{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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Script MT Bold" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Script MT Bold" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 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 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 38 "Section \+ 1.4: The Gramm-Schmidt Process" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT 264 30 "Maple Packages for Section 1.4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "At this point, we should realize that hav ing a set of orthogonal vectors \{ " }{XPPEDIT 18 0 "theta[1];" "6#&%& thetaG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[2];" "6#&%&thet aG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[3];" "6#&%&thetaG6# \"\"$" }{TEXT -1 124 ", ... \} is an important tool. We understand tha t we can convert these to an orthonormal set by dividing by the norm o f each:" }}{PARA 0 "" 0 "" {TEXT -1 27 " " } {XPPEDIT 18 0 "phi[n] = theta[n]/abs(theta[n]);" "6#/&%$phiG6#%\"nG*&& %&thetaG6#F'\"\"\"-%$absG6#&F*6#F'!\"\"" }{TEXT -1 2 " ." }}{PARA 0 " " 0 "" {TEXT -1 59 "Such a conversion produces a set of orthonormal ve ctors: \{ " }{XPPEDIT 18 0 "phi[1];" "6#&%$phiG6#\"\"\"" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "phi[2];" "6#&%$phiG6#\"\"#" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "phi[3];" "6#&%$phiG6#\"\"$" }{TEXT -1 8 ", ... \}." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "In the p revious Section 1.3, we saw that the closest element in the span of so me finite number of \{ " }{XPPEDIT 18 0 "phi[1];" "6#&%$phiG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[2];" "6#&%$phiG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[3];" "6#&%$phiG6#\"\"$" }{TEXT -1 10 ", \+ ... \} is" }}{PARA 0 "" 0 "" {TEXT -1 55 " \+ approximation for f = " }{XPPEDIT 18 0 "sum(a[p]*phi[p],p);" "6#- %$sumG6$*&&%\"aG6#%\"pG\"\"\"&%$phiG6#F*F+F*" }{TEXT -1 2 " ," }} {PARA 0 "" 0 "" {TEXT -1 5 "where" }}{PARA 0 "" 0 "" {TEXT -1 42 " \+ " }{XPPEDIT 18 0 "a[p];" "6#&%\" aG6#%\"pG" }{TEXT -1 8 " = < f, " }{XPPEDIT 18 0 "phi[p];" "6#&%$phiG6 #%\"pG" }{TEXT -1 3 " >." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 151 "To be sure the conversion to orthonormal vectors \+ is not a problem, we say the redundant: we get the best approximation \+ if the collection is orthogonal," }}{PARA 0 "" 0 "" {TEXT -1 54 " \+ approximation for f = " }{XPPEDIT 18 0 "sum (b[p]*theta[p],p);" "6#-%$sumG6$*&&%\"bG6#%\"pG\"\"\"&%&thetaG6#F*F+F* " }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 6 "choose" }}{PARA 0 "" 0 "" {TEXT -1 40 " " }{XPPEDIT 18 0 "b[p];" "6#&%\"bG6#%\"pG" }{TEXT -1 8 " = < f, " }{XPPEDIT 18 0 " theta[p];" "6#&%&thetaG6#%\"pG" }{TEXT -1 3 " > " }{TEXT 265 2 "/ " } {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(theta[p])^2;" "6#*$-%$absG6#&%&thet aG6#%\"pG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 114 "This result about approximating function s is why Fourier Series is a topic of study for scientists and enginee rs. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 589 " Convinced that orthogonal families are important, we ask how they can \+ be generated and since, as we have seen, there might be several orthog onal families in the same space, how can we know which ones to choose? Regarding the second question, there is good news: for many linear pa rtial differential equations with boundary conditions, we do not have \+ to choose the orthogonal family. It arises naturally with the problem. Concerning the first question, there are many ways to generate orthog onal families. In this section, we present one classical method known \+ as the Gramm-Schmidt Process." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "It is assumed that a linearly independent set " }}{PARA 0 "" 0 "" {TEXT -1 37 " \+ " }{TEXT 256 1 "V" }{TEXT -1 4 " = \{" }{XPPEDIT 18 0 "v[1],v[2], v[3];" "6%&%\"vG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 8 " , ... \}" }}{PARA 0 "" 0 "" {TEXT -1 69 "is given. The Gramm-Schmidt Process gen erates an orthogonal family \{ " }{XPPEDIT 18 0 "theta[1];" "6#&%&thet aG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[2];" "6#&%&thetaG6# \"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[3];" "6#&%&thetaG6#\"\"$ " }{TEXT -1 32 ", ... \} having these properties:" }}{PARA 0 "" 0 "" {TEXT -1 44 "(1) for each integer n, the generation of \{ " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[2];" "6#&%&thetaG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "th eta[3];" "6#&%&thetaG6#\"\"$" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "th eta[n];" "6#&%&thetaG6#%\"nG" }{TEXT -1 8 "\} uses \{" }{XPPEDIT 18 0 "v[1],v[2],v[3];" "6%&%\"vG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 8 " , ... ," }{XPPEDIT 18 0 "v[n];" "6#&%\"vG6#%\"nG" }{TEXT -1 6 "\}, an d" }}{PARA 0 "" 0 "" {TEXT -1 33 "(2) for each n, the collection \{ " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "theta[2];" "6#&%&thetaG6#\"\"#" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "theta[3];" "6#&%&thetaG6#\"\"$" }{TEXT -1 7 ", ..., " } {XPPEDIT 18 0 "theta[n];" "6#&%&thetaG6#%\"nG" }{TEXT -1 39 "\}spans e xactly the same subspace that \{" }{XPPEDIT 18 0 "v[1],v[2],v[3];" "6% &%\"vG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 8 " , ... ," }{XPPEDIT 18 0 "v[n];" "6#&%\"vG6#%\"nG" }{TEXT -1 8 "\} spans." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "Below, we present the process and give an example. We generate an orthogonal family, not ne cessarily an orthonormal family. We begin with the linearly independen t vectors as described in the set " }{TEXT 258 1 "V" }{TEXT -1 36 " an d generate the orthogonal family." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT 259 25 "The Gramm-Schmidt Process" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Take " } {XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 7 " to be " }{XPPEDIT 18 0 "v[1];" "6#&%\"vG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\" \"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "v[1];" "6#&%\"vG6#\"\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Form " }{XPPEDIT 18 0 "theta[2];" "6#&%&thetaG6#\"\"#" } {TEXT -1 6 " from " }{XPPEDIT 18 0 "v[2];" "6#&%\"vG6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "theta[2]; " "6#&%&thetaG6#\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "v[2]-*theta[1]/;" "6#,&&%\"vG6#\"\"#\"\"\"*(-%$<, >G6$&F%6#F'6#&%&thetaG6#F(F(&F16#F(F(-F+6$&F16#F(6#&F16#F(!\"\"F<" } {TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Form " }{XPPEDIT 18 0 "theta[3];" "6#&%&thetaG6#\"\"$" } {TEXT -1 6 " from " }{XPPEDIT 18 0 "v[3];" "6#&%\"vG6#\"\"$" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "theta[2];" "6#&%&thetaG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "theta[3];" "6 #&%&thetaG6#\"\"$" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "v[3]-*theta[1]/-*theta[2]/;" "6#,(&%\"vG6#\"\"$\"\"\"*(-%$<,>G6$&F%6#F'6#&%&thetaG6#F(F(& F16#F(F(-F+6$&F16#F(6#&F16#F(!\"\"F<*(-F+6$&F%6#F'6#&F16#\"\"#F(&F16#F EF(-F+6$&F16#FE6#&F16#FEF = 0. We compute:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " < " }{XPPEDIT 18 0 "theta[2];" "6#&%&thetaG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "th eta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 8 " > = < " }{XPPEDIT 18 0 " v[2]-*theta[1]/;" "6#,&&%\"vG6#\"\"# \"\"\"*(-%$<,>G6$&F%6#F'6#&%&thetaG6#F(F(&F16#F(F(-F+6$&F16#F(6#&F16#F (!\"\"F<" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6# \"\"\"" }{TEXT -1 2 "> " }}{PARA 0 "" 0 "" {TEXT -1 4 "= < " } {XPPEDIT 18 0 "v[2];" "6#&%\"vG6#\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "v[1];" "6#&%\"vG6#\"\"\"" }{TEXT -1 8 " > - <" }{XPPEDIT 18 0 "v[2];" "6#&%\"vG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 4 "> < " }{XPPEDIT 18 0 "theta[1];" "6 #&%&thetaG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "v[1];" "6#&%\"vG6 #\"\"\"" }{TEXT -1 6 "> / < " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6 #\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\" \"\"" }{TEXT -1 1 ">" }}{PARA 0 "" 0 "" {TEXT -1 3 "= 0" }}{PARA 0 "" 0 "" {TEXT -1 46 "The remaining two dot products work similarly." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 178 "To illustrate these i deas, we choose a function in C([-1, 1]) and ask what cubic polynomial makes the best approximation, in the sense that the usual norm is as \+ small as possible" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 23 "An Example \+ in C([-1,1])" }}{PARA 0 "" 0 "" {TEXT -1 62 " We perform the Gramm Schmidt Process on the functions 1, " }{XPPEDIT 18 0 "x;" "6#%\"xG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "x^3;" "6#*$%\"xG\"\"$" }{TEXT -1 42 " in C([-1,1]). \+ Since we form the quotient" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ " }{XPPEDIT 18 0 "*g/;" "6#*(-%$<,>G6$%\"fG6#%\"gG\"\" \"F)F*-F%6$F)6#F)!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "so many times, why not make that a special function of f and g? In th is case, the dot product is an integral from -1 to 1. Thus, for conven ience, we define a function of f and g that we will call P." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "P:=(f,g)->g(x)*int(f(t)*g(t),t=-1.. 1)/int(g(t)^2,t=-1..1);" }}}{PARA 0 "" 0 "" {TEXT -1 29 "Next we defin e the functions " }{XPPEDIT 18 0 "v[0];" "6#&%\"vG6#\"\"!" }{TEXT -1 9 " through " }{XPPEDIT 18 0 "v[3];" "6#&%\"vG6#\"\"$" }{TEXT -1 2 " . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "v[0]:=x->1;\nv[1]:=x->x; \nv[2]:=x->x^2;\nv[3]:=x->x^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "theta[0]:=v[0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " form:=v[1](x)-P(v[1],theta[0]);\ntheta[1]:=unapply(form,x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "form:=v[2](x)-P(v[2],theta[0 ])\n -P(v[2],theta[1]);\ntheta[2]:=unapply(form,x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "form:=v[3](x)-P(v[3],theta[ 0])\n -P(v[3],theta[1])\n -P(v[3],theta[2]);\n theta[3]:=unapply(form,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Re ality check that we did the Gramm Schmidt process correctly. Compute \+ the dot product of each pair of the theta functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "int( theta[1](x)*theta[2](x), x=-1..1 ); \nint( theta[2](x)*theta[3](x), x=-1..1 );\nint( theta[1](x)*theta[3]( x), x=-1..1 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 " We find the \"best approximation\" in C([-1, 1]) fo r the absolute value function with a polynomial of degree three. We fo llow these four steps." }}{PARA 0 "" 0 "" {TEXT -1 48 "STEP 1: Perform the Gramm Schmidt Process on 1, " }{XPPEDIT 18 0 "x;" "6#%\"xG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "x^3;" "6#*$%\"xG\"\"$" }{TEXT -1 2 " ." }}{PARA 0 " " 0 "" {TEXT -1 25 "We did that above. We got" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "theta[0](x);\ntheta[1](x);\ntheta[2](x);\ntheta[ 3](x);" }}}{PARA 0 "" 0 "" {TEXT -1 38 "STEP 2: Form the Fourier coeff icients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "a[0]:=int(abs(t )*theta[0](t),t=-1..1)/int(theta[0](t)^2,t=-1..1);\na[1]:=int(abs(t)*t heta[1](t),t=-1..1)/int(theta[1](t)^2,t=-1..1);\na[2]:=int(abs(t)*thet a[2](t),t=-1..1)/int(theta[2](t)^2,t=-1..1);\na[3]:=int(abs(t)*theta[3 ](t),t=-1..1)/int(theta[3](t)^2,t=-1..1);" }}}{PARA 0 "" 0 "" {TEXT -1 30 "STEP 3: Write down the answer." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "approx:=x->sum(a[p]*theta[p](x),p=0..3);\napprox(x); " }}}{PARA 0 "" 0 "" {TEXT -1 58 "You might like to see the graph of t hese two superimposed." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "pl ot([abs(x),approx(x)],x=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 16 "Unassisted Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 231 "Perhaps it is no surprise that Maple alr eady knows these polynomials on [-1, 1]. After all, polynomial approxi mations for functions are commonly made. Here is how to access these i n Maple. The routine we're interested in is called " }{TEXT 266 1 "P" }{TEXT -1 111 ". We form a listing of the first four for comparison w ith the ones we generated. These polynomials are called " }{TEXT 263 20 "Legendre Polynomials" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "with( orthopoly );" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "theta[0](x);P(0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "theta[1](x);P(1,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "theta[2](x);P(2,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "theta[3](x);P(3,x);" }}}{PARA 0 "" 0 "" {TEXT -1 191 "We see that Maple's listing of the Legendre Polynomials differs from \+ our computations only by a constant factor. If we normalized both Mapl e's listing and ours we would have the same results." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "EMAIL: herod@math.gatech.edu or \+ jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 38 "URL: http://www.math .gatech.edu/~herod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 36 "Copyright \251 2003 by James V. Herod" }}{PARA 256 " " 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }