F,7$Ff^l$!+/#flu*F,7$F^bn$!+#fg-!**F,7$F[_l$!+N (zU)**F,7$$\"+J#)*pP(F]o$!+Iqf)***F,7$$\"+IYN;uF]o$!+UdW(***F,7$$\"+H5 rbuF]o$!+vY$3)**F,7$F`_l$!+Y4*)[**F,7$$\"+E-ytvF]o$!+m()3S)*F,7$Fe_l$! +O#zUn*F,7$Fj_l$!+EhhY\"*F,7$F_`l$!+*oM7W)F,7$Fi`l$!+wQR9nF,7$Fcal$!+H KUQ_F,7$F]bl$!+b%)ewQF,7$Fgbl$!+))GboFF,7$F_el$!+z6#Q%>F,7$Fiel$!+R+)[ L\"F,7$Fdfl$!+1/^G\")F07$F^gl$!+shzHVF07$Fhgl$!+N;3svF\\hl7$Fchl$\"+]J (>R#F07$F]il$\"+sqZ5hF07$Fgil$\"+z,:C5F,7$Fajl$\"+W`Am:F,7$F_\\m$\"+G8 =cAF,7$Fc]m$\"+2iP4KF,7$F]^m$\"+<]?.WF,7$Fg^m$\"+iU,8fF,7$Fa_m$\"+;l.& e(F,7$F[`m$\"+L&pc**)F,7$F``m$\"+.FJ\"f*F,7$Fe`m$\"+w(od$**F,7$$\"+N#[ 8Z\"F`fl$\"+1'oE(**F,7$$\"+z2Jv9F`fl$\"+e')3%***F,7$$\"+BLFz9F`fl$\"+ \"4j)****F,7$Fj`m$\"+9q%**)**F,7$$\"+6%)>([\"F`fl$\"+duTk**F,7$$\"+b4; \"\\\"F`fl$\"+<=ZB**F,7$$\"+*\\B^\\\"F`fl$\"+oUUn)*F,7$F_am$\"+x(*p'z* F,7$Fdam$\"+,CPW$*F,7$Fiam$\"+loN$p)F,7$Fcbm$\"+GzTZrF,7$Fadm$\"+KeIZa F,-Ffdm6&FhdmF(F(Fi`o-%+AXESLABELSG6$%\"uGQ#cn6\"-%&TITLEG6#%Dcn(u;k)| +~for~k=0.1,~0.6,~0.9,~0.995G-%%VIEWG6$;F(Fadm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "The third ellip tic function, the elliptic \"dn\" function, is somewhat different from the other two," }}{PARA 0 "" 0 "" {TEXT -1 81 "and, in fact never bec omes negative---at least for real values of its arguments, " }}{PARA 0 "" 0 "" {TEXT -1 33 "varying only between 1-k^2 and 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "plot([JacobiDN(u,0.1),JacobiDN(u,0 .6),JacobiDN(u,0.9),JacobiDN(u,0.995)],u=0..16,-1..1,color=[black,red, green,blue],labels=[u,\"dn\"],title=`dn(u;k)\\n for k=0.1, 0.6, 0.9, 0 .995`);;" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6)-%'C URVESG6$7[x7$$\"\"!F)$\"+++++5!\"*7$$\"+LBxVvtF0$\"+!f%Rx**F07$$\"+\\:MG#)F0$\"+xU8t**F07$$\"+2I\\\"3*F0$\"+Xp #*o**F07$$\"+mWkM**F0$\"+_S*['**F07$$\"+<^Mz5F,$\"+q#G6'**F07$$\"+(yD_ ;\"F,$\"+5&**F07$$\"+:@$z]\"F,$\"+Hh4]**F07$$\"+i$o1b\" F,$\"+8K!*\\**F07$$\"+4YS$f\"F,$\"+%p\"*)\\**F07$$\"+c39O;F,$\"+i;1]** F07$$\"+.r()y;F,$\"+0>T]**F07$$\"+!*\\OP=F,$\"+M$GK&**F07$$\"+wG&e*>F, $\"+bJ>e**F07$$\"+!R0z2#F,$\"+hMXh**F07$$\"+.z&*f@F,$\"+eV2l**F07$$\"+ !)***F07$$\"+!p7lU$F,$\"+I.E'***F07$$\"+4#\\J]$F,$\"+^K+%* **F07$$\"+FdyzNF,$\"+P3I\"***F07$$\"+YAUcOF,$\"+ok@))**F07$$\"+:opUPF, $\"+bvP%)**F07$$\"+$Qr*GQF,$\"+\">g-)**F07$$\"+_fC:RF,$\"+Aj)f(**F07$$ \"+?0_,SF,$\"+dBor**F07$$\"+S$\\\")3%F,$\"+[&eu'**F07$$\"+g\"yZ<%F,$\" +M#eM'**F07$$\"+!)pShUF,$\"+e-!)f**F07$$\"++e.[VF,$\"+vJfc**F07$$\"+S; +:XF,$\"+g6-_**F07$$\"+![n>o%F,$\"+6K'*\\**F07$$\"+CF()>ZF,$\"+>`()\\* *F07$$\"+pzxdZF,$\"+@.$*\\**F07$$\"+8Ko&z%F,$\"+.z7]**F07$$\"+e%)eL[F, $\"+LpY]**F07$$\"+Z*)R4\\F,$\"+r3c^**F07$$\"+O%4_)\\F,$\"+\\t=`**F07$$ \"+SaNv]F,$\"+:Fwb**F07$$\"+W9]l^F,$\"+,e&*e**F07$$\"+[ukb_F,$\"+QTmi* *F07$$\"+`MzX`F,$\"+F'on'**F07$$\"+U*e@U&F,$\"+&)yXq**F07$$\"+KW_)\\&F ,$\"+\\:Du**F07$$\"+A**)[d&F,$\"+@=1y**F07$$\"+8aD^cF,$\"+(Q+=)**F07$$ \"+Jm3SdF,$\"+]T%f)**F07$$\"+]y\"*GeF,$\"+HNu*)**F07$$\"+p!\\x\"fF,$\" +p!zI***F07$$\"+*G!e1gF,$\"+xd%e***F07$$\"+p4@&3'F,$\"+p(\\x***F07$$\" +];%Q;'F,$\"+9C4****F07$$\"+\"*p:.iF,$\"+'GU&****F07$$\"+JBZUiF,$\"+.0 %)****F07$$\"+swy\"G'F,$\"+?_)*****F07$$\"+8I5@jF,$\"+Rb(*****F07$$\"+ ]Ok$\\'F,$\"+9#H\")***F07$$\"+*G%=mmF,$\"+qjb$***F07$$\"+LSL[nF,$\"+go c!***F07$$\"+yP[IoF,$\"+S$er)**F07$$\"+BNj7pF,$\"+UDU$)**F07$$\"+pKy%* pF,$\"+!*)f%z**F07$$\"+4z\\!3(F,$\"+a&)>v**F07$$\"+]D@mrF,$\"++0$4(**F 07$$\"+\">F>D(F,$\"+g,ym**F07$$\"+L=kPtF,$\"+n$oG'**F07$$\"+IYN;uF,$\" +wQef**F07$$\"+Gu1&\\(F,$\"+6rnc**F07$$\"+E-ytvF,$\"+H#>U&**F07$$\"+BI \\_wF,$\"+k.F_**F07$$\"+5aRPxF,$\"+(R$z]**F07$$\"+&z(HAyF,$\"+Xe+]**F0 7$$\"+))*[Z'yF,$\"+#Qy)\\**F07$$\"+\"=+s!zF,$\"+(4I*\\**F07$$\"+t8l\\z F,$\"+01;]**F07$$\"+mD5#*zF,$\"+(Go0&**F07$$\"+$G#H!3)F,$\"+wu'>&**F07 $$\"+,?[o\")F,$\"+uV2a**F07$$\"+=l)F,$\"+kwWu**F07$$\"+K@&[t)F,$\"+b1 ey**F07$$\"+(>lx\"))F,$\"+w\\h#)**F07$$\"+h#y1!*)F,$\"+D.W')**F07$$\"+ E8f$)*)F,$\"+d>&**)**F07$$\"+N'[#p!*F,$\"+w#[J***F07$$\"+Vf!\\:*F,$\"+ jN\"e***F07$$\"+_KcS#*F,$\"+1(py***F07$$\"+f0AE$*F,$\"+VkD****F07$$\"+ U+7o$*F,$\"+kyn****F07$$\"+D&>+T*F,$\"+,h#*****F07$$\"+2!>>X*F,$\"+8%* ******F07$$\"+!\\=Q\\*F,$\"+hs*)****F07$$\"+buhx&*F,$\"+I2<****F07$$\" +>kTh'*F,$\"+1pw(***F07$$\"+Fi\\U(*F,$\"+M4!e***F07$$\"+MgdB)*F,$\"+'Q *G$***F07$$\"+Tel/**F,$\"+<#)H!***F07$$\"+\\ct&)**F,$\"+4f!p)**F07$$\" +&4wv+\"!\")$\"+rwx#)**F07$$\"+D'yl,\"Fgbm$\"+2zRy**F07$$\"+c6eD5Fgbm$ \"+Yx!R(**F07$$\"+'o$eM5Fgbm$\"+,'**F07$$\"+37&)e5Fgbm$\"+&z/(e**F07$$\"+\"QSp1\"F gbm$\"+F(\\e&**F07$$\"+?U@%3\"Fgbm$\"+l!)[^**F07$$\"+g!)[,6Fgbm$\"+Gy( )\\**F07$$\"+E29<6Fgbm$\"+#)*e4&**F07$$\"+#R$zK6Fgbm$\"+bTNa**F07$$\"+ T'\\89\"Fgbm$\"+u%zq&**F07$$\"+!*e!*\\6Fgbm$\"+xLKg**F07$$\"+R@Ye6Fgbm $\"+B?*R'**F07$$\"+)Q=q;\"Fgbm$\"+#=zz'**F07$$\"+_\"p]<\"Fgbm$\"+r#=>( **F07$$\"+:*>J=\"Fgbm$\"+He$f(**F07$$\"+y1<\">\"Fgbm$\"+!\\G*z**F07$$ \"+U9A*>\"Fgbm$\"+nLz$)**F07$$\"+5oj27Fgbm$\"+x&)e()**F07$$\"+y@0;7Fgb m$\"+_#G5***F07$$\"+YvYC7Fgbm$\"+S_,%***F07$$\"+8H)GB\"Fgbm$\"+9^Y'*** F07$$\"+t/6T7Fgbm$\"+XWF)***F07$$\"+L!Q$\\7Fgbm$\"+4^X****F07$$\"+8=X` 7Fgbm$\"+-!*z****F07$$\"+$flvD\"Fgbm$\"+@^(*****F07$$\"+t$z;E\"Fgbm$\" +yA)*****F07$$\"+`Jzl7Fgbm$\"+M\"Fgbm$\"+fp@t**F07$$\"+QxR]8Fgbm$\"+y1.p**F07$$\"+59))e8Fgbm$ \"+AR,l**F07$$\"+\"3ltO\"Fgbm$\"+(>\"Gh**F07$$\"+/&ydP\"Fgbm$\"+=Y'z&* *F07$$\"+E>>%Q\"Fgbm$\"+[Z7b**F07$$\"+[`g#R\"Fgbm$\"+h4%G&**F07$$\"+q( =5S\"Fgbm$\"+Xq<^**F07$$\"+,V)[S\"Fgbm$\"+HQj]**F07$$\"+K)\\(39Fgbm$\" +[ZB]**F07$$\"+(f#o59Fgbm$\"+;+4]**F07$$\"+i`h79Fgbm$\"+M@)*\\**F07$$ \"+G\"[XT\"Fgbm$\"+_7\"*\\**F07$$\"+$*3[;9Fgbm$\"+$[x)\\**F07$$\"+CkM? 9Fgbm$\"+49#*\\**F07$$\"+a>@C9Fgbm$\"+vO6]**F07$$\"+&[x!G9Fgbm$\"+>JX] **F07$$\"+;I%>V\"Fgbm$\"+Yx$4&**F07$$\"+&o.3W\"Fgbm$\"+m&zD&**F07$$\"+ aVm\\9Fgbm$\"+Fo\"\\&**F07$$\"+A]_e9Fgbm$\"+Zr(y&**F07$$\"+\"p&Qn9Fgbm $\"+x(o8'**F07$$\"+z2Jv9Fgbm$\"+FO&['**F07$$\"+neB$[\"Fgbm$\"+&R!fo**F 07$$\"+b4;\"\\\"Fgbm$\"+?h[s**F07$$\"+Vg3*\\\"Fgbm$\"+JPWw**F07$$\"+Se `2:Fgbm$\"+F/i!)**F07$$\"+Oc)f^\"Fgbm$\"+cnj%)**F07$$\"+KaVC:Fgbm$\"+' ey$))**F07$$\"+H_)G`\"Fgbm$\"+#[R<***F07$$\"+iF(4a\"Fgbm$\"+-5^%***F07 $$\"+'Hg!\\:Fgbm$\"+6Mx'***F07$$\"+Iy9d:Fgbm$\"+bvY)***F07$$\"+j`Bl:Fg bm$\"+[\"\\&****F07$$\"+_\"3uc\"Fgbm$\"++8t****F07$$\"+U4ep:Fgbm$\"+cn ')****F07$$\"+KPvr:Fgbm$\"+f_&*****F07$$\"+Al#Rd\"Fgbm$\"+`m******F07$ $\"+7$*4w:Fgbm$\"+Q3******F07$$\"+-@Fy:Fgbm$\"+Uy$*****F07$$\"+#*[W!e \"Fgbm$\"+ix$)****F07$$\"+#o$\"+gT!ye*F07$FC$\" +3I%)o%*F07$FH$\"+I6aT$*F07$FM$\"+e!\\<>*F07$FR$\"+H!>'Q!*F07$FW$\"+C> (e)))F07$Ffn$\"+:n.P()F07$F[o$\"+)3rWf)F07$F`o$\"+V\"*=i%)F07$Feo$\"+w r%GM)F07$Fjo$\"+!**H(Q#)F07$Fdp$\"+U_)R3)F07$Fhq$\"+mBV2!)F07$$\"+u!* \\=F,$\"+l8TR!)F07$Fbr$\"+p-b&3)F07$Fgr$ \"+v-3^\")F07$F\\s$\"+0FmL#)F07$Fas$\"+[*p>L)F07$Ffs$\"+kFOW%)F07$F[t$ \"+;]Lt&)F07$F`t$\"+k8x7()F07$Fet$\"+\"G?)f))F07$Fjt$\"+VsJ6!*F07$F_u$ \"+:NMj\"*F07$Fdu$\"+#*4y7$*F07$Fiu$\"+Jp\"fX*F07$F^v$\"+4'H!*e*F07$Fh v$\"+``p6(*F07$Fbw$\"+-iG;)*F07$Fgw$\"+\"G%e**)*F07$F\\x$\"+J!e*e**F07 $$\"+JW>)Q$F,$\"+-d*p(**F07$Fax$\"+Hq*)*)**F07$$\"+\\4$[Y$F,$\"+g)zv** *F07$Ffx$\"+\\^******F07$$\"+ouYTNF,$\"+Tu7(***F07$F[y$\"+x]**))**F07$ $\"+()R5=OF,$\"+\")*\\c(**F07$F`y$\"+$3xr&**F07$Fey$\"++@Z(*)*F07$Fjy$ \"+*p4V\")*F07$F_z$\"+SX*F07$F^ [l$\"+'\\q]I*F07$Fc[l$\"+3![<:*F07$Fh[l$\"+uT+'**)F07$$\"+?(=:V%F,$\"+ '4Ds%))F07$F]\\l$\"+F--.()F07$$\"+gX[)f%F,$\"+V9Qm&)F07$Fb\\l$\"+O]+S% )F07$Ff]l$\"+WIBW#)F07$F`^l$\"+Y&[85)F07$Fe^l$\"+^o![/)F07$Fj^l$\"+RV# 3,)F07$$\"+Xz.)=&F,$\"+4]$f+)F07$$\"+YWd5_F,$\"+U;]-!)F07$$\"+Z46L_F,$ \"+p!G0+)F07$F__l$\"+4l,+!)F07$$\"+\\R=y_F,$\"+Hv'4+)F07$$\"+]/s+`F,$ \"+o+Q.!)F07$$\"+^pDB`F,$\"+@9D2!)F07$Fd_l$\"+bsd7!)F07$F^`l$\"+BbN'3) F07$Fh`l$\"+\")**\\A#)F07$F]al$\"+2TcF$)F07$Fbal$\"+=(>#\\%)F07$Fgal$ \"+\"[))\\e)F07$F\\bl$\"+ue'>t)F07$Fabl$\"+,*o(o))F07$Ffbl$\"+\\QA4!*F 07$F`cl$\"+R^f]\"*F07$Fjcl$\"+#G(***G*F07$$\"+JLP2kF,$\"+<89P%*F07$F_d l$\"+OpPu&*F07$$\"+pR\"*zlF,$\"+0R&yp*F07$Fddl$\"+[@(R!)*F07$F^el$\"+V X*p%**F07$Fhel$\"+?!y)****F07$Fbfl$\"+S3__**F07$F\\gl$\"+1oU1)*F07$Fag l$\"+3iz5(*F07$Ffgl$\"+a(*Q+'*F07$F[hl$\"+%)R!zZ*F07$F`hl$\"+#zHiM*F07 $Fehl$\"+\"RUt>*F07$Fjhl$\"+Y7,X!*F07$Fdil$\"+?$4H*))F07$F^jl$\"+RY_W( )F07$Fcjl$\"+)p%z(f)F07$Fhjl$\"+x\\$=Y)F07$F][m$\"+df_R$)F07$Fb[m$\"+k ]LL#)F07$F\\\\m$\"+j0y#4)F07$Ff\\m$\"+YM!\\,)F07$$\"+Mtms')F,$\"+^,Y4! )F07$$\"++cR$p)F,$\"+:_C0!)F07$$\"+mQ79()F,$\"+^EE-!)F07$F[]m$\"+k_^+! )F07$$\"+k'3jx)F,$\"+*\\J2+)F07$F`]m$\"+4J*e+)F07$Fe]m$\"+Xa&4.)F07$Fj ]m$\"+b)>`2)F07$F_^m$\"+18pS\")F07$Fd^m$\"+$\\A[A)F07$Fi^m$\"+\">XiK)F 07$F^_m$\"+i;7V%)F07$Fh_m$\"+^^Hq&)F07$Fb`m$\"+IFt2()F07$Fg`m$\"+!\\(p _))F07$F\\am$\"+cN:-!*F07$Faam$\"+Il$z9*F07$Ffam$\"+WUo\"H*F07$F[bm$\" +YO8I%*F07$F`bm$\"+'=++c*F07$Febm$\"+!oY-p*F07$F[cm$\"+U#R=!)*F07$F`cm $\"+K?7\"*)*F07$Fecm$\"+Uf3b**F07$$\"+t#G'Q5Fgbm$\"+-@'\\(**F07$Fjcm$ \"+C/8*)**F07$$\"+ZurY5Fgbm$\"+^/\\(***F07$F_dm$\"+IF)*****F07$$\"+GVy _5Fgbm$\"+o7-****F07$$\"+@m![0\"Fgbm$\"+*\\*e'***F07$$\"+9*Go0\"Fgbm$ \"+X()F07$F`]n$\"+^Z#Q'))F07$Fe]n$\"+\"Rw>,*F07$Fj]n$\"+!H &3h\"*F07$F_^n$\"+_rv2$*F07$Fd^n$\"+!=\"e^%*F07$Fi^n$\"+\"[Qae*F07$F^_ n$\"+WJo0(*F07$Fc_n$\"+-^\"*3)*F07$F]`n$\"+&**F07$Fg`n$\"+)))o**** *F07$Febn$\"+>S.b**F07$Ficn$\"+:kOH)*F07$F^dn$\"+Ed`D(*F07$Fcdn$\"+$GD Dg*F07$Fhdn$\"+C&[TY*F07$F]en$\"+fQ`9$*F07$Fben$\"+%>*ou\"*F07$Fgen$\" +b4LK!*F07$F\\fn$\"+,NV!*))F07$Fafn$\"+2b#=v)F07$Fffn$\"+D[i5')F07$F[g n$\"+2v0z%)F07$F`gn$\"+Wppf$)F07$Fegn$\"+4(pZD)F07$F_hn$\"+$\\(p+\")F0 7$Fihn$\"+iGn:!)F07$F^in$\"+po$)4!)F07$Fcin$\"+p,N0!)F07$Fhin$\"+Lu@-! )F07$F]jn$\"+K>W+!)F07$Fgjn$\"+!fo4+)F07$Fa[o$\"+az#p+)F07$Ff[o$\"+fG- N!)F07$F[\\o$\"+j]D%3)F0-F`\\o6&Fb\\o$\"*++++\"FgbmF(F(-F$6$7cuF'7$F.$ \"+!eM$y)*F07$F4$\"+8@SI&*F07$F>$\"+3\"R*y!*F07$FH$\"+M%*zJ&)F07$FM$\" +tcR)>)F07$FR$\"+@LtbyF07$FW$\"+FnZ5vF07$Ffn$\"+!HK&orF07$F[o$\"+k3\"G $oF07$F`o$\"+be$*4lF07$Feo$\"+>-W.iF07$Fjo$\"+7K7;fF07$Fdp$\"+vJI4aF07 $Fhq$\"+YXh(*\\F07$F]r$\"+J4r0ZF07$Fbr$\"+G\"=?]%F07$Fgr$\"+CGSJWF07$F \\s$\"+JEd%Q%F07$$\"+gT)4?#F,$\"+g\\2qVF07$Fas$\"+A4_hVF07$$\"+tm.$G#F ,$\"+q)4*eVF07$Ffs$\"+z:CiVF07$$\"+VB\\mBF,$\"+`'Q>P%F07$F[t$\"+DI*zQ% F07$$\"+q6N^CF,$\"+'z0/T%F07$F`t$\"+'3y\"RWF07$Fet$\"+RL!e^%F07$Fjt$\" +D(fyh%F07$Fdu$\"+c[%p*[F07$F^v$\"+G(y`F&F07$Fhv$\"+J#=v]&F07$Fbw$\"+U ]%Rw&F07$Fgw$\"+*ygK/'F07$F\\x$\"+f.bVjF07$Fax$\"+)[`\")>#) F07$F_z$\"+K@0c&)F07$Fdz$\"+%e(zu))F07$F^[l$\"+F(o)H%*F07$Fh[l$\"+*)*Q %>)*F07$Figo$\"+A>XK**F07$F]\\l$\"+t9S\"***F07$$\"++JucXF,$\"+uJ#***** F07$Faho$\"+TBM%***F07$$\"+?gASYF,$\"+Rcqu**F07$Fb\\l$\"+HnzO)F07$Fd_l$\"+quh4!)F07$Fi_l$\"+_k)4q(F07$F^`l$\"+kt[#R(F 07$Fc`l$\"+nh7)3(F07$Fh`l$\"+w[P\"z'F07$F]al$\"+kfjfkF07$Fbal$\"+=r%f9 'F07$Fgal$\"+3JF`eF07$F\\bl$\"+yU'Qe&F07$Ffbl$\"+@,xo^F07$Fjcl$\"+E&Rs $[F07$F_dl$\"+7UvsXF07$Fddl$\"+>cD8WF07$$\"+g\"fsq'F,$\"+UQy!R%F07$Fid l$\"+26FuVF07$$\"+0*3%*y'F,$\"+1krjVF07$F^el$\"+c*=\"fVF07$$\"+^'e:(oF ,$\"+!Qy/O%F07$Fcel$\"+&z%znVF07$$\"+'R3P&pF,$\"+t(o5Q%F07$Fhel$\"+P7I +WF07$Fbfl$\"+Ha\"\\a%F07$F\\gl$\"+\")yA$z%F07$Ffgl$\"+T;z6^F07$F`hl$ \"+o6n9bF07$Fehl$\"+[2HldF07$Fjhl$\"+h!*pPgF07$Fdil$\"+n47IjF07$F^jl$ \"+Z*>-k'F07$Fcjl$\"+,*Hy(pF07$Fhjl$\"+wa4FtF07$F][m$\"+Hs!Ho(F07$Fb[m $\"+gt?R!)F07$Fg[m$\"+BuNW$)F07$F\\\\m$\"+**e^R')F07$Fa\\m$\"+)3Q#>*)F 07$Ff\\m$\"+&=xy<*F07$F`]m$\"+KH9Q'*F07$Fj]m$\"++))yA**F07$$\"+\")*>k- *F,$\"+))[)H'**F07$F_^m$\"+S'e'))**F07$$\"+!Hx?6*F,$\"+-`e****F07$Fd^m $\"+V(oc***F07$Fi^m$\"+EQdV**F07$F^_m$\"+nl=M)*F07$Fb`m$\"+<*p,Z*F07$F \\am$\"+Y()fY*)F07$Faam$\"+mUj_')F07$Ffam$\"+6e;T$)F07$F[bm$\"+^4h=!)F 07$F`bm$\"+a!\\4p(F07$Febm$\"+$H(pFtF07$F[cm$\"+)o`7(pF07$F`cm$\"+qZ/F mF07$Fecm$\"+Gq\\*H'F07$F_dm$\"+F;PjdF07$Fidm$\"+0$HgI&F07$F^em$\"+^;d 8\\F07$Fcem$\"+y0bCYF07$Fhem$\"+=JR`WF07$F]fm$\"+2&z(oVF07$$\"+;:2P6Fg bm$\"+F,rgVF07$Fbfm$\"+zH5fVF07$$\"+mxiX6Fgbm$\"+Kz&RO%F07$Fgfm$\"+%Rv _P%F07$F\\gm$\"+[&[%F07$F[hm$\"+?n/$o%F07$Fehm$\"+d >.s\\F07$F_im$\"+u5op`F07$Fiim$\"+\"=,-'eF07$F^jm$\"+d[.JhF07$Fcjm$\"+ C=,?kF07$F][n$\"+&RI[s'F07$Fg[n$\"+WWd***F07$Fh_n$\"+msdx**F07$$ \"+:_)*z8Fgbm$\"+HkFX**F07$F]`n$\"+jN6**)*F07$Fb`n$\"+*QYow*F07$Fg`n$ \"+@u1&e*F07$Febn$\"+F@\\T\"*F07$Ficn$\"+`7(Hf)F07$F^dn$\"+m!)3\\#)F07 $Fcdn$\"+@\\2%*yF07$Fhdn$\"+^t\\NvF07$F]en$\"+8N0!=(F07$Fben$\"+)*=^po F07$Fgen$\"+*3V&plF07$F\\fn$\"+oR,$G'F07$Fafn$\"+)zZA,'F07$F[gn$\"+*y? a\\&F07$Fegn$\"+Z!y+2&F07$F_hn$\"+c6KaZF07$Fihn$\"+V)G-`%F07$F]jn$\"+r 3&zW%F07$Fa[o$\"+=)eBR%F07$$\"+iK'pe\"Fgbm$\"+(RoXP%F07$Ff[o$\"+pnWjVF 07$$\"+?Wl&f\"Fgbm$\"+6I**eVF07$F[\\o$\"+Kn?hVF0-F`\\o6&Fb\\oF(FhfqF(- F$6$7aqF'7$$\"+K3VfV!#6$\"+&*)*f!***F07$$\"+l;')=()F[is$\"+j%)[i**F07$ $\"+]#HyI\"F0$\"+z-$f\"**F07$F.$\"+0'f8&)*F07$$\"++&ech#F0$\"+-Yqq'*F0 7$F4$\"+=@%pU*F07$F>$\"+*Qgt())F07$FH$\"+vB87#)F07$FR$\"+vR:*Q(F07$Ffn $\"+847ZlF07$F`o$\"+9?`FdF07$Fjo$\"+0f`m\\F07$Fdp$\"+$=_MG%F07$Fhq$\"+ 7uJ!o$F07$F]r$\"+2V&>>$F07$Fbr$\"+tk6oFF07$F\\s$\"+m'>>R#F07$Ffs$\"++% =M2#F07$Fjt$\"+NJ9s:F07$F^v$\"+q$*yY7F07$Fbw$\"+cW&e8\"F07$F\\x$\"+wq% )e5F07$Ffx$\"+N)et,\"F07$F`y$\"+oOe&***F[is7$Fjy$\"+F!)Q25F07$Fdz$\"+j 5(\\/\"F07$F^[l$\"+B]r86F07$Fh[l$\"+L&pa@\"F07$Fb\\l$\"+(G*)f^\"F07$F` ^l$\"+B&f*H>F07$Fd_l$\"+f]/UEF07$F^`l$\"+L])*HIF07$Fh`l$\"+/0EwMF07$Fb al$\"+I6vtSF07$F\\bl$\"+k>!zv%F07$Ffbl$\"+!GEHV&F07$Fjcl$\"+5'yN;'F07$ F_dl$\"+\"*3)p+(F07$Fddl$\"+_QpayF07$F^el$\"+V968')F07$Fhel$\"+N\\1h#* F07$Fbfl$\"+5)>\"\\(*F07$F\\gl$\"+mmV%)**F07$$\"+J#)*pP(F,$\"+K5h)***F 07$Fagl$\"+<7Z(***F07$$\"+H5rbuF,$\"+Hg-\")**F07$Ffgl$\"+c?S\\**F07$F[ hl$\"+4npT)*F07$F`hl$\"+JCex'*F07$Fjhl$\"+/9_b\"*F07$F^jl$\"+Q>?e%)F07 $Fhjl$\"+0]`DwF07$Fb[m$\"+&Qj]v'F07$F\\\\m$\"+/P76gF07$Ff\\m$\"+?p02`F 07$F`]m$\"+?k&*3YF07$Fj]m$\"+>9T%)RF07$Fd^m$\"+jTY=MF07$F^_m$\"+)Rv,$H F07$Fb`m$\"++))>ADF07$F\\am$\"+fBvw@F07$F`bm$\"+S`\"=m\"F07$Fecm$\"+S* f^G\"F07$F_dm$\"+Q+\"F07$F]fm$\"+harE5F07$Fgfm$\"+9dJ#3\"F07$Fagm$ \"+rRDp6F07$F[hm$\"+E2z\"G\"F07$Fehm$\"+Iq&oU\"F07$Fiim$\"+-2(4&=F07$F cjm$\"+Ry]F@F07$Fg[n$\"+$eYqX#F07$Fa\\n$\"+md`lGF07$F[]n$\"+=4(eM$F07$ Fe]n$\"+@`.#)QF07$F_^n$\"+/ie$\\%F07$Fi^n$\"+!Q3f>&F07$Fc_n$\"+:)=w'fF 07$F]`n$\"+%p:By'F07$Fg`n$\"+'R4Hh(F07$Febn$\"+U\"H\"[$)F07$Ficn$\"+k2 C1!*F07$Fcdn$\"+/`Z&f*F07$F]en$\"+E:TO**F07$$\"+N#[8Z\"Fgbm$\"+.;%H(** F07$Fben$\"+Uw9%***F07$$\"+BLFz9Fgbm$\"+aW')****F07$Fgen$\"+Qt/!***F07 $F\\fn$\"+*4QU#**F07$Fafn$\"+A&[()z*F07$F[gn$\"+e#R6N*F07$Fegn$\"+i)et q)F07$F_hn$\"+SG4szF07$Fihn$\"+4&p9=(F07$Fa[o$\"+9)GrK'F07$F[\\o$\"+$G ?8^&F0-F`\\o6&Fb\\oF(F(Fhfq-%+AXESLABELSG6$%\"uGQ#dn6\"-%&TITLEG6#%Ddn (u;k)|+~for~k=0.1,~0.6,~0.9,~0.995G-%%VIEWG6$;F(F[\\o;$!\"\"F)$\"\"\"F )" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "A s discussed slightly above, the elliptic sine and elliptic cosine do n ot share the \"intimate\"" }}{PARA 0 "" 0 "" {TEXT -1 44 "relationship that the trigonometric ones do:" }}{PARA 0 "" 0 "" {TEXT -1 93 "for t he trigonometric sine and cosine, one simply shifts one of them over b y a quarter period" }}{PARA 0 "" 0 "" {TEXT -1 46 " \+ and they are identical;" }}{PARA 0 "" 0 "" {TEXT -1 80 "this is no t true for the more general elliptic ones, when k is different from 0: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "plot([JacobiSN(u,0.6),JacobiCN(u-EllipticK(0.6),0.6) ],u=0..3.5,color=[red,green],title=`comparison of sn(u;k=0.6) and cn(u ;k=0.6)\\n with cn moved over, for a quarter period\\n sn is the \"fat ter\" one`);" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6' -%'CURVESG6$7S7$$\"\"!F)F(7$$\"+fR+Hw!#6$\"+]F&*=wF-7$$\"+^fpE9!#5$\"+ >Q9?9F37$$\"+xM?t@F3$\"+Xa=]@F37$$\"+S$fY#HF3$\"+X;.pGF37$$\"+kOasOF3$ \"+(*fejNF37$$\"+o\"GfO%F3$\"+lk1&=%F37$$\"+gw)Q3&F3$\"+0]T-[F37$$\"+$ R-k#eF3$\"+=L!)4aF37$$\"+se`mlF3$\"+V6=\")fF37$$\"+IT&yK(F3$\"+?'e7`'F 37$$\"+lOU)*zF3$\"+TF$G)pF37$$\"+QhK`()F3$\"+VeH`uF37$$\"+8$G8^*F3$\"+ whp%)yF37$$\"+'Q!=C5!\"*$\"+;oMh#)F37$$\"+kX^!4\"Fjo$\"+fw@q&)F37$$\"+ =BRp6Fjo$\"+dyt'*))F37$$\"+A@@O7Fjo$\"+]s5R\"*F37$$\"+Q%RRJ\"Fjo$\"+xj 7#Q*F37$$\"+%GTFQ\"Fjo$\"+g.(Gc*F37$$\"+9yAe9Fjo$\"+g>lC(*F37$$\"+S)3, `\"Fjo$\"+*z7P%)*F37$$\"+^\"4^g\"Fjo$\"+$4D?$**F37$$\"+\\G)Rn\"Fjo$\"+ ?D8\")**F37$$\"+PCF[Fjo$\"+?4X_)*F37$$\"+[26S?Fjo$\"++ nhI(*F37$$\"+U[V8@Fjo$\"+rTav&*F37$$\"+u'zV=#Fjo$\"+&[67R*F37$$\"+8=:j AFjo$\"+]zOY\"*F37$$\"+%3KRL#Fjo$\"+t:r*)))F37$$\"+J^]4CFjo$\"+'R&yw&) F37$$\"+Wb)zZ#Fjo$\"+gN7e#)F37$$\"+CF&Gb#Fjo$\"+*HY7(yF37$$\"+0pHBEFjo $\"+8(G/Z(F37$$\"+s8$pp#Fjo$\"+'HAN,(F37$$\"+BD#*oFFjo$\"+>G')HlF37$$ \"+'e!HWGFjo$\"+5ZR&)fF37$$\"+))*yo\"HFjo$\"+K&3cU&F37$$\"+966\"*HFjo$ \"+?qE>[F37$$\"+g&GZ1$Fjo$\"+]Nq'=%F37$$\"+[`PKJFjo$\"+9$=2e$F37$$\"++ i!*4KFjo$\"+Q?(3'GF37$$\"+=2DzKFjo$\"+(Gez>#F37$$\"+Rk=`LFjo$\"+zB&eZ \"F37$$\"+dB&RU$Fjo$\"+Yd'\\u(F-7$$\"#N!\"\"$\"+[[g2:!#7-%'COLOURG6&%$ RGBG$\"*++++\"!\")F(F(-F$6$7S7$F($\"+ev!)G=!#>7$F+$\"+w3a,hF-7$F1$\"+H ?ES6F37$F7$\"+GXkM(e*e(*F37$Far$\"+_JR%*)*F37$Ffr$\"+pjcq** F37$F[s$\"+7#p*****F37$F`s$\"+9E9s**F37$Fes$\"+68z**)*F37$Fjs$\"+))=Fs (*F37$F_t$\"+X*Q$)e*F37$Fdt$\"+5VKf$*F37$Fit$\"+197%4*F37$F^u$\"+cp'Gv )F37$Fcu$\"+x$)f2%)F37$Fhu$\"+n`N-!)F37$F]v$\"+nQy0wF37$Fbv$\"+cg*Q9(F 37$Fgv$\"+tDa&o'F37$F\\w$\"+Ts6&='F37$Faw$\"+H)3yn&F37$Ffw$\"+_mdI^F37 $F[x$\"+Ed`!f%F37$F`x$\"+['eu-%F37$Fex$\"+[#>.Y$F37$Fjx$\"+4f1LHF37$F_ y$\"+:#zJK#F37$Fdy$\"+'[fQx\"F37$Fiy$\"+R$Q`=\"F37$F^z$\"+`Ln-iF-7$Fcz $\"+)=%317Fhz-Fjz6&F\\[lF(F][lF(-%+AXESLABELSG6$Q\"u6\"Q!F]el-%&TITLEG 6#%fqcomparison~of~sn(u;k=0.6)~and~cn(u;k=0.6)|+~with~cn~moved~over,~f or~a~quarter~period|+~sn~is~the~\"fatter\"~oneG-%%VIEWG6$;F(Fcz%(DEFAU LTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "We may also ask fo r the values, of each, at some arbitrary point on the graph above." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "evalf([JacobiSN(1,0.6),Jaco biCN(1-EllipticK(0.6),0.6)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\" +KT\"49)!#5$\"+!o&ojuF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "and fo r some other value of k:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "evalf([JacobiSN(1,0.9),JacobiCN(1-EllipticK(0.9),0.9)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+)G@kx(!#5$\"+QJoXZF&" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 192 "plot(\{JacobiSN(u,0.9),JacobiCN(u-EllipticK (0.9),0.9)\},u=0..2*EllipticK(0.9),title=`comparison of sn(u;k=0.9) an d cn(u;k=0.9)\\n with cn moved over, for a half period\\n sn is the \" fatter\" one`);" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6'-%'CURVESG6$7W7$$\"\"!F)$!+QK%pH#!#>7$$\"+Hi*=%**!#6$\"+TI*zL%F07$$ \"+)pG#f=!#5$\"+$)*=I8)F07$$\"+4c0KGF6$\"+(=(fW7F67$$\"+\\5L6QF6$\"+s \"=eo\"F67$$\"+k@&fy%F6$\"+W'HT8#F67$$\"+(p]&*o&F6$\"+78gfDF67$$\"+SY< DmF6$\"+91X6IF67$$\"+g#)z#f(F6$\"+!zL=\\$F67$$\"+)p=tb)F6$\"+(4gV)RF67 $$\"+ClW\\&*F6$\"+#4E\\]%F67$$\"+D8LU5!\"*$\"+]R7u\\F67$$\"+A!329\"F^o $\"+@n37bF67$$\"+i')[R7F^o$\"+@'y*egF67$$\"+8@oM8F^o$\"+)>jte'F67$$\"+ &)o7@9F^o$\"+&o#)H1(F67$$\"+\"3=R_\"F^o$\"+%Gh\\h(F67$$\"+3d*4h\"F^o$ \"+Yr&G1)F67$$\"+)o(G7!=F^o$\"+&QQ,%*)F67 $$\"+0,K+>F^o$\"+Gl<7$*F67$$\"+qL*R*>F^o$\"+(>*)4g*F67$$\"+&fJ<4#F^o$ \"+Yf&R#)*F67$$\"+'p3m8#F^o$\"+L$er*)*F67$$\"+'z&[\"=#F^o$\"++=5^**F67 $$\"+7=*)HAF^o$\"+%z!=()**F67$$\"+FyHyAF^o$\"+`Y(*****F67$$\"+u!y&GBF^ o$\"+\"\\v%))**F67$$\"+?$e)yBF^o$\"+d4%=&**F67$$\"+AuiACF^o$\"+YLy**)* F67$$\"+ClRmCF^o$\"+jYQH)*F67$$\"+#=S4c#F^o$\"+iZ^<'*F67$$\"+KIheEF^o$ \"+BRh>$*F67$$\"+ho;aFF^o$\"+@T\\g*)F67$$\"+/,iYGF^o$\"+*=M*f&)F67$$\" +nOF\\HF^o$\"+Il+n!)F67$$\"+fC^TIF^o$\"+o68#f(F67$$\"+dq**RJF^o$\"+OE% G1(F67$$\"+T(Q#HKF^o$\"+KTtrlF67$$\"+tM!oK$F^o$\"+!GN,.'F67$$\"+$G/'=M F^o$\"+U&G>_&F67$$\"+\"eiX^$F^o$\"+'>Bp*\\F67$$\"+c$z$3OF^o$\"+n,5$\\% F67$$\"+Dof1PF^o$\"+F8*z(RF67$$\"+@>>,QF^o$\"+7*))\\\\$F67$$\"+q!Hz*QF ^o$\"+\"*3k9IF67$$\"+'=lQ*RF^o$\"+\"R'R^DF67$$\"+90-#3%F^o$\"+W`PO@F67 $$\"+nk0$=%F^o$\"+*Qf&*>Y%F^o$\"+Q?=CVF07$$\"+w#)4hXF^oF*-%'COLOURG6&%$R GBG$\"#5!\"\"F(F(-F$6$7W7$F(F(7$$\"+9\"[4(\\F0$\"+:fCn\\F07$F.$\"+jrN7 **F07$$\"+g\"4nU\"F6$\"+5@,=9F67$F4$\"+A'y+%=F67$F:$\"+w+XlFF67$F?$\"+ Eie_OF67$FD$\"+/rM![%F67$FI$\"+P,p\">&F67$FN$\"+-%eq'eF67$FS$\"+*e^!) \\'F67$FX$\"+db+fqF67$Fgn$\"+J^YnvF67$F\\o$\"+3\\zgzF67$Fbo$\"+Bx#oM)F 67$Fgo$\"+`x3z')F67$F\\p$\"+Sd&>&*)F67$Fap$\"+wRjj\"*F67$Ffp$\"+=:]v$* F67$F[q$\"+6!G[_*F67$F`q$\"+1![xm*F67$Feq$\"+]]hp(*F67$Fjq$\"+L\"zv&)* F67$F_r$\"+f!e.#**F67$Fdr$\"+$=Be'**F67$F^s$\"+tQl!***F67$Fhs$\"+%>&** ****F67$Fbt$\"+xfz!***F67$F\\u$\"+g#**o'**F67$Fau$\"+yY\"Q#**F67$Ffu$ \"+qPDf)*F67$F[v$\"+J)[Yx*F67$F`v$\"+V5]q'*F67$Fev$\"+sp7E&*F67$Fjv$\" +k+Qn$*F67$F_w$\"+sedj\"*F67$Fdw$\"+Af[W*)F67$Fiw$\"+L+\"Hm)F67$F^x$\" +<4J`$)F67$Fcx$\"+f.\\yzF67$Fhx$\"+r\\zcvF67$F]y$\"+=)pA0(F67$Fby$\"+% 48>]'F67$Fgy$\"+s)Q:(eF67$F\\z$\"+vwmy^F67$Faz$\"+)3%H%[%F67$Ffz$\"+S2 QDOF67$F[[l$\"+0U*p!GF67$F`[l$\"+)**yB*=F67$$\"+`^)eT%F^o$\"+2L'HW\"F6 7$Fe[l$\"+V*[5))*F07$$\"+Ona6XF^o$\"+aq[^\\F07$Fj[lF(-F]\\l6&F_\\lF(F` \\lF(-%+AXESLABELSG6$Q\"u6\"Q!F_gl-%&TITLEG6#%cqcomparison~of~sn(u;k=0 .9)~and~cn(u;k=0.9)|+~with~cn~moved~over,~for~a~half~period|+~sn~is~th e~\"fatter\"~oneG-%%VIEWG6$;F(Fj[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "evalf([JacobiSN(1,0.995),JacobiCN(1-Ellip ticK(0.995),0.995)]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "plot(\{Ja cobiSN(u,0.995),JacobiCN(u-EllipticK(0.995),0.995)\},u=0..2*EllipticK( 0.995),title=`comparison of sn(u;k=0.995) and cn(u;k=0.995)\\n with cn moved over, for a half period\\n sn is the \"fatter\" one`);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+**eXCw!#5$\"+eozo6F&" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6'-%'CURVESG6$7in7$$\"\" !F)F(7$$\"+5c8e!)!#6$\"+#)e#3/)F-7$$\"+Ari6;!#5$\"+*e&)yf\"F37$$\"+$>h FJ#F3$\"+1kesAF37$$\"+k_*Q,$F3$\"+k3EEHF37$$\"+cQR-QF3$\"+E.(*HOF37$$ \"+\\C*3f%F3$\"+(*[y%H%F37$$\"+@xh%Q&F3$\"+E@J?\\F37$$\"+%*HMyhF3$\"+y l8*\\&F37$$\"+HeHopF3$\"+)3ku-'F37$$\"+k'[#exF3$\"+R')33lF37$$\"+jgj! \\)F3$\"+!zV>\"pF37$$\"+iM-B#*F3$\"+4+8xsF37$$\"+#HrR2\"!\"*$\"+>m@>zF 37$$\"+Lv#3B\"Feo$\"+[)*HS%)F37$$\"+M2=(Q\"Feo$\"+H6\")R))F37$$\"+?#4! [:Feo$\"+3r(3:*F37$$\"+Pqm*o\"Feo$\"+DC&)e$*F37$$\"+)=S\"\\=Feo$\"+Sg(*F37$$\"+p?r.B Feo$\"+)Q#)R#)*F37$$\"+F8MqCFeo$\"+z%H*z)*F37$$\"+m\")\\6EFeo$\"+$\\[Z \"**F37$$\"+ErpvFFeo$\"+Uc\\W**F37$$\"+H5/@HFeo$\"+?Hbj**F37$$\"+Dj]!3 $Feo$\"+$yj&y**F37$$\"+)*\\NKKFeo$\"+4@X))**F37$$\"+eJz!R$Feo$\"+g:=&* **F37$$\"+[))GONFeo$\"+yGq)***F37$$\"+0cA$p$Feo$\"+U$*******F37$$\"+!z Qi&QFeo$\"+5Fs)***F37$$\"+YA9)*RFeo$\"+6kL&***F37$$\"+n=S^TFeo$\"+)\\w *))**F37$$\"+^St4VFeo$\"+'\\O)y**F37$$\"+/7jkWFeo$\"+b\"[W'**F37$$\"+k @]9YFeo$\"+.U.X**F37$$\"+8$34y%Feo$\"+MY.:**F37$$\"+R;VI\\Feo$\"+6k#z( )*F37$$\"+/(z+4&Feo$\"+$*f'R#)*F37$$\"+!eWZB&Feo$\"+Hv,e(*F37$$\"++:!H R&Feo$\"+)\\+6m*F37$$\"+P[rTbFeo$\"+#f#eQ&*F37$$\"+_wE(p&Feo$\"+3!\\vO *F37$$\"+h*[$\\eFeo$\"+G8%[9*F37$$\"+)*Rc3gFeo$\"+.mON))F37$$\"+-q!>;' Feo$\"+%HaKW)F37$$\"+qBs=jFeo$\"+3b=BzF37$$\"+H#RUZ'Feo$\"+V!)*QE(F37$ $\"+]6pXlFeo$\"+u(pm!pF37$$\"+rI9h7lF37$$\"+O].*p'Feo$\"+Og %R,'F37$$\"++q#4y'Feo$\"+Iw+kaF37$$\"+\\@^0qFeo$\"+c+F*p$F37$$\"+6lg$3(Feo$\"+Cx81IF37$$ \"+9INerFeo$\"+^Bk6BF37$$\"+;&*4LsFeo$\"+EY*Gf\"F37$$\"+S[U8tFeo$\"+#y 'Q:!)F-7$$\"+k,v$R(FeoF(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7en7$F($! +=bF1]!#?7$F1$\"+(yZkh\"F-7$F<$\"+B;'\\0$F-7$FF$\"+?jUWZF-7$FP$\"+T\"R ;c'F-7$FZ$\"+p[!*H&)F-7$F^o$\"+!>iP0\"F37$Fco$\"+$3bXG\"F37$Fio$\"+J+y _:F37$F^p$\"+ibhb=F37$Fcp$\"+\"[d.@#F37$Fhp$\"+`&=Wc#F37$F]q$\"+9l^:IF 37$Fbq$\"+DI3INF37$Fgq$\"+'Q9&)3%F37$F\\r$\"+V86]YF37$Far$\"+DXl#Q&F37 $Ffr$\"+,,r]gF37$F[s$\"+V8YkoF37$F`s$\"+]0P#f(F37$Fes$\"+]s&*e$)F37$Fj s$\"+x$*f4!*F37$F_t$\"+0g/\\&*F37$$\"+.5ajMFeo$\"+@,xL(*F37$Fdt$\"+/0U s)*F37$$\"+PI_vNFeo$\"+ZR\"o#**F37$$\"+Esv9OFeo$\"+1qPm**F37$$\"+:9*Rl $Feo$\"+]>\"3***F37$Fit$\"+5M$*****F37$$\"++*yRt$Feo$\"+a/7$***F37$$\" +(>KZx$Feo$\"+;rwp**F37$$\"+$\\&[:QFeo$\"+!ek+$**F37$F^u$\"+.bLu)*F37$ $\"+=0>FRFeo$\"+@qYS(*F37$Fcu$\"+p>hi&*F37$Fhu$\"+PlU[!*F37$F]v$\"+ZV2 v$)F37$Fbv$\"+'Q!RKwF37$Fgv$\"+^rG#)oF37$F\\w$\"+E-CdgF37$Faw$\"+oMY]` F37$Ffw$\"+\")p$*\\YF37$F[x$\"+S40rSF37$F`x$\"+\"))39]$F37$Fex$\"+TdDC IF37$Fjx$\"+H)>De#F37$F_y$\"+=2(=?#F37$Fdy$\"+GZ]^=F37$Fiy$\"+)eEYb\"F 37$F^z$\"+\"HbiG\"F37$Fcz$\"+Jlu\\5F37$F][l$\"+^B7S&)F-7$Fg[l$\"+(Ga?] 'F-7$Fa\\l$\"+5#yW#[F-7$F[]l$\"+[gUYJF-7$Fe]l$\"+o&o7h\"F-7$F_^lF\\_l- Fb^l6&Fd^lF(Fe^lF(-%+AXESLABELSG6$Q\"u6\"Q!F[[m-%&TITLEG6#%gqcompariso n~of~sn(u;k=0.995)~and~cn(u;k=0.995)|+~with~cn~moved~over,~for~a~half~ period|+~sn~is~the~\"fatter\"~oneG-%%VIEWG6$;F(F_^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Now, let's actually make some graphs of the behavior of an actual pendulum, moving to large values \+ of the maximum angle," }}{PARA 0 "" 0 "" {TEXT -1 126 "relative to the small-angle approximation in a region where it is not particulary a g ood approximation to the actual behavior." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "nn:=evalf(EllipticK(0.6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nnG$\"+.Qv] " 0 "" {MPLTEXT 1 0 5 "4*nn;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7_,.q!\"* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3`=$G'!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "tmax:=2*arcsin(0.6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tmaxG$\"+=A+(G\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "tmaxdegr:=tmax*180/evalf(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %)tmaxdegrG$\"+J&zRP(!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "theta:=2*arcsin(k*JacobiSN(t,k)); apprtheta:=2*k*sin(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG,$*&\"\"#\"\"\"-%'arcsinG6#*&%\"kGF (-%)JacobiSNG6$%\"tGF-F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*app rthetaG,$*(\"\"#\"\"\"%\"kGF(-%$sinG6#%\"tGF(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 174 "plot(subs(k=0.6,\{theta,apprtheta\}),t=0..15, labels=[\"omega0*t\",\"theta\"],title=`true pendulum for max angle=74 \+ degrees\\n versus linear approx. (the one with shorter period)` );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6'-%'CURVESG6$7er7$$\"\"!F)F(7$$\"+ilyM;!#5$\"+.U .`>F-7$$\"+DJdpKF-$\"+SwZaQF-7$$\"+s@*>p%F-$\"+;m8HaF-7$$\"+>7T9hF-$\" +o!>'**oF-7$$\"+/-29xF-$\"+1D!*)R)F-7$$\"+)=HPJ*F-$\"+k;C3(*F-7$$\"+v \"*R#4\"!\"*$\"+oMM\"3\"FL7$$\"+JaU`7FL$\"+Q(>!p6FL7$$\"+ejo89FL$\"+oq hK7FL7$$\"+%GZRd\"FL$\"+6#3?F\"FL7$$\"+\"[Q#[;FL$\"+^*e>G\"FL7$$\"+y'H Ds\"FL$\"+b)>mG\"FL7$$\"+w_nf\"FL7$$\"+'H>zL#FL$\"+K3*H7\" FL7$$\"+(fHq\\#FL$\"+9$[P-\"FL7$$\"+G'Hcl#FL$\"+/4]L!*F-7$$\"+f'HU\"GF L$\"+wR\\KwF-7$$\"+'Gpt(HFL$\"+%3k3,'F-7$$\"+7*309$FL$\"+iWNRUF-7$$\"+ %Q-UG$FL$\"+/pD(e#F-7$$\"+ce*yU$FL$\"+OxVD))!#67$$\"+s1m*e$FL$!+S!ok0 \"F-7$$\"+)[D9v$FL$!+m0.oHF-7$$\"+DX&Q\"RFL$!+]2w5[F-7$$\"+iNGwSFL$!+? v'>`'F-7$$\"+PS\"GB%FL$!+1\"Q*R!)F-7$$\"+7XM*Q%FL$!+e'4?P*F-7$$\"+!=*[ JXFL$!+g^9T5FL7$$\"+ZQjtYFL$!+C2oF6FL7$$\"+0wlU[FL$!+,@!o?\"FL7$$\"+i8 o6]FL$!+F'e#f7FL7$$\"+4SF$3&FL$!+)>/LF\"FL7$$\"+cm'[:&FL$!+v+X#G\"FL7$ $\"+!)Hm!>&FL$!+&3\"=&G\"FL7$$\"+.$fkA&FL$!+iEo'G\"FL7$$\"+FcDi_FL$!+u U&pG\"FL7$$\"+]>0)H&FL$!+:e*fG\"FL7$$\"+=7L\"Q&FL$!+#R!zoFL$!+/!3R[\"F-7$$\"+9diEqFL$ \"+#)p*H$GFhs7$$\"+4A@urFL$\"+=YOW?F-7$$\"+#=/ML(FL$\"+-3[$*QF-7$$\"+c hf#\\(FL$\"+e&\\ok&F-7$$\"+w=&zl(FL$\"+7mjCtF-7$$\"+(f2L#yFL$\"+7kF?)) F-7$$\"+P-DnzFL$\"+Y\")p^**F-7$$\"+yG>6\")FL$\"+#fj44\"FL7$$\"+t[lm#)F L$\"+#\\&yt6FL7$$\"+po6A%)FL$\"+[dMM7FL7$$\"+&f?C])FL$\"+FLsc7FL7$$\"+ AVs#e)FL$\"+x)oHF\"FL7$$\"+&=wGi)FL$\"+:[x%G\"FL7$$\"+rSt/\"F_el$\"+Qp!)ePFhs7$$\"+0Q*>1\"F_el$!+=H \">Q\"F-7$$\"+sn.y5F_el$!+g[aoKF-7$$\"+R(zS4\"F_el$!+yO%H2&F-7$$\"+?\\ <46F_el$!+?qudmF-7$$\"+-,FC6F_el$!+!o%G-\")F-7$$\"+;*[+9\"F_el$!+I^'fV *F-7$$\"+Jx#e:\"F_el$!+Wh&o0\"FL7$$\"+1WDr6F_el$!+#yvm9\"FL7$$\"+\"3\" o'=\"F_el$!+UZ$[@\"FL7$$\"+Iiv%>\"F_el$!+q`lT7FL7$$\"+!QJG?\"F_el$!+1m Ii7FL7$$\"+Il!4@\"F_el$!+^guw7FL7$$\"+!o\")*=7F_el$!+Ic%\\G\"FL7$$\"+W .(GA\"F_el$!+yKm'G\"FL7$$\"+4!fnA\"F_el$!+T%HpG\"FL7$$\"+uwkI7F_el$!++ Su&G\"FL7$$\"+Qj`M7F_el$!+gu5$G\"FL7$$\"+mOJU7F_el$!+Mi[t7FL7$$\"+&*44 ]7F_el$!+CE3e7FL7$$\"+zy*fE\"F_el$!+/?q37FL7$$\"+jZ!>G\"F_el$!+lVoN6FL 7$$\"+I*zwH\"F_el$!+Y7kS5FL7$$\"+(4bMJ\"F_el$!+%*>rT#*F-7$$\"+Q3&zK\"F _el$!++\"Hj*zF-7$$\"+ylWU8F_el$!+Yvm+mF-7$$\"+K.1f8F_el$!+[FmW[F-7$$\" +'3ucP\"F_el$!+AFsgHF-7$$\"+EO`!R\"F_el$!+;1H17F-7$$\"+lJR09F_el$\"+7& [fu&Fhs7$$\"+MlB@9F_el$\"+?/aeCF-7$$\"+-*zqV\"F_el$\"+7F]bm7$$\"3')*****\\7t&pKF]bm$\"3 yoMlfi&R&QF]bm7$$\"3=+](=<#*>p%F]bm$\"3IP)RADmgU&F]bm7$$\"3$****\\(=7T 9hF]bm$\"33LF1Ipd))oF]bm7$$\"3p**\\7.-29xF]bm$\"3!3I2%pftl$)F]bm7$$\"3 X****\\(=HPJ*F]bm$\"3)ppZ9bz#H'*F]bm7$$\"3;+++v\"*R#4\"!#<$\"3KFK)Rt!G l5F\\dm7$$\"3;++DJaU`7F\\dm$\"3WRbz(*42S6F\\dm7$$\"3,]7`%*ebL8F\\dm$\" 3!Rqc%y#)Qm6F\\dm7$$\"32+D\"yN'o89F\\dm$\"3y4oR8.A&=\"F\\dm7$$\"3+DJX* e^PX\"F\\dm$\"3-np;(p*y\">\"F\\dm7$$\"39]P4@o\"Q\\\"F\\dm$\"3%o&f&)[iW '>\"F\\dm7$$\"31vVt_?)Q`\"F\\dm$\"35U2m%\\#=*>\"F\\dm7$$\"3)***\\P%GZR d\"F\\dm$\"3wDZ)G/%***>\"F\\dm7$$\"3'**\\7G)G46;F\\dm$\"33w]PWe-*>\"F \\dm7$$\"3%****\\7[Q#[;F\\dm$\"3mx7kDMS'>\"F\\dm7$$\"3#**\\(ozSQ&o\"F \\dm$\"3->)oYSI@>\"F\\dm7$$\"3\"***\\7y'HDs\"F\\dm$\"3wT[nwE@'=\"F\\dm 7$$\"3')*****\\(3#oz\"F\\dm$\"3Ix!>Z,y%p6F\\dm7$$\"3%)**\\(=276(=F\\dm $\"3Qhc%*ezL#F\\dm$\"3IJ8% 4(f#*Q')F]bm7$$\"3/+](ofHq\\#F\\dm$\"3)*=0HQ1>5sF]bm7$$\"3E+]7G'Hcl#F \\dm$\"3G;Cj\"H@Zg&F]bm7$$\"3.+]Pf'HU\"GF\\dm$\"3gh]'eAm&eQF]bm7$$\"3% )*\\PfGpt(HF\\dm$\"3Y()=blV$='>F]bm7$$\"33++]7*309$F\\dm$\"3%y:#=`!*[+ 8!#?7$$\"3!***\\P%Q-UG$F\\dm$!3d/AD/?_0&F]bm7$$\"3;++]([D9 v$F\\dm$!3ugE\"p@lF(oF]bm7$$\"3k*****\\_aQ\"RF\\dm$!3i\\*=vL(3t$)F]bm7 $$\"3c****\\iNGwSF\\dm$!3K#y[$[f)Hl*F]bm7$$\"3&)****\\PS\"GB%F\\dm$!3M bK?]$HY1\"F\\dm7$$\"37++]7XM*Q%F\\dm$!3\">Ut\"**z#z8\"F\\dm7$$\"35]P4Y oTgWF\\dm$!37LBz[r5i6F\\dm7$$\"34+voz\"*[JXF\\dm$!3)3f8Vk=/=\"F\\dm7$$ \"3kuV[Y`-nXF\\dm$!3%f8Gx&RM(=\"F\\dm7$$\"32]7G8:c-YF\\dm$!3CyrBG+x#> \"F\\dm7$$\"3^D\"y+o(4QYF\\dm$!3o?gT/+p'>\"F\\dm7$$\"3/+](o%QjtYF\\dm$ !3w)ekl$*)4*>\"F\\dm7$$\"30D\"Gjy*)er%F\\dm$!3wd@iYE***>\"F\\dm7$$\"31 ]7yDd9eZF\\dm$!3sx)48,W()>\"F\\dm7$$\"31vVBl;S+[F\\dm$!3[3[\")f_N&>\"F \\dm7$$\"31+vo/wlU[F\\dm$!3+_22UC$)*=\"F\\dm7$$\"31]Pf$[pr#\\F\\dm$!3? qqpRyUs6F\\dm7$$\"32++]i8o6]F\\dm$!3-5YPQWlY6F\\dm7$$\"3I++Dcm'[:&F\\d m$!3I'GYnDKW3\"F\\dm7$$\"3i******\\>0)H&F\\dm$!3Z-z`t\\,+5F\\dm7$$\"3a *\\Pf[5YY&F\\dm$!37\")pE/O2i()F]bm7$$\"3Y**\\(=-p6j&F\\dm$!3AJ]eyO[\"G (F]bm7$$\"3'**\\P%[:gydF\\dm$!3*4o'QwtJ,eF]bm7$$\"3d*****\\2Mg#fF\\dm$ !3o$o*Gc)z_>%F]bm7$$\"3%)*\\PM#4z(3'F\\dm$!3#er7'\\6%)HBF]bm7$$\"35+]( =xZ&\\iF\\dm$!3`**45tIuNS!#>7$$\"38++vV&yNS'F\\dm$\"3L\"fK-GJ7W\"F]bm7 $$\"3;+]i:$4wb'F\\dm$\"3!3Xry^5>D$F]bm7$$\"3k*\\(=nUK=nF\\dm$\"3KW'*e \"*pVe]F]bm7$$\"3-++v=#R!zoF\\dm$\"3*3'ev(e)eMnF]bm7$$\"3\"3]iSrDm-(F \\dm$\"3)3oT.e%*=7)F]bm7$$\"3q+]P4A@urF\\dm$\"3.&*zR(=7EL*F]bm7$$\"3&4 ]7G=/ML(F\\dm$\"3#4E#>L&Q5/\"F\\dm7$$\"3I++Dchf#\\(F\\dm$\"3U8x(*G%*[A 6F\\dm7$$\"3)*\\iS;SFvvF\\dm$\"3qOGN^Tp`6F\\dm7$$\"3m*\\il(=&zl(F\\dm$ \"3W;#*GSr,x6F\\dm7$$\"3#\\iSm!3H*p(F\\dm$\"39!Q\"Rc5n&=\"F\\dm7$$\"3@ ](=ntH1u(F\\dm$\"3!>O#Hf!*H#>\"F\\dm7$$\"3guozm'o>y(F\\dm$\"3a-O'R#)*) o>\"F\\dm7$$\"3))**\\(of2L#yF\\dm$\"3mR\"F\\dm7$$\"3e]7.dKHfyF \\dm$\"3K,h.sI)**>\"F\\dm7$$\"3S+v=<*y_*yF\\dm$\"3m7>1mo(*)>\"F\\dm7$$ \"3B]PMxXEJzF\\dm$\"3'3%GF\">=k>\"F\\dm7$$\"31++]P-DnzF\\dm$\"3%*RErg. J#>\"F\\dm7$$\"3q*\\7yb@#R!)F\\dm$\"3I*3=q]q%z6F\\dm7$$\"3M**\\7yG>6\" )F\\dm$\"35I%f]yB0;\"F\\dm7$$\"3;*\\PM([lm#)F\\dm$\"3[\"z>FOi#*4\"F\\d m7$$\"3w++voo6A%)F\\dm$\"3<-$Ru&p[65F\\dm7$$\"3A+](=KCFe)F\\dm$\"3%R$z j;t9_*)F]bm7$$\"3q*****\\xJLu)F\\dm$\"3g_!y\"oD+fvF]bm7$$\"3'4+vV$[X+* )F\\dm$\"3*fWx&*4Ev+'F]bm7$$\"3W++v$*ydd!*F\\dm$\"3J71%fwU!3VF]bm7$$\" 3>+D\"G`-'4#*F\\dm$\"39Rt(ywDAc#F]bm7$$\"3#***\\(=t/\"Fd]o$!3Tmy 7))4'*R5F\\dm7$$\"3%**\\(o/Q*>1\"Fd]o$!3/\"3F`FHj6\"F\\dm7$$\"33DJ?)G: +2\"Fd]o$!3A$*[eXa,[6F\\dm7$$\"31](=\"F\\dm7$$\"3g(ozT``y4\"Fd]o$!3K!p5)3e#)*>\"F\\dm7$$\"3/v$4'Hti,6Fd]o $!37Ke!)RHu*>\"F\\dm7$$\"3li!R]7,a5\"Fd]o$!3o8)Gel^z>\"F\\dm7$$\"33](o /#\\<46Fd]o$!3S)oRw]aW>\"F\\dm7$$\"37D\"G8^An6\"Fd]o$!3ObB>g\\O#=\"F\\ dm7$$\"3***\\(=-,FC6Fd]o$!3K^pCMJaj6F\\dm7$$\"3.+Dc;*[+9\"Fd]o$!37DB8VgX!*F]bm7$$\"3!****\\73\"o'=\"Fd]o$!3c#)*yEJrls(F]bm7$$\"3 '*\\(o/QJG?\"Fd]o$!3OP!G'p*>'\\hF]bm7$$\"3-+voz;)*=7Fd]o$!3!3LRV5:ET%F ]bm7$$\"3/]PMPj`M7Fd]o$!3<5Ls7(\\0j#F]bm7$$\"31+++&*44]7Fd]o$!33(pvp\" 3n\\yFecn7$$\"33+D1zy*fE\"Fd]o$\"37RSB_$e;7\"F]bm7$$\"35+]7jZ!>G\"Fd]o $\"3mxZ!\\Pi***HF]bm7$$\"35]i:I*zwH\"Fd]o$\"3CC_n\"RH!)y%F]bm7$$\"34+v =(4bMJ\"Fd]o$\"3E*)H#*o.>dkF]bm7$$\"38]PMP3&zK\"Fd]o$\"3sx\\'=Q@0&yF]b m7$$\"3;++]xlWU8Fd]o$\"3UbrM-3=z!*F]bm7$$\"3;+DcJ.1f8Fd]o$\"3@noyg(y^- \"F\\dm7$$\"39+]i&3ucP\"Fd]o$\"3S$)Q?4y?96F\\dm7$$\"3-+DJDO`!R\"Fd]o$ \"3?H1_*)o*y;\"F\\dm7$$\"3\"******\\;$R09Fd]o$\"3U*pn6`Xe>\"F\\dm7$$\" 3[P%[r+a$49Fd]o$\"3oiC'))>e))>\"F\\dm7$$\"30voH\\[J89Fd]o$\"3sYvD6.*** >\"F\\dm7$$\"3X7`W\"pvsT\"Fd]o$\"3C'e'\\#4S#*>\"F\\dm7$$\"3-]PfLlB@9Fd ]o$\"3U0#*Q>(3m>\"F\\dm7$$\"3*\\i!*y@e\"H9Fd]o$\"3w&4**\\'>s&=\"F\\dm7 $$\"38+v=-*zqV\"Fd]o$\"3[aw5I$)Rn6F\\dm7$$\"35++]FSC_9Fd]o$\"3;\\g.%*[ .76F\\dm7$$\"33+D\"G:3uY\"Fd]o$\"3Jn&p<5\\6.\"F\\dm7$$\"3/]iSwSq$[\"Fd ]o$\"3w#))>0g\"3z\"*F]bm7$F\\am$\"3d-a)=3aM!yF]bm-Faam6&FcamF(FdamF(-% +AXESLABELSG6$Q)omega0*t6\"Q&thetaF][p-%&TITLEG6#%fptrue~pendulum~for~ max~angle=74~degrees|+~versus~linear~approx.~(the~one~with~shorter~per iod)G-%%VIEWG6$;F(F\\am%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 63 "and let's try it again for a larger value for the maxim um angle" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "2*arcsin(.95); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'zrk]#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "%*180/evalf(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aD5O9!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "plot(subs(k=0.95,\{theta,apprtheta\}),t=0..15,labels=[\"omega0* t\",\"theta\"],title=`true pendulum for max angle=143 degrees\\n versu s linear approx. (the one with shorter period)` );" }}{PARA 13 "" 1 " " {GLPLOT2D 495 495 495 {PLOTDATA 2 "6'-%'CURVESG6$7io7$$\"\"!F)F(7$$ \"+ilyM;!#5$\"+iQM#4$F-7$$\"+DJdpKF-$\"+wh:/hF-7$$\"+s@*>p%F-$\"+#QOLg )F-7$$\"+>7T9hF-$\"+`d+&4\"!\"*7$$\"+)=HPJ*F-$\"+%z@^b\"F?7$$\"+JaU`7F ?$\"+&)**Q;>F?7$$\"+%GZRd\"F?$\"++Raz@F?7$$\"+s?6r=F?$\"+ec!zM#F?7$$\" +M0'\\-#F?$\"+oWu4CF?7$$\"+(**3)y@F?$\"+#)pubCF?7$$\"+'H>zL#F?$\"+3,a( [#F?7$$\"+(fHq\\#F?$\"+1d!R]#F?7$$\"+G'Hcl#F?$\"+uT>0DF?7$$\"+f'HU\"GF ?$\"+C%3:\\#F?7$$\"+'Gpt(HF?$\"+Wc^hCF?7$$\"+7*309$F?$\"+e%fZT#F?7$$\" +ce*yU$F?$\"+i%e&)G#F?7$$\"+)[D9v$F?$\"+=7Ur?F?7$$\"+iNGwSF?$\"+UW8h0)H&F?$!+#4qtB#F-7$$\"+'[5YY&F?$!+1q^N`F-7$$ \"+A!p6j&F?$!+?d#RH)F-7$$\"+\\:gydF?$!+3+\\u5F?7$$\"+vS.EfF?$!+$Ry0I\" F?7$$\"+sxa\\iF?$!+@()eA#R!zoF?$!+#o g'eAF?7$$\"+4A@urF?$!+EQn)R#F?7$$\"+#=/ML(F?$!+-&)>\\CF?7$$\"+chf#\\(F ?$!+/9_$[#F?7$$\"+w=&zl(F?$!+%>UF]#F?7$$\"+(f2L#yF?$!+Y(Hc]#F?7$$\"+P- DnzF?$!+)=/\\\\#F?7$$\"+yG>6\")F?$!+osnrCF?7$$\"+t[lm#)F?$!+Ik5KCF?7$$ \"+po6A%)F?$!+K,xwBF?7$$\"+vF? 7$$\"+srih$*F?$!+o%p(Q;F?7$$\"+9YUI&*F?$!+akf>9F?7$$\"+c?A*p*F?$!+!f$R s6F?7$$\"+oS*3&)*F?$!+cl?z#*F-7$$\"+3mD+5!\")$!+#4_:l'F-7$$\"+K3X;5Fjz $!+Ess\"p$F-7$$\"+c]kK5Fjz$!+'zIt/\"Fjz$\"+?UEX@F-7$$\" +0Q*>1\"Fjz$\"+E-B$)[F-7$$\"+sn.y5Fjz$\"+e(4,w(F-7$$\"+R(zS4\"Fjz$\"+! G:e/\"F?7$$\"+?\\<46Fjz$\"+L6`z7F?7$$\"+-,FC6Fjz$\"+_C_\"\\\"F?7$$\"+J x#e:\"Fjz$\"+H0lh=F?7$$\"+\"3\"o'=\"Fjz$\"+uizJ@F?7$$\"+!o\")*=7Fjz$\" +%pe%GBF?7$$\"+Qj`M7Fjz$\"+KNO&R#F?7$$\"+&*44]7Fjz$\"+#fBeW#F?7$$\"+zy *fE\"Fjz$\"+)eq8[#F?7$$\"+jZ!>G\"Fjz$\"+wYP,DF?7$$\"+I*zwH\"Fjz$\"+K'f i]#F?7$$\"+(4bMJ\"Fjz$\"+5m\"F?7$$\"+`\"3uY\"Fjz$\"+fe@p9F?7$$\"+wSq$[\"Fjz$\"+U\\vO7F?7$$\" #:F)$\"+Uh\\$z*F--%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7_sF'7$$\"3$**** *\\ilyM;!#=$\"3#)**RqTxF#4$Fadl7$$\"3')*****\\7t&pKFadl$\"3k#*z66u4-hF adl7$$\"3=+](=<#*>p%Fadl$\"3%*3k/m:F\"f)Fadl7$$\"3$****\\(=7T9hFadl$\" 3mK*fsM\"p!4\"!#<7$$\"3p**\\7.-29xFadl$\"3eAG#='[dC8Fcel7$$\"3X****\\( =HPJ*Fadl$\"3i=#z*GfjC:Fcel7$$\"3;+++v\"*R#4\"Fcel$\"33VMZ&\\%p'o\"Fce l7$$\"3;++DJaU`7Fcel$\"3!3FwJT7^!=Fcel7$$\"3,]7`%*ebL8Fcel$\"39\"y*)3W \"yY=Fcel7$$\"32+D\"yN'o89Fcel$\"3s[\\aH))fw=Fcel7$$\"3+DJX*e^PX\"Fcel $\"3#ypZQN+q)=Fcel7$$\"39]P4@o\"Q\\\"Fcel$\"39l_NFKP%*=Fcel7$$\"31vVt_ ?)Q`\"Fcel$\"33vha;cq)*=Fcel7$$\"3)***\\P%GZRd\"Fcel$\"3a:3!zc!****=Fc el7$$\"3'**\\7G)G46;Fcel$\"3y.s#ped%)*=Fcel7$$\"3%****\\7[Q#[;Fcel$\"3 %H-KRU0V*=Fcel7$$\"3#**\\(ozSQ&o\"Fcel$\"3)oH#R2)Rv)=Fcel7$$\"3\"***\\ 7y'HDs\"Fcel$\"3+L=Sr+'\\\"=Fcel7$$\"3!***\\PM0'\\-#Fcel $\"3Y\\q'fA$R2yi7\\f:Fcel7$$\"3*** *\\(oH>zL#Fcel$\"37W!*RX*HyO\"Fcel7$$\"3/+](ofHq\\#Fcel$\"31K$Hx^8;9\" Fcel7$$\"3E+]7G'Hcl#Fcel$\"3keY3X?9u))Fadl7$$\"3.+]Pf'HU\"GFcel$\"3%zM 'GClR4hFadl7$$\"3%)*\\PfGpt(HFcel$\"3[brXXxB1JFadl7$$\"33++]7*309$Fcel $\"3g;%Qvm2\"f?!#?7$$\"3!***\\P%Q-UG$Fcel$!3R!*f1S)4/q#Fadl7$$\"3:++Dc e*yU$Fcel$!3)G*zQq!\\dO&Fadl7$$\"3Q+](=ng'*e$Fcel$!3%4,DdWw7B)Fadl7$$ \"3;++]([D9v$Fcel$!3Er6,Ey=)3\"Fcel7$$\"3k*****\\_aQ\"RFcel$!3G+0WG)Qd K\"Fcel7$$\"3c****\\iNGwSFcel$!3j!*=#=W*QG:Fcel7$$\"3&)****\\PS\"GB%Fc el$!3)4#=#y8jco\"Fcel7$$\"37++]7XM*Q%Fcel$!3z%eC?L>)GSm7'*z=Fcel7$$\"32]7G8:c-YFcel$!3el)3Za_&))=Fcel7$$ \"3^D\"y+o(4QYFcel$!3G*peOOb\"=Fcel7$$\"3I++Dcm'[:&Fcel$!3V'G[)Rx,<0)H&Fcel$!3A7]VTqN$e\"Fcel7$$\"3a*\\Pf[5YY&Fcel$!3I0cn?$GtQ\"Fcel7$ $\"3Y**\\(=-p6j&Fcel$!33IEW#e,H:\"Fcel7$$\"3'**\\P%[:gydFcel$!3%>\"*y# z\">a=*Fadl7$$\"3d*****\\2Mg#fFcel$!3ZK`7Rk_UmFadl7$$\"3%)*\\PM#4z(3'F cel$!3WmMb`^\"*)o$Fadl7$$\"35+](=xZ&\\iFcel$!35;*4C`E**Q'!#>7$$\"38++v V&yNS'Fcel$\"3KO\\.F&\\>G#Fadl7$$\"3;+]i:$4wb'Fcel$\"3#*HJ'*>$e)[^Fadl 7$$\"3k*\\(=nUK=nFcel$\"34q-N.>>4!)Fadl7$$\"3-++v=#R!zoFcel$\"3S6!G(f) 4j1\"Fcel7$$\"3\"3]iSrDm-(Fcel$\"3qKu)=kmfG\"Fcel7$$\"3q+]P4A@urFcel$ \"3#e,QjfjwZ\"Fcel7$$\"3&4]7G=/ML(Fcel$\"3MYx)3,6$[;Fcel7$$\"3I++Dchf# \\(Fcel$\"3)Hr9#H\\Fxy(Fcel$\"3?P !4Ybv]*=Fcel7$$\"3))**\\(of2L#yFcel$\"3g(ee$=i5**=Fcel7$$\"3e]7.dKHfyF cel$\"3'Q\\!R(>t***=Fcel7$$\"3S+v=<*y_*yFcel$\"3ch8$z.!Q)*=Fcel7$$\"3B ]PMxXEJzFcel$\"3)[m[G!)GV*=Fcel7$$\"31++]P-DnzFcel$\"3C8]zPZ#y)=Fcel7$ $\"3q*\\7yb@#R!)Fcel$\"3C3`Wp\\\\n=Fcel7$$\"3M**\\7yG>6\")Fcel$\"3Qk2, $*f\\P=Fcel7$$\"3;*\\PM([lm#)Fcel$\"3G.I(4u)\\S\"Fcel7$$\"3'4+vV$[X+*)Fcel$\"3pb4LCj\">^*Fadl 7$$\"3W++v$*ydd!*Fcel$\"3o-$RFrn5#oFadl7$$\"3>+D\"G`-'4#*Fcel$\"35(y0e Ydo0%Fadl7$$\"3#***\\(=\"Fadl7$$\"3T+D19YUI&*Fce l$!3w$=rk`aN+#Fadl7$$\"35***\\i0A#*p*Fcel$!3M6^m\"pD#\\^Fadl7$$\"3-** \\ilS*3&)*Fcel$!3m+U+]%4M&yFadl7$$\"3*)****\\2mD+5!#;$!3-Xk(>jFx.\"Fce l7$$\"3&)**\\(=$3X;5Fh_n$!3!)*[KMlo2G\"Fcel7$$\"3*****\\i0XE.\"Fh_n$!3 iErt'[%H!\\\"Fcel7$$\"3'*\\(o/V>t/\"Fh_n$!3!>7py*[gY;Fcel7$$\"3%**\\(o /Q*>1\"Fh_n$!3o67g_8_nC(**=Fcel7$$\"3/v$4'Hti,6Fh_n$!3/,f-))Hf**=Fcel7$$\"3li!R]7,a5\"Fh _n$!3c@cR)ycn*=Fcel7$$\"33](o/#\\<46Fh_n$!3Mth4P'>7*=Fcel7$$\"37D\"G8^ An6\"Fh_n$!39'*yj`y2s=Fcel7$$\"3***\\(=-,FC6Fh_n$!3wRVAHmFU=Fcel7$$\"3 .+Dc;*[+9\"Fh_n$!3M\"=1OFeju\"Fcel7$$\"33+v$4tFe:\"Fh_n$!3'=;v%)z]qg\" Fcel7$$\"3)*\\P41WDr6Fh_n$!3pWH#Ho?AV\"Fcel7$$\"3!****\\73\"o'=\"Fh_n$ !3AsT<\"ztLA\"Fcel7$$\"3'*\\(o/QJG?\"Fh_n$!3%>RW_G)*ot*Fadl7$$\"3-+voz ;)*=7Fh_n$!3w11(=eSm)pFadl7$$\"3/]PMPj`M7Fh_n$!3?T_9Xq.lTFadl7$$\"31++ +&*44]7Fh_n$!3'>:@gakGC\"Fadl7$$\"33+D1zy*fE\"Fh_n$\"3G&*Qqd!ffx\"Fadl 7$$\"35+]7jZ!>G\"Fh_n$\"3-kD$pUS*\\ZFadl7$$\"35]i:I*zwH\"Fh_n$\"3!\\5> o`Y5e(Fadl7$$\"34+v=(4bMJ\"Fh_n$\"3Rt%z]Z)QA5Fcel7$$\"38]PMP3&zK\"Fh_n $\"33)G&z=#**HC\"Fcel7$$\"3;++]xlWU8Fh_n$\"3.L;Pgp`P9Fcel7$$\"3;+DcJ.1 f8Fh_n$\"3P1#zXq*>B;Fcel7$$\"39+]i&3ucP\"Fh_n$\"3G:GdkB;k[@>)*=Fcel7$$\"30voH\\[J89Fh_n$\"3# e6\"\\fY)***=Fcel7$$\"3X7`W\"pvsT\"Fh_n$\"3!\\f>J\"oz)*=Fcel7$$\"3-]Pf LlB@9Fh_n$\"3#>CmBZIY*=Fcel7$$\"3*\\i!*y@e\"H9Fh_n$\"3Ao&[76$Rx=Fcel7$ $\"38+v=-*zqV\"Fh_n$\"3_>rL*o!Q[=Fcel7$$\"35++]FSC_9Fh_n$\"3#z2db\">sg " 0 "" {MPLTEXT 1 0 37 "simplify(JacobiSN(EllipticF(u,k),k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"uG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sim plify(sin(arcsin(x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "It doesn't work the other directi on because, of course, the arc function is multivalued, so it's unwill ing to make a " }}{PARA 0 "" 0 "" {TEXT -1 14 "choice for us." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "simplify(EllipticF(JacobiSN( u,k),k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*EllipticFG6$-%)JacobiS NG6$%\"uG%\"kGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplif y(arcsin(sin(x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arcsinG6#-%$s inG6#%\"xG" }}}{EXCHG }{EXCHG }{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 97 "Let us look at the over the top case, where sin(theta/2 ) = sn(omega0*t/k;k), with k = 2mgL/E < 1." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "bg:=2*arcsin(JacobiSN(t/k,k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bgG,$*&\"\"#\"\"\"-%'arcsinG6#-%)JacobiSNG6$*&%\"tGF (%\"kG!\"\"F1F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "th:=t- >arcsin(JacobiSN(t/.8,.8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#thGf *6#%\"tG6\"6$%)operatorG%&arrowGF(-%'arcsinG6#-%)JacobiSNG6$,$*&$\"+++ +]7!\"*\"\"\"9$F7F7$\"\")!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "nq8:=evalf(.8*EllipticK(.8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nq8G$\"+AAC'f\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plot(th,t=0..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6%-%'CURVESG6$7fn7$$\"\"!F)F(7$$\"+4x&)*3\"!#5$\" +cNF>FF-7$$\"+uq8Q?F-$\"+a$*eg]F-7$$\"+(RwX5$F-$\"+eH(4k(F-7$$\"+sZ3yT F-$\"+9F%f,\"!\"*7$$\"+]4\\Y_F-$\"+:>#pD\"F?7$$\"+U-/PiF-$\"+7Pqq9F?7$ $\"+fmpisF-$\"+b9\"=o\"F?7$$\"+#*>VB$)F-$\"+6Q=*)=F?7$$\"+Mbw!Q*F-$\"+ Y&)G&3#F?7$$\"+0j$o/\"F?$\"+w'*)oF#F?7$$\"+_>jU6F?$\"+Q^;QCF?7$$\"+j^Z ]7F?$\"+E!)o7EF?7$$\"+)=h(e8F?$\"+E[.#y#F?7$$\"+Q[6j9F?$\"+'Q78%HF?7$$ \"+%R'\\5:F?$\"+ej\"G,$F?7$$\"+\\z(yb\"F?$\"+[@.%3$F?7$$\"+iK'>d\"F?$ \"+31<0JF?7$$\"+w&[ge\"F?$\"+m>IEJF?7$$\"+*)Q8+;F?$\"+C`vNJF?7$$\"+-#> Uh\"F?$\"+Idi9JF?7$$\"+G)*QU;F?$\"+kmMsIF?7$$\"+b/cq;F?$\"+IF,IIF?7$$ \"+')))G=F?$\"+'\\Y%fDF?7$$\"+\"f#=$3#F?$\"+q3y#Q#F?7$$\"+t(pe= #F?$\"+AG:2AF?7$$\"+uI,$H#F?$\"+Gf'[,#F?7$$\"+rSS\"R#F?$\"+MH?H=F?7$$ \"+`?`(\\#F?$\"+,kY=;F?7$$\"++#pxg#F?$\"+JWq(Q\"F?7$$\"+g4t.FF?$\"+G/7 x6F?7$$\"+!Hst!GF?$\"+#ps>S*F-7$$\"+ERW9HF?$\"+-d*Q'oF-7$$\"+KE>>IF?$ \"+%pX3J%F-7$$\"+#RU07$F?$\"+$z-qz\"F-7$$\"+?S2LKF?$!+A4Y95F-7$$\"+$p) =MLF?$!+Uf$3`$F-7$$\"+*=]@W$F?$!+SNByhF-7$$\"+]$z*RNF?$!+)Gt)>&)F-7$$ \"+kC$pk$F?$!+:@u*4\"F?7$$\"+3qcZPF?$!+P>iB8F?7$$\"+/\"fF&QF?$!+9:=Z:F ?7$$\"+0OgbRF?$!+O#*4bF?7$$\"+&)*pp;%F?$!+-c^] @F?7$$\"+ye,tUF?$!+[/OMBF?7$$\"+fO=yVF?$!+OCn3DF?7$$\"+E>#[Z%F?$!+'3_I m#F?7$$\"+(G!e&e%F?$!+)p#yMGF?7$$\"+&)Qk%o%F?$!+%y'=&)HF?7$$\"+8^XPZF? $!+=;lkIF?7$$\"+UjE!z%F?$!+GBGRJF?7$$\"+FM\"3%[F?$!+3sUjIF?7$$\"+60O\" *[F?$!+==P()HF?7$$\"\"&F)$!+5 " 0 "" {MPLTEXT 1 0 31 "plot(JacobiSN(t/.8,.8),t=0..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)F(7$$\"+] i9Rl!#6$\"+Eo.f\")F-7$$\"+WA)GA\"!#5$\"+F:\"*=:F37$$\"+Qeui=F3$\"+o#3X H#F37$$\"+i3&o]#F3$\"+%[!*>0$F37$$\"+pX*y9$F3$\"+)y)4wPF37$$\"+WTAUPF3 $\"+=*fgT%F37$$\"+%*zhdVF3$\"+u].V]F37$$\"+%>fS*\\F3$\"+*)[D]cF37$$\"+ >$f%GcF3$\"+pQz6iF37$$\"+Dy,\"G'F3$\"+)3sFu'F37$$\"+7S$*F37$$\"+:o?&=\"Fdp$\"+F[=-& *F37$$\"+a&4*\\7Fdp$\"+9F=^'*F37$$\"+j=_68Fdp$\"+]_um(*F37$$\"+Wy!eP\" Fdp$\"+sCWh)*F37$$\"+UC%[V\"Fdp$\"+a\"*=E**F37$$\"+J#>&)\\\"Fdp$\"+Fy1 t**F37$$\"+>:mk:Fdp$\"+&=%>(***F37$$\"+w&QAi\"Fdp$\"+<*)4)***F37$$\"+u LU%o\"Fdp$\"+d:3y**F37$$\"+bjm[Fdp$\"+gk$ol*F37$$\"+:K^+?Fd p$\"+J#Q\">&*F37$$\"+7,Hl?Fdp$\"+QF/V$*F37$$\"+4w)R7#Fdp$\"+[+(e:*F37$ $\"+y%f\")=#Fdp$\"+RS>>*)F37$$\"+/-a[AFdp$\"+Vu&Rm)F37$$\"+ial6BFdp$\" +BAIh$)F37$$\"+i@OtBFdp$\"+5'Qz-)F37$$\"+fL'zV#Fdp$\"+%\\@sj(F37$$\"+! *>=+DFdp$\"+1y!*=sF37$$\"+E&4Qc#Fdp$\"+G^AZnF37$$\"+%>5pi#Fdp$\"+?]%\\ B'F37$$\"+bJ*[o#Fdp$\"+!p#>DdF37$$\"+r\"[8v#Fdp$\"+\\p'f4&F37$$\"+Ijy5 GFdp$\"+\\_A%\\%F37$$\"+/)fT(GFdp$\"+290:QF37$$\"+1j\"[$HFdp$\"+p,UKJF 37$$\"\"$F)$\"+IqmoBF3-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$ Q\"t6\"Q!Fb[l-%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 76 "ff:=t->if t%#ffGf*6 #%\"tG6\"6$%)operatorG%&arrowGF(@%29$%$nq8G,$*&\"\"#\"\"\"-%#thG6#F.F3 F3@%2F.,$*&F2F3F/F3F3,&*&F2F3%#PiGF3F3*&F2F3F4F3!\"\"^#F3F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 125 "plot(ff,0..4,labels=[\"omega0*t\",\"theta\"],thick ness=2,title=`angle versus time\\n for one revolution\\n k = 0.8 or E \+ = 5mgL/2`);" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6'- %'CURVESG6$7O7$\"\"!$F(F(7$$\"+m;')=()!#6$\"+e8'p<#!#57$$\"+e'40j\"F0$ \"+c[QeSF07$$\"+<6m$[#F0$\"+3NrYhF07$$\"+( >%F0$\"+f;M?5!\"*7$$\"+\">K'*)\\F0$\"+s4%**>\"FE7$$\"+Dt:5eF0$\"+Z4vz8 FE7$$\"+\"fX(emF0$\"+$Hu(e:FE7$$\"+DCh/vF0$\"+%)f2I#FE7$$\"+ ]*3q3\"FE$\"+=KJXBFE7$$\"+q=\\q6FE$\"+]k!R[#FE7$$\"+fBIY7FE$\"+%=bgg#F E7$$\"+j$[kL\"FE$\"+Ws`ZFFE7$$\"+`Q\"GT\"FE$\"+k/\"\\'GFE7$$\"+s]k,:FE $\"+Ga[**HFE7$$\"+`dF!e\"FE$\"+G>kFE$\"+!H<[i$FE7$$ \"+Uc-)*>FE$\"+5)4.w$FE7$$\"+f`@'3#FE$\"+=7[0RFE7$$\"+nZ)H;#FE$\"+?!*> OSFE7$$\"+Ky*eC#FE$\"+%=\\D=%FE7$$\"+S^bJBFE$\"+_)o*RVFE7$$\"+0TN:CFE$ \"+57d+XFE7$$\"+7RV'\\#FE$\"+g6[iYFE7$$\"+:#fke#FE$\"+2(H*\\[FE7$$\"+` 4NnEFE$\"+3[>D]FE7$$\"+],s`FFE$\"+x0<>_FE7$$\"+zM)>$GFE$\"+C^d+aFE7$$ \"+qfa8iFE7$$\"+](\\_<$FE$ \"+.25SiFE7$$\"+;m,'=$FE$\"+_e,niFE7$$\"+K$3()=$FE$\"+\\]utiFE7$$\"+\\ +S\">$FE$\"+AVZ!G'FE7$%*undefinedGF_y-%'COLOURG6&%$RGBG$\"#5!\"\"F)F)- %+AXESLABELSG6$Q)omega0*t6\"Q&thetaF[z-%&TITLEG6#%gnangle~versus~time| +~for~one~revolution|+~k~=~0.8~or~E~=~5mgL/2G-%*THICKNESSG6#\"\"#-%%VI EWG6$;F)$\"\"%F(%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " th95:=t->arcsin(JacobiSN(t/.95,.95));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "nq95:=evalf(.95*EllipticK(.95));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "ff95:=t->if t " 0 "" {MPLTEXT 1 0 149 "plot(ff9 5,0..2.2*nq95,labels=[\"omega0*t\",\"theta\"],thickness=2,title=`angle versus time\\n for one revolution\\n k = 0.95 or E = 40mgL/19`,color= green);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%th95Gf*6#%\"tG6\"6$%)ope ratorG%&arrowGF(-%'arcsinG6#-%)JacobiSNG6$,$*&$\"+z:j_5!\"*\"\"\"9$F7F 7$\"#&*!\"#F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nq95G$\"+p1^gC !\"*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%ff95Gf*6#%\"tG6\"6$%)operat orG%&arrowGF(@%29$%%nq95G,$*&\"\"#\"\"\"-%%th95G6#F.F3F3@%2F.,$*&F2F3F /F3F3,&*&F2F3%#PiGF3F3*&F2F3F4F3!\"\"^#F3F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6(-%'CURVESG6#7T7$\"\"!$F(F(7$$\"+p o!*z6!#5$\"+1DFyCF-7$$\"+Ev`1AF-$\"+9z83YF-7$$\"+t54hLF-$\"+ObqYpF-7$$ \"+]xHBXF-$\"+-s*\\@*F-7$$\"+h1)*zcF-$\"+1AiO6!\"*7$$\"+&ytBv'F-$\"+s@ HD8FD7$$\"+]ZxiyF-$\"+;R#*3:FD7$$\"+\"G\"e#FD7$$\"+-+f3=FD$\"+cJM#o#FD7$$\"+eR$ >\">FD$\"+58uiFFD7$$\"+[v9K?FD$\"+[\\M^GFD7$$\"+WpbQ@FD$\"+WOGEHFD7$$ \"+;\\IbAFD$\"+]!\\d+$FD7$$\"+%fwkO#FD$\"+#['ozIFD7$$\"+yDZ#[#FD$\"+74 .cJFD7$$\"+,J**)e#FD$\"+EaGEKFD7$$\"+v)*)Qq#FD$\"+5X;.LFD7$$\"+L`BBGFD $\"+7TG&Q$FD7$$\"+/f7FHFD$\"+)=G&fMFD7$$\"+o0LRIFD$\"+K&GNa$FD7$$\"+l* [_:$FD$\"+/\\\\NOFD7$$\"+,DloKFD$\"+gQiJPFD7$$\"+`iPyLFD$\"+qQVJQFD7$$ \"+td?+NFD$\"+#es7&RFD7$$\"+9]n4OFD$\"+e\"z!oSFD7$$\"+'zcls$FD$\"+3dQ. UFD7$$\"+i!pC$QFD$\"+b0EOVFD7$$\"+W\"f#[RFD$\"+r'pM\\%FD7$$\"+(e3s0%FD $\"+V2J`YFD7$$\"+[@4rTFD$\"+f#HI$[FD7$$\"+WTV#G%FD$\"+'y\\6-&FD7$$\"+' *))**)R%FD$\"+`7pI_FD7$$\"+1YE6XFD$\"+g " 0 "" {MPLTEXT 1 0 40 "th995:=t->arcsin(JacobiSN(t/.995,.9 95));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "nq995:=evalf(.995*Elliptic K(.995));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "ff995:=t->if t " 0 "" {MPLTEXT 1 0 154 "plot(ff995,0..2.2*nq995,labels=[ \"omega0*t\",\"theta\"],thickness=2,title=`angle versus time\\n for on e revolution\\n k = 0.995 or E = 400mgL/199`,color=black);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&th995Gf*6#%\"tG6\"6$%)operatorG%&arrowGF( -%'arcsinG6#-%)JacobiSNG6$,$*&$\"+E^-05!\"*\"\"\"9$F7F7$\"$&**!\"$F(F( F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&nq995G$\"+22RyO!\"*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&ff995Gf*6#%\"tG6\"6$%)operatorG%&arrowGF( @%29$%&nq995G,$*&\"\"#\"\"\"-%&th995G6#F.F3F3@%2F.,$*&F2F3F/F3F3,&*&F2 F3%#PiGF3F3*&F2F3F4F3!\"\"^#F3F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6(-%'CURVESG6#7T7$\"\"!$F(F(7$$\"+&)e#Rw\"!#5 $\"+[hLFNF-7$$\"+m$3()H$F-$\"+[`^8lF-7$$\"+%yJZ-&F-$\"+/b:+(*F-7$$\"+0 k>inF-$\"+o0>m7!\"*7$$\"+hJS\"\\)F-$\"+$RyD`\"F?7$$\"+=*f%45F?$\"+(3-I v\"F?7$$\"+z:Yv6F?$\"+e;qa>F?7$$\"+N29Z8F?$\"+!)\\]O@F?7$$\"+8$p#=:F?$ \"+Mm)HH#F?7$$\"+!4'H%p\"F?$\"+#)=0JCF?7$$\"+s0M\\=F?$\"+_F\"e`#F?7$$ \"+NR)Q-#F?$\"+Y7XPEF?7$$\"+\"*R9*>#F?$\"+M7oCFF?7$$\"+H&R!oBF?$\"+'QG pz#F?7$$\"+fGT@DF?$\"+5L7aGF?7$$\"+#e)y.FF?$\"+E^`8HF?7$$\"+UZGeGF?$\" +Es*y&HF?7$$\"+/0+QIF?$\"+;52/IF?7$$\"+0%zq>$F?$\"+G`>TIF?7$$\"+sThrLF ?$\"+)yN!zIF?7$$\"+WD\"y`$F?$\"+ozF8JF?7$$\"+5JA6PF?$\"+-b=[JF?7$$\"+V \"o/(QF?$\"+1/R!=$F?7$$\"+CaBUSF?$\"+/wD;KF?7$$\"+aLl?UF?$\"+-X'eD$F?7 $$\"+(omfP%F?$\"+m!eKH$F?7$$\"+U$4Pa%F?$\"+qSxPLF?7$$\"+=R+ " 0 " " {MPLTEXT 1 0 34 "th3:=t->arcsin(JacobiSN(t/.3,.3));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "nq3:=evalf(.3*EllipticK(.3));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "ff3:=t->if t " 0 "" {MPLTEXT 1 0 144 "p lot(ff3,0..2.2*nq3,labels=[\"omega0*t\",\"theta\"],thickness=2,title=` angle versus time\\n for one revolution\\n k = 0.3 or E = 20mgL/3`,col or=blue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$th3Gf*6#%\"tG6\"6$%)op eratorG%&arrowGF(-%'arcsinG6#-%)JacobiSNG6$,$*&$\"+LLLLL!\"*\"\"\"9$F7 F7$\"\"$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nq3G$\"+ge9C [!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ff3Gf*6#%\"tG6\"6$%)operato rG%&arrowGF(@%29$%$nq3G,$*&\"\"#\"\"\"-%$th3G6#F.F3F3@%2F.,$*&F2F3F/F3 F3,&*&F2F3%#PiGF3F3*&F2F3F4F3!\"\"^#F3F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6(-%'CURVESG6#7T7$\"\"!$F(F(7$$\"+A $eLJ#!#6$\"+*[,@a\"!#57$$\"+H!*>EVF-$\"+YoB$)GF07$$\"+?*[)*e'F-$\"+IT3 !R%F07$$\"+_P]o))F-$\"+?Gs/fF07$$\"+aGj86F0$\"+S!*H4uF07$$\"+T#*)QK\"F 0$\"+%[]6!))F07$$\"+_vfT:F0$\"+QU(Q-\"!\"*7$$\"+)z^nw\"F0$\"+-Z7s6FO7$ $\"+sR=\"*>F0$\"+ncV>8FO7$$\"+]+/AAF0$\"+_[Xq9FO7$$\"+3#y`U#F0$\"+@W.. ;FO7$$\"+9))GaEF0$\"+^Sz^FO7$$\"+HHk0JF0$\" +=%F0$\"+E0*)RFFO7$$ \"+MP\"=U%F0$\"+/pl&)GFO7$$\"+G+yRYF0$\"+?oLCIFO7$$\"+*)\\?n[F0$\"+'ew *oJFO7$$\"+)4_g2&F0$\"+W<\"=I$FO7$$\"+>>K,`F0$\"+Q>G&=H%F O7$$\"+\"3,E'oF0$\"+F?BYWFO7$$\"+J&Hs2(F0$\"+GmT&e%FO7$$\"+77R1tF0$\"+ F'HXt%FO7$$\"+bd/9vF0$\"+q35q[FO7$$\"+?q1TxF0$\"+q@!)=]FO7$$\"+QjnazF0 $\"+Ho;f^FO7$$\"+C'fz<)F0$\"+0qK1`FO7$$\"+[)fiR)F0$\"++?g]aFO7$$\"+_** zC')F0$\"+d2--cFO7$$\"+N7\"\\%))F0$\"+qg;[dFO7$$\"+8o+q!*F0$\"+$Gvy*eF O7$$\"+'[QKH*F0$\"+TF`YgFO7$$\"+[=!eR*F0$\"+%ew[6'FO7$$\"+5_O)\\*F0$\" +V!QK='FO7$$\"+.)Rrb*F0$\"+1#=CA'FO7$$\"+(R9fh*F0$\"+j/ghiFO7$$\"+W!31 j*F0$\"+'>'RriFO7$$\"+$p,`k*F0$\"+a>>\"G'FO7$%*undefinedGFhz-%+AXESLAB ELSG6$Q)omega0*t6\"Q&thetaF][l-%'COLOURG6&%$RGBGF)F)$\"*++++\"!\")-%&T ITLEG6#%hnangle~versus~time|+~for~one~revolution|+~k~=~0.3~or~E~=~20mg L/3G-%*THICKNESSG6#\"\"#-%%VIEWG6$;F)$\"+*3781\"FO%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "pp995:=plot(ff995,0..2.2*nq995,labels=[\"omega0*t\", \"theta\"],thickness=2,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "pp95:=plot(ff95,0..2.2*nq95,labels=[\"omega0*t\",\"th eta\"],thickness=2,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "pp8:=plot(ff,0..4,labels=[\"omega0*t\",\"theta\"],thi ckness=2,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "pp3 :=plot(ff3,0..2.2*nq3,labels=[\"omega0*t\",\"theta\"],thickness=2,colo r=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" } }{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been \+ redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display([pp9 95,pp95,pp8,pp3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6(-%'CURVESG6%7T7$\"\"!$F(F(7$$\"+&)e#Rw\"!#5$\"+[hLFNF-7 $$\"+m$3()H$F-$\"+[`^8lF-7$$\"+%yJZ-&F-$\"+/b:+(*F-7$$\"+0k>inF-$\"+o0 >m7!\"*7$$\"+hJS\"\\)F-$\"+$RyD`\"F?7$$\"+=*f%45F?$\"+(3-Iv\"F?7$$\"+z :Yv6F?$\"+e;qa>F?7$$\"+N29Z8F?$\"+!)\\]O@F?7$$\"+8$p#=:F?$\"+Mm)HH#F?7 $$\"+!4'H%p\"F?$\"+#)=0JCF?7$$\"+s0M\\=F?$\"+_F\"e`#F?7$$\"+NR)Q-#F?$ \"+Y7XPEF?7$$\"+\"*R9*>#F?$\"+M7oCFF?7$$\"+H&R!oBF?$\"+'QGpz#F?7$$\"+f GT@DF?$\"+5L7aGF?7$$\"+#e)y.FF?$\"+E^`8HF?7$$\"+UZGeGF?$\"+Es*y&HF?7$$ \"+/0+QIF?$\"+;52/IF?7$$\"+0%zq>$F?$\"+G`>TIF?7$$\"+sThrLF?$\"+)yN!zIF ?7$$\"+WD\"y`$F?$\"+ozF8JF?7$$\"+5JA6PF?$\"+-b=[JF?7$$\"+V\"o/(QF?$\"+ 1/R!=$F?7$$\"+CaBUSF?$\"+/wD;KF?7$$\"+aLl?UF?$\"+-X'eD$F?7$$\"+(omfP%F ?$\"+m!eKH$F?7$$\"+U$4Pa%F?$\"+qSxPLF?7$$\"+=R+\"G\"e#F?7$$\"+-+f3=F? $\"+cJM#o#F?7$$\"+eR$>\">F?$\"+58uiFF?7$$\"+[v9K?F?$\"+[\\M^GF?7$$\"+W pbQ@F?$\"+WOGEHF?7$$\"+;\\IbAF?$\"+]!\\d+$F?7$$\"+%fwkO#F?$\"+#['ozIF? 7$$\"+yDZ#[#F?$\"+74.cJF?7$$\"+,J**)e#F?$\"+EaGEKF?7$$\"+v)*)Qq#F?$\"+ 5X;.LF?7$$\"+L`BBGF?$\"+7TG&Q$F?7$$\"+/f7FHF?$\"+)=G&fMF?7$$\"+o0LRIF? $\"+K&GNa$F?7$$\"+l*[_:$F?$\"+/\\\\NOF?7$$\"+,DloKF?$\"+gQiJPF?7$$\"+` iPyLF?$\"+qQVJQF?7$$\"+td?+NF?$\"+#es7&RF?7$$\"+9]n4OF?$\"+e\"z!oSF?7$ $\"+'zcls$F?$\"+3dQ.UF?7$$\"+i!pC$QF?$\"+b0EOVF?7$$\"+W\"f#[RF?$\"+r'p M\\%F?7$$\"+(e3s0%F?$\"+V2J`YF?7$$\"+[@4rTF?$\"+f#HI$[F?7$$\"+WTV#G%F? $\"+'y\\6-&F?7$$\"+'*))**)R%F?$\"+`7pI_F?7$$\"+1YE6XF?$\"+g(>%F-$\"+f;M?5F?7$$\"+\">K'*)\\F-$\"+s4%**>\"F?7$$\"+Dt:5eF-$ \"+Z4vz8F?7$$\"+\"fX(emF-$\"+$Hu(e:F?7$$\"+DCh/vF-$\"+%)f2I#F ?7$$\"+]*3q3\"F?$\"+=KJXBF?7$$\"+q=\\q6F?$\"+]k!R[#F?7$$\"+fBIY7F?$\"+ %=bgg#F?7$$\"+j$[kL\"F?$\"+Ws`ZFF?7$$\"+`Q\"GT\"F?$\"+k/\"\\'GF?7$$\"+ s]k,:F?$\"+Ga[**HF?7$$\"+`dF!e\"F?$\"+G>kF?$\"+!H<[i $F?7$$\"+Uc-)*>F?$\"+5)4.w$F?7$$\"+f`@'3#F?$\"+=7[0RF?7$$\"+nZ)H;#F?$ \"+?!*>OSF?7$$\"+Ky*eC#F?$\"+%=\\D=%F?7$$\"+S^bJBF?$\"+_)o*RVF?7$$\"+0 TN:CF?$\"+57d+XF?7$$\"+7RV'\\#F?$\"+g6[iYF?7$$\"+:#fke#F?$\"+2(H*\\[F? 7$$\"+`4NnEF?$\"+3[>D]F?7$$\"+],s`FF?$\"+x0<>_F?7$$\"+zM)>$GF?$\"+C^d+ aF?7$$\"+qfa8iF?7$$\"+](\\_ <$F?$\"+.25SiF?7$$\"+;m,'=$F?$\"+_e,niF?7$$\"+K$3()=$F?$\"+\\]utiF?7$$ \"+\\+S\">$F?$\"+AVZ!G'F?Ffz-Fiz6&F[[lFejlF)F)F\\[l-F$6%7TF'7$$\"+A$eL J#F^[m$\"+*[,@a\"F-7$$\"+H!*>EVF^[m$\"+YoB$)GF-7$$\"+?*[)*e'F^[m$\"+IT 3!R%F-7$$\"+_P]o))F^[m$\"+?Gs/fF-7$$\"+aGj86F-$\"+S!*H4uF-7$$\"+T#*)QK \"F-$\"+%[]6!))F-7$$\"+_vfT:F-$\"+QU(Q-\"F?7$$\"+)z^nw\"F-$\"+-Z7s6F?7 $$\"+sR=\"*>F-$\"+ncV>8F?7$$\"+]+/AAF-$\"+_[Xq9F?7$$\"+3#y`U#F-$\"+@W. .;F?7$$\"+9))GaEF-$\"+^Sz^F?7$$\"+HHk0JF-$ \"+=%F-$\"+E0*)RFF?7$ $\"+MP\"=U%F-$\"+/pl&)GF?7$$\"+G+yRYF-$\"+?oLCIF?7$$\"+*)\\?n[F-$\"+'e w*oJF?7$$\"+)4_g2&F-$\"+W<\"=I$F?7$$\"+>>K,`F-$\"+Q>G&=H% F?7$$\"+\"3,E'oF-$\"+F?BYWF?7$$\"+J&Hs2(F-$\"+GmT&e%F?7$$\"+77R1tF-$\" +F'HXt%F?7$$\"+bd/9vF-$\"+q35q[F?7$$\"+?q1TxF-$\"+q@!)=]F?7$$\"+QjnazF -$\"+Ho;f^F?7$$\"+C'fz<)F-$\"+0qK1`F?7$$\"+[)fiR)F-$\"++?g]aF?7$$\"+_* *zC')F-$\"+d2--cF?7$$\"+N7\"\\%))F-$\"+qg;[dF?7$$\"+8o+q!*F-$\"+$Gvy*e F?7$$\"+'[QKH*F-$\"+TF`YgF?7$$\"+[=!eR*F-$\"+%ew[6'F?7$$\"+5_O)\\*F-$ \"+V!QK='F?7$$\"+.)Rrb*F-$\"+1#=CA'F?7$$\"+(R9fh*F-$\"+j/ghiF?7$$\"+W! 31j*F-$\"+'>'RriF?7$$\"+$p,`k*F-$\"+a>>\"G'F?Ffz-Fiz6&F[[lF)F)FejlF\\[ l-%+AXESLABELSG6%Q)omega0*t6\"Q&thetaF^hn-%%FONTG6#%(DEFAULTG-%%VIEWG6 $;F)$\"+b&fC4)F?Fchn" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "?phase plane" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "?display" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(arcsin(.9));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+:&p( >6!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(arcsin(-.9) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+:&p(>6!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "56 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }