{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 18 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 "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2 " -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 18 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 "R3 Font 3" -1 258 1 {CSTYLE " " -1 -1 "Courier" 1 14 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 "R3 Font 4" -1 259 1 {CSTYLE "" -1 -1 "Courier" 1 14 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 " R3 Font 5" -1 260 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 18 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 261 "" 0 "" {TEXT 256 56 "Guessing the Legendre P olynomial Expansion of a Function" }}{PARA 0 "" 0 "" {TEXT 257 33 "by \+ Corinne Manogue & Kerry Browne" }}{PARA 0 "" 0 "" {TEXT -1 30 "Copyrig ht 2000 Corinne Manogue" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart: with(orthopoly):with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f(z):=(1/8)-z-(33/4)*z^2+(105/8)*z^4;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 1 "\n" }{TEXT 259 39 "Look at the function f(z) plotted bel ow" }{TEXT -1 2 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(f(z),z= -1..1,color=red);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 30 "It is a linear combination of " }{XPPEDIT 18 0 "P( l,z);" "6#-%\"PG6$%\"lG%\"zG" }{TEXT -1 25 " for different values of \+ " }{XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 7 ", i.e\n\n" }{XPPEDIT 18 0 "f(z) = c[0]*P[0](z)+c[1]*P[1](z)+c[2]*P[2](z)+c[3]*P[3](z);" "6#/-%\" fG6#%\"zG,**&&%\"cG6#\"\"!\"\"\"-&%\"PG6#F-6#F'F.F.*&&F+6#F.F.-&F16#F. 6#F'F.F.*&&F+6#\"\"#F.-&F16#F>6#F'F.F.*&&F+6#\"\"$F.-&F16#FF6#F'F.F." }{TEXT -1 1 " " }{TEXT 258 5 "+ ..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 52 "Your job is to guess the values of the c oefficients " }{XPPEDIT 18 0 "c[l];" "6#&%\"cG6#%\"lG" }{TEXT -1 174 " . To make your job a little easier, I have put in only 3 nonzero term s, all of the coefficients are positive or negative integers, and I ha ve not included any terms for n>4." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 260 12 "Look at the " }{XPPEDIT 261 0 "P[l](z);" "6#- &%\"PG6#%\"lG6#%\"zG" }{TEXT 262 3 "'s " }}{PARA 0 "" 0 "" {TEXT -1 36 "You might want to start by plotting " }{XPPEDIT 18 0 "P[l](z);" "6 #-&%\"PG6#%\"lG6#%\"zG" }{TEXT -1 23 " for various values of " } {XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 53 " if you don't remember what \+ this function looks like:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(s ubs(el=2,P(el,z)),z=-1..1,color=blue);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 263 31 "Guess the most important terms " } {TEXT -1 181 " \nPlot your guess and the function f(z) on the same gra ph. Keep guessing more and more terms until the graph for your guess a nd the graph for f(z) completely coincide. For example:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "myguess:=2*P(2,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plotmyguess:=plot(myguess,z=-1..1,color=blue):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plotf:=plot(f(z),z=-1..1,color=red) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display([plotmyguess, plotf]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }