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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 36 "Calculation of an Integral for F
AME\n" }{TEXT 288 154 "(Integral encountered in the memo by Bob Reasen
berg, \"Bias in the Estimate of Star Coordinates due to Spatial Variat
ion in Detector Sensitivity\", TM97-01)" }}{PARA 19 "" 0 "" {TEXT -1 
15 "Marc A. Murison" }}{PARA 256 "" 0 "" {TEXT -1 83 "Astronomical App
lications Department\nUnited States Naval Observatory\nWashington, DC
" }}{PARA 258 "" 0 "" {TEXT -1 57 "murison@aa.usno.navy.mil\nhttp://aa
.usno.navy.mil/murison/" }}{PARA 257 "" 0 "" {TEXT -1 16 "February 9, \+
1997" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "1.  A Calculation of the \+
Integral " }{XPPEDIT 258 0 "I[v]" "6#&%\"IG6#%\"vG" }{TEXT -1 25 " (eq
. 8) in Memo TM97-01." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Start wit
h a clean slate." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "restart;" "
6#%(restartG" }}}{SECT 1 {PARA 4 "" 0 "1.1.  Assumptions and Definitio
ns." {TEXT -1 34 "1.1.  Assumptions and Definitions." }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 134 "Make some reasonable assumptions on the variab
les we'll be using, in order to help out the integrator as much as pos
sible.  We'll use " }{TEXT 267 1 "g" }{TEXT -1 12 " instead of " }
{XPPEDIT 18 0 "gamma" "6#%&gammaG" }{TEXT -1 8 ", since " }{XPPEDIT 
18 0 "gamma" "6#%&gammaG" }{TEXT -1 40 " in Maple is a reserved numeri
cal value." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "assume( S > 0, g \+
>= 0, g < 1, p,real, lambda > 0, \n        k > 0, phi,real, f,real );
" "6#-%'assumeG6-2\"\"!%\"SG1F'%\"gG2F*\"\"\"%\"pG%%realG2F'%'lambdaG2
F'%\"kG%$phiGF.%\"fGF." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "about
(S,g,p,lambda,k,phi,f);" "6#-%&aboutG6)%\"SG%\"gG%\"pG%'lambdaG%\"kG%$
phiG%\"fG" }}{PARA 6 "" 1 "" {TEXT -1 25 "Originally S, renamed S~:" }
}{PARA 6 "" 1 "" {TEXT -1 47 "  is assumed to be: RealRange(Open(0),in
finity)" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 
25 "Originally g, renamed g~:" }}{PARA 6 "" 1 "" {TEXT -1 40 "  is ass
umed to be: RealRange(0,Open(1))" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}
{PARA 6 "" 1 "" {TEXT -1 25 "Originally p, renamed p~:" }}{PARA 6 "" 
1 "" {TEXT -1 24 "  is assumed to be: real" }}{PARA 6 "" 1 "" {TEXT 
-1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 35 "Originally lambda, renamed lamb
da~:" }}{PARA 6 "" 1 "" {TEXT -1 47 "  is assumed to be: RealRange(Ope
n(0),infinity)" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" 
{TEXT -1 25 "Originally k, renamed k~:" }}{PARA 6 "" 1 "" {TEXT -1 47 
"  is assumed to be: RealRange(Open(0),infinity)" }}{PARA 6 "" 1 "" 
{TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 29 "Originally phi, renamed p
hi~:" }}{PARA 6 "" 1 "" {TEXT -1 24 "  is assumed to be: real" }}
{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 25 "Originall
y f, renamed f~:" }}{PARA 6 "" 1 "" {TEXT -1 24 "  is assumed to be: r
eal" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 77 "For part of the integrand, we need to differentiate the followi
ng expression." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Js := (sin(Pi
*p*S/lambda)-sin(Pi*p*S*g/lambda))/(Pi*p/lambda):" "6#>%#JsG*&,&-%$sin
G6#**%#PiG\"\"\"%\"pGF,%\"SGF,%'lambdaG!\"\"F,-F(6#*,F+F,F-F,F.F,%\"gG
F,F/F0F0F,*(F+F,F-F,F/F0F0" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "d
Js := diff(Js,p);" "6#>%$dJsG-%%diffG6$%#JsG%\"pG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$dJsG,&*&*&,&*&*(-%$cosG6#*&*(%#PiG\"\"\"%#p|irGF1%#S
|irGF1\"\"\"%(lambda|irG!\"\"F1F0F4F3F4F4F5F6F1*&**-F,6#*&**F0F4F2F4F3
F4%#g|irGF1F4F5F6F1F0F4F3F4F=F4F4F5F6!\"\"F1F5F1F4*&F0\"\"\"F2\"\"\"F6
F1*&*&,&-%$sinGF-F1-FFF:F>F1F5F4F4*&F0\"\"\")F2\"\"#F4F6F>" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "dJs := termfunc( dJs, factor );" "6#>%$
dJsG-%)termfuncG6$F$%'factorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dJ
sG,&*&*&%#S|irG\"\"\",&-%$cosG6#*&*(%#PiG\"\"\"%#p|irGF1F(F1F)%(lambda
|irG!\"\"!\"\"*&-F,6#*&**F0F)F2F)F(F)%#g|irGF1F)F3F4F1F;F)F1F1F)F2F4F5
*&*&,&-%$sinGF-F5-F@F8F1F1F3F1F)*&F0\"\"\")F2\"\"#F)F4F1" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 16 "The integrand is" }}}{EXCHG {PARA 0 "" 0 
"" {XPPEDIT 19 1 "integrand := Js*dJs*sin(k*p+phi);" "6#>%*integrandG*
(%#JsG\"\"\"%$dJsGF'-%$sinG6#,&*&%\"kGF'%\"pGF'F'%$phiGF'F'" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%*integrandG*&**,&-%$sinG6#*&*(%#PiG\"\"\"%
#p|irGF.%#S|irGF.\"\"\"%(lambda|irG!\"\"F.-F)6#*&**F-F1F/F1F0F1%#g|irG
F.F1F2F3!\"\"F.F2F1,&*&*&F0F1,&-%$cosGF*F9*&-F?F5F.F8F1F.F.F1F/F3F9*&*
&,&F(F9F4F.F.F2F.F1*&F-\"\"\")F/\"\"#F1F3F.F.-F)6#,&*&%#k|irGF.F/F1F.%
%phi|irGF.F.F1*&F-\"\"\"F/\"\"\"F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 41 "Let's get some idea what this looks like." }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "F := fn(integrand,g,phi,lambda,k,S,p);" "6#>%\"F
G-%#fnG6)%*integrandG%\"gG%$phiG%'lambdaG%\"kG%\"SG%\"pG" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%\"FGR6(%#g|irG%%phi|irG%(lambda|irG%#k|irG%#S|
irG%#p|irG6\"6$%)operatorG%&arrowGF-*&**,&-%$sinG6#*&*(%#PiG\"\"\"9)F:
9(F:\"\"\"9&!\"\"F:-F56#*&**F9F=F;F=F<F=9$F:F=F>F?!\"\"F:F>F=,&*&*&F<F
=,&-%$cosGF6FE*&-FKFAF:FDF=F:F:F=F;F?FE*&*&,&F4FEF@F:F:F>F:F=*&F9\"\"
\")F;\"\"#F=F?F:F:-F56#,&*&9'F:F;F=F:9%F:F:F=*&F9\"\"\"F;\"\"\"F?F-F-F
-" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "dp := 0.000007:" "6#>%#dpG
$\"\"(!\"'" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "plot( F( 0.1, Pi/4, 5500*
10^(-10), 0.1, 0.6, x ), x=-dp..dp, \n      color=blue, numpoints=100,
 axes=normal, thickness=2 );" "6#-%%plotG6(-%\"FG6($\"\"\"!\"\"*&%#PiG
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "Hmm, this appears to be an odd fu
nction, which ought to integrate to zero for integration limits symmet
ric about " }{XPPEDIT 18 0 "p=0" "6#/%\"pG\"\"!" }{TEXT -1 25 ".  Let'
s expand the term " }{XPPEDIT 18 0 "sin(k*p+phi)" "6#-%$sinG6#,&*&%\"k
G\"\"\"%\"pGF)F)%$phiGF)" }{TEXT -1 18 " and consider the " }{XPPEDIT 
18 0 "sin(k*p)" "6#-%$sinG6#*&%\"kG\"\"\"%\"pGF(" }{TEXT -1 5 " and " 
}{XPPEDIT 18 0 "cos(k*p)" "6#-%$cosG6#*&%\"kG\"\"\"%\"pGF(" }{TEXT -1 
18 " terms separately." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "loc :
= location(integrand,sin(k*p+phi)):" "6#>%$locG-%)locationG6$%*integra
ndG-%$sinG6#,&*&%\"kG\"\"\"%\"pGF/F/%$phiGF/" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "subsop( loc=expand(op(loc,integrand)), integrand );" 
"6#-%'subsopG6$/%$locG-%'expandG6#-%#opG6$F'%*integrandGF." }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#*&**,&-%$sinG6#*&*(%#PiG\"\"\"%#p|irGF,%#S|i
rGF,\"\"\"%(lambda|irG!\"\"F,-F'6#*&**F+F/F-F/F.F/%#g|irGF,F/F0F1!\"\"
F,F0F/,&*&*&F.F/,&-%$cosGF(F7*&-F=F3F,F6F/F,F,F/F-F1F7*&*&,&F&F7F2F,F,
F0F,F/*&F+\"\"\")F-\"\"#F/F1F,F,,&*&-F'6#*&%#k|irGF,F-F/F,-F=6#%%phi|i
rGF,F,*&-F=FJF,-F'FNF,F,F,F/*&F+\"\"\"F-\"\"\"F1" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "integrand_s := coeff( %, sin(k*p), 1 )*sin(k*p);
" "6#>%,integrand_sG*&-%&coeffG6%%\"%G-%$sinG6#*&%\"kG\"\"\"%\"pGF/\"
\"\"F/-F+6#*&F.F/F0F/F/" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "integrand_c \+
:= coeff( %%, cos(k*p), 1 )*cos(k*p);" "6#>%,integrand_cG*&-%&coeffG6%
%#%%G-%$cosG6#*&%\"kG\"\"\"%\"pGF/\"\"\"F/-F+6#*&F.F/F0F/F/" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%,integrand_sG*&*,,&-%$sinG6#*&*(%#PiG\"\"
\"%#p|irGF.%#S|irGF.\"\"\"%(lambda|irG!\"\"F.-F)6#*&**F-F1F/F1F0F1%#g|
irGF.F1F2F3!\"\"F.F2F1,&*&*&F0F1,&-%$cosGF*F9*&-F?F5F.F8F1F.F.F1F/F3F9
*&*&,&F(F9F4F.F.F2F.F1*&F-\"\"\")F/\"\"#F1F3F.F.-F?6#%%phi|irGF.-F)6#*
&%#k|irGF.F/F1F.F1*&F-\"\"\"F/\"\"\"F3" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%,integrand_cG*&*,,&-%$sinG6#*&*(%#PiG\"\"\"%#p|irGF.%#S|irGF.
\"\"\"%(lambda|irG!\"\"F.-F)6#*&**F-F1F/F1F0F1%#g|irGF.F1F2F3!\"\"F.F2
F1,&*&*&F0F1,&-%$cosGF*F9*&-F?F5F.F8F1F.F.F1F/F3F9*&*&,&F(F9F4F.F.F2F.
F1*&F-\"\"\")F/\"\"#F1F3F.F.-F)6#%%phi|irGF.-F?6#*&%#k|irGF.F/F1F.F1*&
F-\"\"\"F/\"\"\"F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Guard again
st typos..." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "expand( simplify
( integrand_s + integrand_c - integrand ) );" "6#-%'expandG6#-%)simpli
fyG6#,(%,integrand_sG\"\"\"%,integrand_cGF+%*integrandG!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
72 "Make the two integrand contributions into functions so we can plot
 them." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Fs := fn( integrand_s
, g, phi, lambda, k, S, p ):" "6#>%#FsG-%#fnG6)%,integrand_sG%\"gG%$ph
iG%'lambdaG%\"kG%\"SG%\"pG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "Fc := fn(
 integrand_c, g, phi, lambda, k, S, p ):" "6#>%#FcG-%#fnG6)%,integrand
_cG%\"gG%$phiG%'lambdaG%\"kG%\"SG%\"pG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 13 "The argument " }{XPPEDIT 18 0 "k*p" "6#*&%\"kG\"\"\"%\"pG
F%" }{TEXT -1 95 " is mighty small for the numerical values we have to
 use in order to see the \"wiggles\", so the " }{XPPEDIT 18 0 "sin(k*p
)" "6#-%$sinG6#*&%\"kG\"\"\"%\"pGF(" }{TEXT -1 44 " term ought to be m
iniscule compared to the " }{XPPEDIT 18 0 "cos(k*p)" "6#-%$cosG6#*&%\"
kG\"\"\"%\"pGF(" }{TEXT -1 6 " term." }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "plot( Fs( 0.1, Pi/4, 5500*10^(-10), 0.1, 0.6, x ), \n  \+
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\"\"\"!\"\"$\"\"'!\"\"%\"xG/F=;,$%#dpGF0FA/%%axesG%'normalG/%*numpoint
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ocb@F*$\"1=EN()*Qe@\"F-7$$\"1onn(o(p!>#F*$\"1IX`!\\IrE*F37$$\"1100lj3k
AF*$!1(R'zV(fyU'FM7$$\"1VUUU]ZPBF*$!1u!GT@iM^*F37$$\"1%HHz'=FrBF*$!1G8
&[#G[\\6F-7$$\"1WVV$po]S#F*$!1$)*4LEW^:\"F-7$$\"1%RR*=b')QCF*$!1EWwq!y
kW*F37$$\"1XWWWBmsCF*$!1[&eWy=C5&F37$$\"1**)*)*)[()=a#F*$\"1Vxm47#*G5F
-7$$\"1```LE66EF*$\"1^6&>2(Q5KF-7$$\"1eee=aa(o#F*$\"14i:+@\\\\eF-7$$\"
1jjj.#yRw#F*$\"1OXCP.Xy!)F-7$$\"1www;+e)z#F*$\"1KX-v[h%y)F-7$$\"1*)*)*
)H==LGF*$\"1t)*f#*o3Q#*F-7$$\"1...VOynGF*$\"185JC!e\"3%*F-7$$\"1;;;caQ
-HF*$\"1v()[Y+Yx#*F-7$$\"1OOO\"Hsx$HF*$\"1qrF(fG$H))F-7$$\"1dccE\"fJ(H
F*$\"17BtiDxu!)F-7$$\"1ywwhfa3IF*$\"1`!4)=*>)RqF-7$$\"1)pppzKR/$F*$\"1
(*>n(*\\OjdF-7$$\"1dcc'z#)y=$F*$!1+%pa'\\+(y(F37$$\"1GFFF=/aKF*$!1ibRO
3X=PF-7$$\"1*zzz&3??LF*$!1%350k$z+hF-7$$\"1<::]DYbLF*$!1uopXUV_qF-7$$
\"1MKKUUs!R$F*$!1)RxnUH>v(F-7$$\"1^\\\\Mf)fU$F*$!1'*\\6J#*))*=)F-7$$\"
1ommEwChMF*$!1!G$\\(*zyo$)F-7$$\"1caa/4C'\\$F*$!1VRYVze.$)F-7$$\"1VUU#
=M7`$F*$!1Cto`uX>!)F-7$$\"1JIIguAmNF*$!1=&peb%4ZvF-7$$\"1===Q2A,OF*$!1
)3@$=l2BpF-7$$\"1RQQ)f%3wOF*$!1==R_'QsE&F-7$$\"1feee%[4v$F*$!1O2mb*)>,
NF-7$$\"1777sD,<QF*$!13zaEogA@F-7$$\"1kll&owI)QF*$!1B9:#=*=s5F-7$$\"1P
QQe4IfRF*$!1vj&zN#HZLF37$$\"1566J__NSF*$!1S1B(HZNy$FM7$$\"1555!H)4tTF*
$\"17eubT\"zz#Fbs7$$\"1,,,T'>#4VF*$\"14\"=Q6k;U%F37$$\"1iiiAQfbWF*$\"1
\"fJdUqr(=F-7$$\"1*zzz6gDg%F*$\"1DsvlP-uMF-7$$\"1#>>>!*=%pYF*$\"1d-V5(
f$QQF-7$$\"1&eeeoxit%F*$\"1]g:m[*G\"QF-7$$\"1aaaW(ys![F*$\"11_f!)QP7LF
-7$$\"1BBB.)z#y[F*$\"1'*)3O$fSiBF-7$$\"1766\"**fo,&F*$!1,(oy:'fdEF37$$
\"1xwwwB3n^F*$!10Xue$*3bGF-7$$\"1srrTW2K_F*$!1n1\\n--'\\$F-7$$\"1nmm1l
1(H&F*$!1iV')=wwbPF-7$$\"1111cArr`F*$!1yE^&pa:g$F-7$$\"1XXX0!ejW&F*$!1
6J+U-FiIF-7$$\"1#GGG')*p'e&F*$!1L!4&)>)R!e\"F-7$$\"1\"444<'oDdF*$!1.UZ
(y(*G\"RF37$$\"1kjjVIykeF*$\"1q&*>2N2tLFM7$$\"1444\\Il1gF*$!17gF7@X1EF
M7$$\"1<==yG.dhF*$\"1!p@%fZ#f7*FM7$$\"1wvvbk\"\\H'F*$\"1AH\")oIYCnF37$
$\"1#>>>DD5V'F*$\"19))Geyp$Q\"F-7$$\"1%\\\\\\JBld'F*$\"1NUh8*=?j\"F-7$
$\"1TTTTYq@nF*$\"1AEn'G[ik*F37$$\"1XVV.N,_oF*$!1![bg/`xt\"F37$$\"1++++
+++qF*$!1B>>&R$eM8F--%(SCALINGG6#%.UNCONSTRAINEDG-%'COLOURG6&%$RGBG\"
\"!F^]o$\"*++++\"!\")-%+AXESLABELSG6%Q\"x6\"%!G-%%FONTG6$%*HELVETICAG
\"#=-%*AXESTICKSG6%%(DEFAULTGF`^o-Fi]o6$F[^o\"#7-%*AXESSTYLEG6#%'NORMA
LG-%*THICKNESSG6#\"\"#-%&TITLEG6$Fg]oFh]o-%%VIEWG6$;$!\"(!\"'$\"\"(Fe_
oF`^o" 1 2 0 1 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 25 "However, notice that the " }{XPPEDIT 18 
0 "sin(k*p)" "6#-%$sinG6#*&%\"kG\"\"\"%\"pGF(" }{TEXT -1 42 " part of \+
the integrand is even, while the " }{XPPEDIT 18 0 "cos(k*p)" "6#-%$cos
G6#*&%\"kG\"\"\"%\"pGF(" }{TEXT -1 88 " part of the integrand is odd. \+
 Hence, for integration limits symmetric about zero, the " }{XPPEDIT 
18 0 "cos(k*p)" "6#-%$cosG6#*&%\"kG\"\"\"%\"pGF(" }{TEXT -1 73 " contr
ibution to the integral, despite the overwhelming magnitude of the " }
{XPPEDIT 18 0 "cos(k*p)" "6#-%$cosG6#*&%\"kG\"\"\"%\"pGF(" }{TEXT -1 
19 " term, will vanish." }}}}{SECT 1 {PARA 4 "" 0 "1.2.  Perform the D
efinite Integration." {TEXT -1 39 "1.2.  Perform the Definite Integrat
ion." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Iv := int( integrand, p=
-infinity..infinity );" "6#>%#IvG-%$intG6$%*integrandG/%\"pG;,$%)infin
ityG!\"\"F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#IvG,$*&*(%#k|irG\"\"
\"%(lambda|irGF),fq*,F*F)%#PiGF)F(F)-%'signumG6#,&*&F-\"\"\"%#S|irGF3
\"\"#*&F(F3F*F3!\"\"F3-%$cosG6#%%phi|irGF3F7*,F*F)F-F)F(F)-F/6#,(*(F-F
)F4F)%#g|irGF3F3F6F7F2F7F3F8F)!\"#*,F*F)F-F)F(F)-%$sinGF:F3-F/6#,(F@F3
F6F3F2F7F3FB*,F*F)F-F)F(F)FDF)F.F)F7*,F*F)F-F)F(F)FDF)F=F)FB*,FDF)F4F)
FAF)-%#lnG6#,(F@F7F6F3F2F7F3F-F)F5*,FDF)F4F)FAF)-FM6#,(F@F7F6F7F2F3F3F
-F)F5*,FDF)F4F)FAF)-FMF>F3F-F)F5*,FDF)F4F)FAF)-FM6#,(F@F3F6F7F2F3F3F-F
)FB*,FDF)F4F)FAF)-FMFGF3F-F)FB*,)F-F5F)FDF)F4F)FAF)-F/6#,&F@F5F6F7F3F5
*,FgnF)FDF)F4F)FAF)F=F)F5*,FgnF)FDF)F4F)FAF)FFF)FB*,FgnF)-F/FXF3F8F)F4
F)FAF)FB*,FgnF)FDF)F4F)FAF)F^oF)FB*,FgnF)FhnF)F8F)F4F)FAF)F5*,FgnF)FFF
)F8F)F4F)FAF)F5*,FgnF)F=F)F8F)F4F)FAF)F5*,FDF)F4F)FAF)-FM6#,&F@FBF6F3F
3F-F)FB*,F*F)F-F)F(F)FDF)F^oF)F5*,F*F)F-F)F(F)F^oF)F8F)F5*,F*F)F-F)F(F
)FFF)F8F)F5*,F*F)F-F)F(F)FhnF)F8F)F7*,F*F)F-F)F(F)FDF)FhnF)F7**FDF)F*F
)F(F)FenF)FB**FDF)F*F)F(F)-FMF0F3F7**FDF)F*F)F(F)-FMFinF3F7**FDF)F*F)F
(F)FWF)F5**FDF)F*F)F(F)FUF)FB**FDF)F*F)F(F)FLF)FB**FDF)F*F)F(F)-FM6#,&
F2FBF6F3F3F3**FDF)F*F)F(F)-FM6#,(F@F7F6F3F2F3F3F5**FDF)F*F)F(F)FQF)F5*
*FDF)F*F)F(F)FdoF)F3**FgnF)FDF)F4F)F^oF)FB**FgnF)FDF)F4F)F=F)FB**FDF)F
4F)FQF)F-F)FB**FDF)F4F)FenF)F-F)F5**FDF)F4F)FepF)F-F)FB**FDF)F4F)FLF)F
-F)F5**FDF)F4F)FipF)F-F)F5**FDF)F4F)FUF)F-F)FB**FDF)F4F)F^pF)F-F)F5**F
DF)F4F)FWF)F-F)FB**FgnF)FDF)F4F)FFF)F5**FgnF)F.F)F8F)F4F)F5**FgnF)FFF)
F8F)F4F)FB**FgnF)FDF)F4F)F.F)F5**FgnF)F=F)F8F)F4F)FB**FgnF)F^oF)F8F)F4
F)FB**F*F)F-F)F(F)F8F)!\"%*,FDF)F4F)FAF)F`pF)F-F)F5*,FDF)F4F)FAF)FipF)
F-F)FBF3F)*$)F-\"\"#F)!\"\"#F7\"\")" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "cost(%);" "6#-%%costG6#%\"%G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,*%0multiplicationsG\"$]%%*additionsG\"$S\"%*functionsG
\"$0\"%*divisionsG\"\"\"" }}}}{SECT 1 {PARA 4 "" 0 "1.3.  Simplificati
on of the Result." {TEXT -1 35 "1.3.  Simplification of the Result." }
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Well, that's a mess.  Let's see h
ow much we can clean it up." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "
Iv := collect(Iv,[sin(phi),cos(phi),signum,ln],factor);" "6#>%#IvG-%(c
ollectG6%F$7&-%$sinG6#%$phiG-%$cosG6#F,%'signumG%#lnG%'factorG" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#IvG,&*&,@*&**%#k|irG\"\"\"%(lambda|
irGF+,&*(%#PiG\"\"\"%#S|irGF0%#g|irGF0\"\"#*&F*F0F,F0!\"\"F0-%'signumG
6#F-F0F+F/!\"\"#F5\"\")*&**F*F+F,F+,(F.F0F4F0*&F/F+F1F+F5F0-F76#F>F0F+
F/F9#F0\"\"%*&**F*F+F,F+,(F.F0F4F5F?F0F0-F76#FFF0F+F/F9FB*&**F*F+F,F+,
&F?F3F4F5F0-F76#FKF0F+F/F9F:*&**F*F+F,F+,(F.F0F4F5F?F5F0-F76#FPF0F+F/F
9#F5FC*&**F*F+F,F+F-F+-%#lnGF8F0F+*$)F/\"\"#F+F9F:*&**F*F+F,F+FFF+-FWF
HF0F+*$)F/\"\"#F+F9FB*&**F*F+F,F+F-F+-FW6#,&F.!\"#F4F0F0F+*$)F/\"\"#F+
F9#F0F;*&**F*F+F,F+FKF+-FW6#,&F?F`oF4F0F0F+*$)F/\"\"#F+F9Fdo*&**F*F+F,
F+FFF+-FW6#,(F.F5F4F0F?F5F0F+*$)F/\"\"#F+F9FS*&**F*F+F,F+F>F+-FW6#,(F.
F5F4F5F?F0F0F+*$)F/\"\"#F+F9FS*&**F*F+F,F+FPF+-FW6#,(F.F5F4F0F?F0F0F+*
$)F/\"\"#F+F9FB*&**F*F+F,F+FKF+-FWFMF0F+*$)F/\"\"#F+F9F:*&**F*F+F,F+F>
F+-FWFAF0F+*$)F/\"\"#F+F9FB*&**F*F+F,F+FPF+-FWFRF0F+*$)F/\"\"#F+F9FSF0
-%$sinG6#%%phi|irGF0F0*&,.F(F:F<FSFDFBFIF:FNFS*&*&)F*F3F+)F,F3F+F+F/F9
#F0F3F0-%$cosGFirF0F0" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "cost(%
);" "6#-%%costG6#%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%*addition
sG\"#%)%0multiplicationsG\"$[#%*functionsG\"#A%*divisionsG\"#@" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Iv := convert(Iv,abs);" "6#>%#Iv
G-%(convertG6$F$%$absG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#IvG,&*&,@
*&*(%#k|irG\"\"\"%(lambda|irGF+),&*(%#PiG\"\"\"%#S|irGF1%#g|irGF1\"\"#
*&F*F1F,F1!\"\"F4F+F+*&F0\"\"\"-%$absG6#F.\"\"\"!\"\"#F6\"\")*&*(F*F+F
,F+),(F/F1F5F1*&F0F+F2F+F6F4F+F+*&F0\"\"\"-F:6#FC\"\"\"F=#F1\"\"%*&*(F
*F+F,F+),(F/F1F5F6FDF1F4F+F+*&F0\"\"\"-F:6#FO\"\"\"F=FJ*&*(F*F+F,F+),&
FDF4F5F6F4F+F+*&F0\"\"\"-F:6#FX\"\"\"F=F>*&*(F*F+F,F+),(F/F1F5F6FDF6F4
F+F+*&F0\"\"\"-F:6#F[o\"\"\"F=#F6FK*&**F*F+F,F+F.F1-%#lnGF;F1F+*$)F0\"
\"#F+F=F>*&**F*F+F,F+FOF1-FeoFSF1F+*$)F0\"\"#F+F=FJ*&**F*F+F,F+F.F+-Fe
o6#,&F/!\"#F5F1F1F+*$)F0\"\"#F+F=#F1F?*&**F*F+F,F+FXF1-Feo6#,&FDFdpF5F
1F1F+*$)F0\"\"#F+F=Fhp*&**F*F+F,F+FOF+-Feo6#,(F/F6F5F1FDF6F1F+*$)F0\"
\"#F+F=Fao*&**F*F+F,F+FCF1-Feo6#,(F/F6F5F6FDF1F1F+*$)F0\"\"#F+F=Fao*&*
*F*F+F,F+F[oF1-Feo6#,(F/F6F5F1FDF1F1F+*$)F0\"\"#F+F=FJ*&**F*F+F,F+FXF+
-FeoFfnF1F+*$)F0\"\"#F+F=F>*&**F*F+F,F+FCF+-FeoFHF1F+*$)F0\"\"#F+F=FJ*
&**F*F+F,F+F[oF+-FeoF_oF1F+*$)F0\"\"#F+F=FaoF1-%$sinG6#%%phi|irGF1F1*&
,.F(F>F@FaoFLFJFUF>FhnFao*&*&)F*F4F+)F,F4F+F+F0F=#F1F4F1-%$cosGF]tF1F1
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Now let's make the substituti
on " }{XPPEDIT 18 0 "f = k*lambda/(Pi*S);" "6#/%\"fG*(%\"kG\"\"\"%'lam
bdaGF'*&%#PiGF'%\"SGF'!\"\"" }{TEXT -1 3 ".  " }}}{EXCHG {PARA 0 "" 0 
"" {XPPEDIT 19 1 "collect(subs(lambda = f*Pi*S/k,Iv),[sin, cos, ln]);
" "6#-%(collectG6$-%%subsG6$/%'lambdaG**%\"fG\"\"\"%#PiGF-%\"SGF-%\"kG
!\"\"%#IvG7%%$sinG%$cosG%#lnG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,&*&,
@*&*(%#f|irG\"\"\"%#S|irGF)),&*(%#PiG\"\"\"F*F/%#g|irGF/\"\"#*(F(F/F.F
)F*F)!\"\"F1F)F)-%$absG6#F,!\"\"#F3\"\")*&*(F(F)F*F)),(F-F/F2F/*&F.F)F
*F)F3F1F)F)-F56#F=F7#F/\"\"%*&*(F(F)F*F)),(F-F/F2F3F>F/F1F)F)-F56#FFF7
FA*&*(F(F)F*F)),&F>F1F2F3F1F)F)-F56#FLF7F8*&*(F(F)F*F)),(F-F/F2F3F>F3F
1F)F)-F56#FRF7#F3FB*&**F(F)F*F)F,F/-%#lnGF6F/F)F.F7F8*&**F(F)F*F)FFF/-
FYFHF/F)F.F7FA*&**F(F)F*F)F,F)-FY6#,&F-!\"#F2F/F/F)F.F7#F/F9*&**F(F)F*
F)FLF/-FY6#,&F>F\\oF2F/F/F)F.F7F]o*&**F(F)F*F)FFF)-FY6#,(F-F3F2F/F>F3F
/F)F.F7FU*&**F(F)F*F)F=F/-FY6#,(F-F3F2F3F>F/F/F)F.F7FU*&**F(F)F*F)FRF/
-FY6#,(F-F3F2F/F>F/F/F)F.F7FA*&**F(F)F*F)FLF)-FYFNF/F)F.F7F8*&**F(F)F*
F)F=F)-FYF@F/F)F.F7FA*&**F(F)F*F)FRF)-FYFTF/F)F.F7FUF/-%$sinG6#%%phi|i
rGF/F/*&,.F&F8F:FUFCFAFIF8FOFU*()F(F1F)F.F))F*F1F)#F/F1F/-%$cosGF]qF/F
/" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "L[1] := location(%,coeff(%
,sin(phi),1));" "6#>&%\"LG6#\"\"\"-%)locationG6$%\"%G-%&coeffG6%F+-%$s
inG6#%$phiG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"LG6#\"\"\"7$
F'F'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "L[2] := location(%%,coe
ff(%%,cos(phi),1));" "6#>&%\"LG6#\"\"#-%)locationG6$%#%%G-%&coeffG6%F+
-%$cosG6#%$phiG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"LG6#\"\"
#7$F'\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subsop(L[1] = ma
p(proc (x) options operator, arrow; simplify(factor(x)) end,coeff(%%%,
sin(phi),1)),L[2] = map(proc (x) options operator, arrow; simplify(fac
tor(x)) end,coeff(%%%,cos(phi),1)),%%%);" "6#-%'subsopG6%/&%\"LG6#\"\"
\"-%$mapG6$R6#%\"xG7\"6$%)operatorG%&arrowG6\"-%)simplifyG6#-%'factorG
6#F0F5F5F5-%&coeffG6%%$%%%G-%$sinG6#%$phiG\"\"\"/&F(6#\"\"#-F,6$R6#F07
\"6$F3F4F5-F76#-F:6#F0F5F5F5-F=6%F?-%$cosG6#FC\"\"\"F?" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#,&*&,@**%#f|irG\"\"\"%#PiGF()%#S|irG\"\"#\"\"\"-%$
absG6#,&%#g|irGF,F'!\"\"F(#F3\"\")**F'F-F)F-F*F--F/6#,(F2F(F'F(F3F(F(#
F(\"\"%**F'F-F)F--F/6#,(F2F(F'F3F(F(F(F*F-F:**F'F-F)F--F/6#,&!\"#F(F'F
(F(F*F-F4**-F/6#,(F2F(F'F3F3F(F(F'F-F)F-F*F-#F3F;**F'F-F*F-F1F(,(-%#ln
G6#F)F(-FM6#F+F(-FMF0F(F(F4**F'F-F*F-F?F(,(FLF(FOF(-FMF>F(F(F:**F'F-F*
F-F1F-,(FLF(FOF(-FM6#,&F2FDF'F(F(F(#F(F5**F'F-F*F-FCF(,(FLF(FOF(-FMFBF
(F(F4**F'F-F*F-F?F-,(FLF(FOF(-FM6#,(F2F3F'F(F3F(F(F(FI**F'F-F*F-F9F(,(
FLF(FOF(-FM6#,(F2F3F'F3F(F(F(F(FI**F'F-F*F-FHF(,(FLF(FOF(-FM6#,(F2F3F'
F(F(F(F(F(F:**F'F-F*F-FCF-,(FLF(FOF(-FM6#,&F,F(F'F3F(F(FZ**F'F-F*F-F9F
-,(FLF(FOF(-FMF8F(F(F:**F'F-F*F-FHF-,(FLF(FOF(-FMFGF(F(FIF(-%$sinG6#%%
phi|irGF(F(*&,.F&F4F6FIF<F:F@F4FEFI*()F'F,F-F)F-F*F-#F(F,F(-%$cosGFdpF
(F(" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Pi*f*S^2*subs(Q = Pi,col
lect(coeff(subs(Pi = Q,factor(8*%/(Pi*f))),S,2),[sin, cos, Q, ln],fact
or))/8;" "6#*,%#PiG\"\"\"%\"fGF%%\"SG\"\"#-%%subsG6$/%\"QGF$-%(collect
G6%-%&coeffG6%-F*6$/F$F--%'factorG6#*(\"\")F%%\"%GF%*&F$F%F&F%!\"\"F'
\"\"#7&%$sinG%$cosGF-%#lnGF8F%\"\")F>" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#,$**%#PiG\"\"\"%#f|irGF&)%#S|irG\"\"#\"\"\",&*&,.-%$absG6#,(%#g|irG
F&F'!\"\"F4F&!\"#-F06#,(F3F&F'F&F4F&F*-F06#,&F3F*F'F4F4-F06#,(F3F&F'F4
F&F&F*-F06#,&F5F&F'F&F4*&,6*&,(F'F*F5F&F3F5F&-%#lnG6#,(F3F4F'F&F4F&F&F
&*&,(F'F5F3F5F*F&F&-FG6#,(F3F4F'F4F&F&F&F&*&F;F&-FG6#,&F3F5F'F&F&F&*&,
(F'F5F5F&F3F*F&-FG6#,(F3F4F'F&F&F&F&F&*&,&F*F&F'F4F&-FGF@F&F&*&FAF&-FG
6#FYF&F&*&,(F3F*F'F*F5F&F&-FGF7F&F&*&,(F3F5F'F*F*F&F&-FGF1F&F&*&FRF&-F
GF:F&F&*&,(F*F&F3F*F'F5F&-FGF=F&F&F+F%!\"\"F&F&-%$sinG6#%%phi|irGF&F&*
&,.F<F*F'\"\"%F9F4F6F5F/F5F?F4F&-%$cosGFfoF&F&F&#F&\"\")" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "simplify(subs(f = k*lambda/(Pi*S),%)-Iv);" "6#-%)simpli
fyG6#,&-%%subsG6$/%\"fG*(%\"kG\"\"\"%'lambdaGF.*&%#PiGF.%\"SGF.!\"\"%
\"%GF.%#IvGF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "Iv := %%:" "6#>%#IvG%#%%G" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "cost(Iv);" "6#-%%costG6#%#IvG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,*%0multiplicationsG\"#U%*additionsG\"#o%*fu
nctionsG\"#A%*divisionsG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 
"Much better.  Notice there are both " }{XPPEDIT 18 0 "sin phi" "6#*&%
$sinG\"\"\"%$phiGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos phi" "6#*
&%$cosG\"\"\"%$phiGF%" }{TEXT -1 54 " terms, and that the latter is mu
ch simpler in form.  " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Iv_sin
 := coeff(Iv,sin(phi),1)*sin(phi);" "6#>%'Iv_sinG*&-%&coeffG6%%#IvG-%$
sinG6#%$phiG\"\"\"\"\"\"-F+6#F-F/" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>
%'Iv_sinG,$*,%#PiG\"\"\"%#f|irGF()%#S|irG\"\"#\"\"\",.-%$absG6#,(%#g|i
rGF(F)!\"\"F4F(!\"#-F06#,(F3F(F)F(F4F(F,-F06#,&F3F,F)F4F4-F06#,(F3F(F)
F4F(F(F,-F06#,&F5F(F)F(F4*&,6*&,(F)F,F5F(F3F5F(-%#lnG6#,(F3F4F)F(F4F(F
(F(*&,(F)F5F3F5F,F(F(-FG6#,(F3F4F)F4F(F(F(F(*&F;F(-FG6#,&F3F5F)F(F(F(*
&,(F)F5F5F(F3F,F(-FG6#,(F3F4F)F(F(F(F(F(*&,&F,F(F)F4F(-FGF@F(F(*&FAF(-
FG6#FYF(F(*&,(F3F,F)F,F5F(F(-FGF7F(F(*&,(F3F5F)F,F,F(F(-FGF1F(F(*&FRF(
-FGF:F(F(*&,(F,F(F3F,F)F5F(-FGF=F(F(F-F'!\"\"F(F(-%$sinG6#%%phi|irGF(#
F(\"\")" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Iv_cos := coeff(Iv,c
os(phi),1)*cos(phi);" "6#>%'Iv_cosG*&-%&coeffG6%%#IvG-%$cosG6#%$phiG\"
\"\"\"\"\"-F+6#F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Iv_cosG,$*,%
#f|irG\"\"\")%#S|irG\"\"#\"\"\"%#PiGF(,.-%$absG6#,(%#g|irGF(F'!\"\"F(F
(F+F'\"\"%-F06#,&F3F+F'F4F4-F06#,(F3F(F'F(F4F(!\"#-F06#,(F3F(F'F4F4F(F
<-F06#,&F<F(F'F(F4F(-%$cosG6#%%phi|irGF(#F(\"\")" }}}}{SECT 1 {PARA 4 
"" 0 "1.4.  Characterization of the sin and cos Terms." {TEXT -1 30 "1
.4.  Characterization of the " }{TEXT 259 3 "sin" }{TEXT -1 5 " and " 
}{TEXT 260 3 "cos" }{TEXT -1 7 " Terms." }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "Let's make the coefficients of " }{XPPEDIT 18 0 "Pi*S^2*s
in(phi)/8;" "6#**%#PiG\"\"\"*$%\"SG\"\"#F%-%$sinG6#%$phiGF%\"\")!\"\"
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi*S^2*cos(phi)/8;" "6#**%#PiG\"
\"\"*$%\"SG\"\"#F%-%$cosG6#%$phiGF%\"\")!\"\"" }{TEXT -1 20 " into fun
ctions of (" }{TEXT 268 1 "g" }{TEXT -1 1 "," }{TEXT 269 1 "f" }{TEXT 
-1 2 ")." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "G[s] := fn(coeff(al
gsubs(Pi*sin(phi)*S^2/8 = Q,Iv_sin),Q,1),g,f):" "6#>&%\"GG6#%\"sG-%#fn
G6%-%&coeffG6%-%(algsubsG6$/**%#PiG\"\"\"-%$sinG6#%$phiGF4%\"SG\"\"#\"
\")!\"\"%\"QG%'Iv_sinGF=\"\"\"%\"gG%\"fG" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "G[c] := fn(coeff(algsubs(Pi*cos(phi)*S^2/8 = Q,Iv_cos),
Q,1),g,f):" "6#>&%\"GG6#%\"cG-%#fnG6%-%&coeffG6%-%(algsubsG6$/**%#PiG
\"\"\"-%$cosG6#%$phiGF4%\"SG\"\"#\"\")!\"\"%\"QG%'Iv_cosGF=\"\"\"%\"gG
%\"fG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Notice that the collecti
on of " }{TEXT 261 2 "ln" }{TEXT -1 14 " terms in the " }{TEXT 262 3 "
sin" }{TEXT -1 52 " term make for a very difficult time of keeping the
 " }{TEXT 263 3 "sin" }{TEXT -1 37 " term purely real.  In fact, setti
ng " }{XPPEDIT 18 0 "g=0" "6#/%\"gG\"\"!" }{TEXT -1 9 " we have " }
{XPPEDIT 19 1 "G[s](0,f);" "6#-&%\"GG6#%\"sG6$\"\"!%\"fG" }{TEXT -1 1 
" " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&%#f|irG\"\"\",,-%$absG6#,&F$\"
\"\"F+F+!\"#-F(6#,&F$F+!\"\"F+\"\"%-F(6#F$F0-F(6#,&F,F+F$F+F0*&,>*&-%#
lnG6#,&\"\"#F+F$F0F+F$F+F+*&-F;F5F+F$F%F0*&-F;6#,$F$F0F+F$F%F+*&-F;6#,
&F+F+F$F0F+F$F%!\"%*&-F;F.F+F$F%F1*&-F;F3F+F$F%F0*&-F;F)F+F$F%F,F@F>*&
-F;6#,&F0F+F$F0F+F$F%F>F:F,FQF>FFF1FOF,FKFIF%%#PiG!\"\"F+F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "where it becomes quite clear we're
 in trouble.  There is no real value of " }{TEXT 264 1 "f" }{TEXT -1 
89 " for which this expression does not have an imaginary component.  \+
On the other hand, the " }{TEXT 270 3 "cos" }{TEXT -1 38 " term is com
paratively well-behaved:  " }{XPPEDIT 19 1 "G[c](g,f);" "6#-&%\"GG6#%
\"cG6$%\"gG%\"fG" }{TEXT -1 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&
%#f|irG\"\"\",.-%$absG6#,(%#g|irGF%F$!\"\"F%F%\"\"#F$\"\"%-F(6#,&F+F-F
$F,F,-F(6#,(F+F%F$F%F,F%!\"#-F(6#,(F+F%F$F,F,F%F5-F(6#,&F5F%F$F%F,F%" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Let's take a look at the " }
{TEXT 265 3 "cos" }{TEXT -1 28 " term for various values of " }{TEXT 
266 1 "g" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 301 "
cosplot := proc( gvals::list, frange::range )\n  local p, k;\n  p := [
];\n  for k from 1 to nops(gvals) do\n    p := [ op(p), plot( G[c](gva
ls[k],f), f=frange, \n                        color=mycolors[(k-1 mod \+
10)+1], \n                        thickness=2 ) ];\n  od;\n  plots[dis
play]( p, axes=normal );\nend:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 
1 "cosplot( [seq(0.1*i,i=0..10)], -0.5..2.5 );" "6#-%(cosplotG6$7#-%$s
eqG6$*&$\"\"\"!\"\"\"\"\"%\"iGF./F/;\"\"!\"#5;,$$\"\"&!\"\"!\"\"$\"#D!
\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 667 399 399 {PLOTDATA 2 "64-%'CURVE
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Hmm, disturbing that it is asymmet
ric.  " }}}}}{SECT 1 {PARA 3 "" 0 "2.  Integration Check." {TEXT -1 
22 "2.  Integration Check." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Sinc
e both Mathcad 5.0 and Mathcad 6.0 were unable to do the " }{XPPEDIT 
18 0 "I[v]" "6#&%\"IG6#%\"vG" }{TEXT -1 166 " integral and therefore p
rovide an independent check, we should differentiate the integral and \+
make sure we recover the integrand.  First, do the indefinite integral
." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Iv_indef := int( integrand
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*&F/\"\"\"F'\"\"\"F?FO*&*&)F-F1F(,**&*&-FNF:F0F`yF(F(*&)F-\"\"#F()F>\"
\"#F(F?F[xF6F[xFDF[xFJFexF0F(*&F/\"\"\"F'\"\"\"F?Fao*&*&)FjpF1F(,**&*&
-FNF_qF0F`yF(F(*&)Fjp\"\"#F()F>\"\"#F(F?F[xF\\qF[xFfqFexFjqF[xF0F(*&F/
\"\"\"F'\"\"\"F?Fex*&*&)FUF1F(,**&*&-FNFenF0F`yF(F(*&)FU\"\"#F()F>\"\"
#F(F?F[xFXF[xF\\oF[xF_oFexF0F(*&F/\"\"\"F'\"\"\"F?Fex*&*&)FbsF1F(,**&*
&-FNFgsF0F`yF(F(*&)Fbs\"\"#F()F>\"\"#F(F?F[xFdsF[xF^tFexFbtF[xF0F(*&F/
\"\"\"F'\"\"\"F?F[x*&*&)F_rF1F(,**&*&-FNFdrF0F`yF(F(*&)F_r\"\"#F()F>\"
\"#F(F?F[xFarF[xF[sF[xF^sFexF0F(*&F/\"\"\"F'\"\"\"F?F[x*&*&)FjuF1F(,**
&*&-FNF_vF0F`yF(F(*&)Fju\"\"#F()F>\"\"#F(F?F[xF\\vF[xFfvFexFjvF[xF0F(*
&F/\"\"\"F'\"\"\"F?FO*&*&)FgtF1F(,**&*&-FNF\\uF0F`yF(F(*&)Fgt\"\"#F()F
>\"\"#F(F?F[xFitF[xFcuF[xFfuFexF0F(*&F/\"\"\"F'\"\"\"F?FaoF0F(F/F?" }}
}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "cost(%);" "6#-%%costG6#%\"%G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%0multiplicationsG\"$)f%*divisionsG
\"$M\"%*additionsG\"$L#%*functionsG\"#%*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 27 "Now differentiate this wrt " }{TEXT 257 1 "p" }{TEXT -1 
2 ". " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "diff( Iv_indef, p );" 
"6#-%%diffG6$%)Iv_indefG%\"pG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&*&%
(lambda|irG\"\"\",D*&*(%#S|irGF&,&*&%#PiG\"\"\"F*F.\"\"#*&%#k|irGF.F%F
.!\"\"F&,**&-%$sinG6#,&*&*&F+F.%#p|irGF.F&F%!\"\"F.%%phi|irGF2F&F;F<F.
*&*&-%$cosGF7F.F%F&F&*&F+\"\"\")F;\"\"#F&F<F.*&*&-F66#F9F.-FA6#F=F.F&F
;F<F2*&*&-FAFIF.-F6FKF.F&F;F<F.F.F&F%F<#F.\"\"%*&**F*F&,&F.F.%#g|irGF.
F.,(*(F-F&F*F&FUF.F.F0F2F,F.F&,**&-F66#,&*&*&FVF.F;F&F&F%F<F.F=F2F&F;F
<F.*&*&-FAFenF.F%F&F&*&FV\"\"\")F;\"\"#F&F<F.*&*&-F66#FgnF.FJF&F&F;F<F
2*&*&-FAFcoF.FOF&F&F;F<F.F.F&F%F<#F2FQ*&*(F*F&,&F,F/F0F.F&,**&-F66#,&*
&*&FjoF.F;F&F&F%F<F.F=F.F&F;F<F.*&*&-FAF^pF.F%F&F&*&Fjo\"\"\")F;\"\"#F
&F<F.*&*&-F66#F`pF.FJF&F&F;F<F2*&*&-FAF\\qF.FOF&F&F;F<F2F.F&F%F<Fgo*&*
*F*F&,&FUF.F2F.F.,(FWF.F0F.F,F2F&,**&-F66#,&*&*&FcqF.F;F&F&F%F<F.F=F.F
&F;F<F.*&*&-FAFgqF.F%F&F&*&Fcq\"\"\")F;\"\"#F&F<F.*&*&-F66#FiqF.FJF&F&
F;F<F2*&*&-FAFerF.FOF&F&F;F<F2F.F&F%F<Fgo*&**F*F&FbqF&,(FWF.F0F2F,F2F&
,**&-F66#,&*&*&F[sF.F;F&F&F%F<F.F=F2F&F;F<F.*&*&-FAF_sF.F%F&F&*&F[s\"
\"\")F;\"\"#F&F<F.*&*&-F66#FasF.FJF&F&F;F<F2*&*&-FAF]tF.FOF&F&F;F<F.F.
F&F%F<FP*&**F*F&FTF&,(FWF.F0F.F,F.F&,**&-F66#,&*&*&FctF.F;F&F&F%F<F.F=
F.F&F;F<F.*&*&-FAFgtF.F%F&F&*&Fct\"\"\")F;\"\"#F&F<F.*&*&-F66#FitF.FJF
&F&F;F<F2*&*&-FAFeuF.FOF&F&F;F<F2F.F&F%F<FP*&**F*F&FUF&,&FWF/F0F2F&,**
&-F66#,&*&*&F[vF.F;F&F&F%F<F.F=F2F&F;F<F.*&*&-FAF_vF.F%F&F&*&F[v\"\"\"
)F;\"\"#F&F<F.*&*&-F66#FavF.FJF&F&F;F<F2*&*&-FAF]wF.FOF&F&F;F<F.F.F&F%
F<FP*&**F*F&FUF&,&FWF/F0F.F&,**&-F66#,&*&*&FcwF.F;F&F&F%F<F.F=F.F&F;F<
F.*&*&-FAFgwF.F%F&F&*&Fcw\"\"\")F;\"\"#F&F<F.*&*&-F66#FiwF.FJF&F&F;F<F
2*&*&-FAFexF.FOF&F&F;F<F2F.F&F%F<Fgo*&*(F%F&)F1F/F&,**&-F66#,&*&F1F&F;
F&F.F=F.F&*&)F1\"\"#F&)F;\"\"$F&F<F.*&F^yF&F;F<#F.F/*&*&-F66#FayF.FJF&
F&F;F<#F2F/*&*&-FAF\\zF.FOF&F&F;F<F]zF.F&F-F<F2*&*&)FjoF/F&,**&*&F]pF.
)F%F/F&F&*&)Fjo\"\"#F&)F;\"\"$F&F<F.F\\pFhyFipF]zF]qF]zF.F&*&F-\"\"\"F
%\"\"\"F<FP*&*&)F+F/F&,**&*&F5F.FgzF&F&*&)F+\"\"#F&)F;\"\"$F&F<F.F4Fhy
FFF]zFLFhyF.F&*&F-\"\"\"F%\"\"\"F<Fgo*&*&)FcqF/F&,**&*&FfqF.FgzF&F&*&)
Fcq\"\"#F&)F;\"\"$F&F<F.FeqFhyFbrF]zFfrF]zF.F&*&F-\"\"\"F%\"\"\"F<Fhy*
&*&)FVF/F&,**&*&FZF.FgzF&F&*&)FV\"\"#F&)F;\"\"$F&F<F.FYFhyF`oF]zFdoFhy
F.F&*&F-\"\"\"F%\"\"\"F<Fhy*&*&)FctF/F&,**&*&FftF.FgzF&F&*&)Fct\"\"#F&
)F;\"\"$F&F<F.FetFhyFbuF]zFfuF]zF.F&*&F-\"\"\"F%\"\"\"F<F]z*&*&)F[sF/F
&,**&*&F^sF.FgzF&F&*&)F[s\"\"#F&)F;\"\"$F&F<F.F]sFhyFjsF]zF^tFhyF.F&*&
F-\"\"\"F%\"\"\"F<F]z*&*&)FcwF/F&,**&*&FfwF.FgzF&F&*&)Fcw\"\"#F&)F;\"
\"$F&F<F.FewFhyFbxF]zFfxF]zF.F&*&F-\"\"\"F%\"\"\"F<FP*&*&)F[vF/F&,**&*
&F^vF.FgzF&F&*&)F[v\"\"#F&)F;\"\"$F&F<F.F]vFhyFjvF]zF^wFhyF.F&*&F-\"\"
\"F%\"\"\"F<FgoF.F&F-F<" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "cost
(%);" "6#-%%costG6#%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%0multip
licationsG\"$1'%*divisionsG\"$v\"%*additionsG\"$\\#%*functionsG\"$-\"
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "This had better be equal to t
he original integrand." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "expan
d( simplify(%%-integrand) );" "6#-%'expandG6#-%)simplifyG6#,&%#%%G\"\"
\"%*integrandG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 209 "Whew.  However, this means only t
hat Maple is able to successfully differentiate the result of an integ
ration.  It does not necessarily mean the integration was correct, tho
ugh it increases that probability.  " }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 48 "3.  Another Approach to Evaluating the Integral." }}
{SECT 1 {PARA 4 "" 0 "3.1.  Simplification of the Indefinite Integral.
" {TEXT -1 48 "3.1.  Simplification of the Indefinite Integral." }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Start with the indefinite integrat
ion result and then insert the limits for " }{TEXT 271 1 "p" }{TEXT 
-1 130 ".  First, let's simplify the form of the indefinite integral r
esult.  Recall that the indefinite integral from the last section is" 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Iv_indef := collect(Iv_indef,
[sin(phi),cos(phi),p,lambda,g],factor);" "6#>%)Iv_indefG-%(collectG6%F
$7'-%$sinG6#%$phiG-%$cosG6#F,%\"pG%'lambdaG%\"gG%'factorG" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%)Iv_indefG,**&,&*&*()%#k|irG\"\"#\"\"\",4-%#C
iG6#*&*&,(*(%#PiG\"\"\"%#S|irGF7%#g|irGF7F7*&F+F7%(lambda|irGF7!\"\"*&
F6F-F8F-F<F7%#p|irGF7F-F;!\"\"F,-F06#*&*&,(F5F7F:F7F=F<F7F>F-F-F;F?F,-
F06#*&*&,(F5F7F:F<F=F7F7F>F-F-F;F?!\"#-F06#*&*&,&F5F,F:F7F7F>F-F-F;F?F
7-F06#*&*&,&F=F,F:F<F7F>F-F-F;F?F7-F06#*&*&,&F=F,F:F7F7F>F-F-F;F?F7-F0
6#*&*&,(F5F7F:F7F=F7F7F>F-F-F;F?FJ-F06#*&F+F-F>F-!\"%-F06#*&*&,&F5F,F:
F<F7F>F-F-F;F?F7F7)F;F,F-F-*$)F6\"\"#F-F?#F<\"\")*&,&*&**F8F-F+F-,.FEF
7F@F7FZF<F/F<FKF7F]oF<F7F9F-F-F6F?#F<\"\"%*&*(F8F-F+F-,.FEF7F@F<FPF<F/
F7FUF7FZF<F7F-F6F?F]pF7F;F-F7F7-%$sinG6#%%phi|irGF7F7*&,&*&*(F*F-,4-%#
SiGFVF<-F\\qFjnF^p-F\\qF1F,-F\\qFFFJ-F\\qFenF,-F\\qFAFJ-F\\qFLF<-F\\qF
QF7-F\\qF^oF7F7FboF-F-*$)F6\"\"#F-F?#F7Fgo*&,&*&**F8F-F+F-,.F`qF7FaqF<
F^qF<F_qF7FbqF<FdqF<F7F9F-F-F6F?#F7F^p*&*(F8F-F+F-,.F_qF<FaqF<FcqF7F^q
F<F[qF7F`qF<F7F-F6F?F]pF7F;F-F7F7-%$cosGFdpF7F7*&*(F+F-,4-Fcr6#*&,***F
>F-F6F-F8F-F9F-F7*(F>F-F+F-F;F-F<*(F>F-F6F-F8F-F7*&FepF7F;F-F<F-F;F?FJ
-Fcr6#*&,*F[sF7F\\sF7F]sF7F^sF7F-F;F?FJ-Fcr6#*&,*F[sF7F\\sF<F]sF<F^sF<
F-F;F?F,-Fcr6#*&,(F]sF,F\\sF<F^sF<F-F;F?F7-Fcr6#*&,*F[sF7F\\sF7F]sF<F^
sF7F-F;F?F,-Fcr6#*&,(F[sF,F\\sF7F^sF7F-F;F?F7-Fcr6#*&,(F]sF,F\\sF7F^sF
7F-F;F?F7-Fcr6#,&F[oF7FepF7F\\o-Fcr6#*&,(F[sF,F\\sF<F^sF<F-F;F?F7F7Fbo
F-F-*&)F6\"\"#F-F>\"\"\"F?Ffo*&*&,4-FcpFhtF\\o-FcpFhsF<-FcpF\\tF,-FcpF
dsFJ-FcpFhrF,-FcpF`sFJ-FcpFdtF7-FcpF[uF<-FcpF`tF7F7FboF-F-*&)F6\"\"#F-
)F>\"\"#F-F?Ffo" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Once again, we
 make the substitution " }{XPPEDIT 18 0 "f = k*lambda/(Pi*S)" "6#/%\"f
G*(%\"kG\"\"\"%'lambdaGF'*&%#PiGF'%\"SGF'!\"\"" }{TEXT -1 42 ".  Addit
ionally, we make the substitution " }{XPPEDIT 18 0 "Q = k*p/f" "6#/%\"
QG*(%\"kG\"\"\"%\"pGF'%\"fG!\"\"" }{TEXT -1 3 ".  " }{TEXT 272 1 "Q" }
{TEXT -1 10 " replaces " }{TEXT 273 1 "p" }{TEXT -1 106 " as our indep
endent variable.  Here is a procedure that does the substitutions in t
he function arguments (" }{TEXT 274 3 "sin" }{TEXT -1 2 ", " }{TEXT 
275 3 "cos" }{TEXT -1 2 ", " }{TEXT 276 2 "Ci" }{TEXT -1 2 ", " }
{TEXT 277 2 "Si" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "collect(subs(lambda = f*Pi*S/k,k = Q*f/p,Iv_indef),[S, Q, sin, c
os, Si, Ci],factor);" "6#-%(collectG6%-%%subsG6%/%'lambdaG**%\"fG\"\"
\"%#PiGF-%\"SGF-%\"kG!\"\"/F0*(%\"QGF-F,F-%\"pGF1%)Iv_indefG7(F/F4%$si
nG%$cosG%#SiG%#CiG%'factorG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,**&,
4*(%#f|irG\"\"\",(%#g|irG\"\"\"F(!\"\"F-F,F,-%#CiG6#*&*&,(*(%#PiGF,%#S
|irGF,F+F,F,*(F(F,F5F)F6F)F-*&F5F)F6F)F-F,%\"QGF,F)*&F5\"\"\"F6\"\"\"!
\"\"F,#F,\"\"%*(F(F),(F+F,F(F,F-F,F,-F/6#*&*&,(F4F,F7F,F8F-F,F9F)F)*&F
5\"\"\"F6\"\"\"F=F,#F-F?*(F(F),&F(F,F+\"\"#F,-F/6#*&*&,&F4FMF7F,F,F9F)
F)*&F5\"\"\"F6\"\"\"F=F,#F-\"\")*(F(F),&!\"#F,F(F,F,-F/6#*&*&,&F8FMF7F
-F,F9F)F)*&F5\"\"\"F6\"\"\"F=F,FV*(F(F),&F(F,FMF,F,-F/6#*&*&,&F8FMF7F,
F,F9F)F)*&F5\"\"\"F6\"\"\"F=F,FV*(F(F),(F(F,F,F,F+F,F,-F/6#*&*&,(F4F,F
7F,F8F,F,F9F)F)*&F5\"\"\"F6\"\"\"F=F,F>*&)F(FMF)-F/6#*&F9F)F(F)F,#F,FM
*(F(F),&F+FMF(F-F,-F/6#*&*&,&F4FMF7F-F,F9F)F)*&F5\"\"\"F6\"\"\"F=F,#F,
FW*(F(F),(F+F,F(F-F,F,F,-F/6#*&*&,(F4F,F7F-F8F,F,F9F)F)*&F5\"\"\"F6\"
\"\"F=F,FJF,-%$sinG6#%%phi|irGF,F,*&,4*(F(F)FAF)-%#SiGFCF,FJ*(F(F)FLF)
-FdrFOF,FV*(F(F)FYF)-FdrFfnF,Faq*(F(F)FhpF)-FdrFjpF,FV*(F(F)FcqF)-FdrF
eqF,F>*(F(F)FhoF)-FdrFjoF,F>*(F(F)F^oF)-FdrF`oF,FV*&FbpF)-FdrFdpF,Ffp*
(F(F)F*F)-FdrF0F,FJF,-%$cosGF^rF,F,*&,4*&F(F)-Ffs6#*&*&,***%#p|irGF,F5
F)F6F)F+F)F,**F`tF)F(F)F5F)F6F)F,*(F`tF)F5F)F6F)F-*&**F_rF,F5F)F6F)F`t
F)F)F9F=F,F,F9F)F)*(F5\"\"\"F6\"\"\"F`t\"\"\"F=F,FJ*&F(F)-Ffs6#*&*&,(F
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&,*F_tF,FatF-FbtF-FctF-F,F9F)F)*(F5\"\"\"F6\"\"\"F`t\"\"\"F=F,FJ*&F(F)
-Ffs6#*&*&,(FbtFMFatF-FctF-F,F9F)F)*(F5\"\"\"F6\"\"\"F`t\"\"\"F=F,FV*&
F(F)-Ffs6#*&*&,(FbtFMFatF,FctF,F,F9F)F)*(F5\"\"\"F6\"\"\"F`t\"\"\"F=F,
FV*&F(F)-Ffs6#,&FepF,F_rF,F,Ffp*&F(F)-Ffs6#*&*&,(F_tFMFatF-FctF-F,F9F)
F)*(F5\"\"\"F6\"\"\"F`t\"\"\"F=F,FV*&F(F)-Ffs6#*&*&,*F_tF,FatF-FbtF,Fc
tF-F,F9F)F)*(F5\"\"\"F6\"\"\"F`t\"\"\"F=F,F>*&F(F)-Ffs6#*&*&,*F_tF,Fat
F,FbtF,FctF,F,F9F)F)*(F5\"\"\"F6\"\"\"F`t\"\"\"F=F,F>F)F9F=F,*&,4-F]rF
cwFfp-F]rF_vFaq-F]rF[tFJ-F]rFeuF>-F]rFaxFJ-F]rF[yF>-F]rFivFV-F]rFgwFaq
-F]rF[uFVF)*$)F9\"\"#F)F=F,F,)F6FMF)" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "collect(termfunc(%, factor),[S, Q, sin, cos, Si, Ci],fa
ctor);" "6#-%(collectG6%-%)termfuncG6$%\"%G%'factorG7(%\"SG%\"QG%$sinG
%$cosG%#SiG%#CiGF*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,**&,4*(%#f|ir
G\"\"\",(%#g|irGF)F(!\"\"F)F)\"\"\"-%#CiG6#*&F*F)%\"QGF)F)#F,\"\"%*(F(
F-,&F+\"\"#F(F,F--F/6#*&F6F)F2F-F)#F)\"\")*(F(F-,(F(F)F)F)F+F)F--F/6#*
&F>F)F2F-F)#F)F4*(F(F-,&F(F)F7F)F--F/6#*&FDF)F2F-F)#F,F<*(F(F-,&!\"#F)
F(F)F--F/6#,$*&FJF)F2F-F,F)FH*(F(F-,&F(F)F+F7F--F/6#*&FQF)F2F-F)FH*(F(
F-,(F+F)F(F)F,F)F--F/6#*&FVF)F2F-F)F3*(F(F-,(F+F)F(F,F,F)F--F/6#*&FenF
)F2F-F)FB*&)F(F7F--F/6#*&F2F-F(F-F)#F)F7F)-%$sinG6#%%phi|irGF)F)*&,4*(
F(F-FQF--%#SiGFSF)FH*(F(F-FVF--FgoFXF)F3*(F(F-FenF--FgoFgnF)F3*(F(F-FD
F--FgoFFF)FH*(F(F-F>F--FgoF@F)FB*(F(F-F*F--FgoF0F)FB*(F(F-F6F--FgoF9F)
FH*(F(F-FJF--Fgo6#FOF)FH*&FjnF--FgoF\\oF)F^oF)-%$cosGFaoF)F)*&,4*&F(F-
-Fjp6#,(F2FKF]oF)FboF)F)FH*&F(F--Fjp6#,(*&F+F)F2F-F7F]oF)FboF)F)FH*&F(
F--Fjp6#,(FeqF7F]oF,FboF,F)FH*&F(F--Fjp6#,&F]oF)FboF)F)F^o*&F(F--Fjp6#
,*FeqF)F]oF,F2F,FboF,F)F3*&F(F--Fjp6#,*FeqF)F]oF)F2F)FboF)F)FB*&F(F--F
jp6#,*FeqF)F]oF)F2F,FboF)F)F3*&F(F--Fjp6#,*FeqF)F]oF,F2F)FboF,F)FB*&F(
F--Fjp6#,(F2F7F]oF)FboF)F)FHF-F2!\"\"F)*&,4-F`oFcqFH-F`oFhrF3-F`oF`rFB
-F`oF\\sF3-F`oF_qFH-F`oF\\rF^o-F`oF`sFH-F`oFhqF;-F`oFdrFBF-*$)F2\"\"#F
-FbsF)F))%#S|irGF7F-" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Iv_inde
f := %:" "6#>%)Iv_indefG%\"%G" }}}}{SECT 1 {PARA 4 "" 0 "3.2.  Substit
ution of the Integration Limits into the Indefinite Integral Expressio
n." {TEXT -1 85 "3.2.  Substitution of the Integration Limits into the
 Indefinite Integral Expression." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
30 "Now do the integration limits " }{XPPEDIT 18 0 "Q=-infinity" "6#/%
\"QG,$%)infinityG!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "Q = +infini
ty" "6#/%\"QG%)infinityG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "I1 := collect(limit(Iv_indef,Q = -infinity),[sin, cos, \+
Pi, f]);" "6#>%#I1G-%(collectG6$-%&limitG6$%)Iv_indefG/%\"QG,$%)infini
tyG!\"\"7&%$sinG%$cosG%#PiG%\"fG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
#I1G,&*(,&*(,4-%'signumG6#,&%#f|irG\"\"\"\"\"#F/#!\"\"\"#;-F+6#,(F.F/F
/F/%#g|irGF/#F/\"\")-F+6#,&!\"#F/F.F/#F/F3-F+6#,(F7F/F.F2F/F/F8-F+6#F.
#F/\"\"%-F+6#,(F7F/F.F2F2F/#F2F9-F+6#,&F7F0F.F2F1-F+6#,(F7F/F.F/F2F/FI
-F+6#,&F.F/F7F0F1F/)%#S|irGF0\"\"\")F.F0FUF/*(,:F:FI*&,&F/F/FJF/F/F7F/
F8*&,&F/F/F?F/F/F7FUFI*&,&F/F/FFF/F/F7FUF8*&,&F/F/FMF/F/F7FUFI*&,&F/F/
FPF/F/F7FUFI*&,&F/F/F4F/F/F7FUF8F?FIF4F8FFFIFMF8F*FIF/FSFUF.F/F/F/%#Pi
GF/-%$sinG6#%%phi|irGF/F/*(,&*(,4FFFIFB#F2FEFPF>F*F>FJF1F?F8FMF8F4FIF:
F>F/FSFUFVFUF/*(,:FMFIF?FIFFFI*&F?F/F7FUFI*&FFF/F7FUF8*&F4F/F7FUFI*&FP
F/F7FUF8F*F8*&FMF/F7FUF8F:FIF4FI*&FJF/F7FUF8F/FSFUF.FUF/F/F_oFU-%$cosG
FboF/F/" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "I2 := collect(limit(
Iv_indef,Q = infinity),[sin(phi), cos(phi), Pi, f]);" "6#>%#I2G-%(coll
ectG6$-%&limitG6$%)Iv_indefG/%\"QG%)infinityG7&-%$sinG6#%$phiG-%$cosG6
#F3%#PiG%\"fG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#I2G,&*(,&*(,4-%'si
gnumG6#,(%#g|irG\"\"\"%#f|irG!\"\"F1F/#F/\"\")-F+6#F0#F1\"\"%-F+6#,&F0
F/\"\"#F/#F/\"#;-F+6#,&F0F/F.F;F<-F+6#,&!\"#F/F0F/#F1F=-F+6#,(F0F/F/F/
F.F/#F1F3-F+6#,(F.F/F0F1F/F/FI-F+6#,&F.F;F0F1F<-F+6#,(F.F/F0F/F1F/F2F/
)%#S|irGF;\"\"\")F0F;FUF/*(,:FFFI*&,&F1F/FPF/F/F.F/F2FJF2*&,&F1F/F>F/F
/F.FUF2F8F2*&,&F1F/FFF/F/F.FUFI*&,&F1F/FJF/F/F.FUF2*&,&F1F/FMF/F/F.FUF
IFAF2F*F2FPFI*&,&F1F/F*F/F/F.FUFIF/FSFUF0F/F/F/%#PiGF/-%$sinG6#%%phi|i
rGF/F/*(,&*(,4FJFIF*F2FAFEF8FEFFF2FPFIF>FEF4#F/F7FMF<F/FSFUFVFUF/*(,:*
&FJF/F.FUF2*&FFF/F.FUF2*&F*F/F.FUFIFFF2FAF2*&FMF/F.FUFIFJF2F*F2FPF2*&F
>F/F.FUFIF8FI*&FPF/F.FUFIF/FSFUF0FUF/F/F_oFU-%$cosGFboF/F/" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 20 "We obtain the result" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "Iv2 := collect(algsubs(k*lambda = Pi*f*S,simplif
y(I2-I1)),[S, Pi, sin, cos, signum],factor);" "6#>%$Iv2G-%(collectG6%-
%(algsubsG6$/*&%\"kG\"\"\"%'lambdaGF.*(%#PiGF.%\"fGF.%\"SGF.-%)simplif
yG6#,&%#I2GF.%#I1G!\"\"7'F3F1%$sinG%$cosG%'signumG%'factorG" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%$Iv2G*(,&*&,4*(%#f|irG\"\"\",&F*F+%#g|irG
\"\"#F+-%'signumG6#F,F+#F+\"\")*(F*\"\"\",&F*F+F.F+F+-F06#F6F+F2*(F*F5
,(F-F+F*F+!\"\"F+F+-F06#F:F+#F+\"\"%*(F*F5,(F-F+F*F;F+F+F+-F06#FAF+F>*
(F*F5,&!\"#F+F*F+F+-F06#FEF+#F;F3*(F*F5,&F-F.F*F;F+-F06#FKF+FI*(F*F5,(
F-F+F*F;F;F+F+-F06#FOF+#F;F?*(F*F5,(F*F+F+F+F-F+F+-F06#FTF+FR*&F*F5-%$
absG6#F*F+#F;F.F+-%$sinG6#%%phi|irGF+F+*&,4F)FIF4FIF9FRF@F>FDFIFJFIFNF
RFSF>FW#F+F.F+-%$cosGFhnF+F+F+%#PiGF+)%#S|irGF.F5" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "convert(%/(Pi*S^2),abs);" "6#-%(convertG6$*&%\"%
G\"\"\"*&%#PiGF(*$%\"SG\"\"#F(!\"\"%$absG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,&*&,4*&*&%#f|irG\"\"\"),&F(F)%#g|irG\"\"#F-\"\"\"F.-%$
absG6#F+!\"\"#F)\"\")*&*&F(F.),&F(F)F-F)F-F.F.-F06#F8F2F3*&*&F(F.),(F,
F)F(F)!\"\"F)F-F.F.-F06#F>F2#F)\"\"%*&*&F(F.),(F,F)F(F?F)F)F-F.F.-F06#
FGF2FB*&*&F(F.),&!\"#F)F(F)F-F.F.-F06#FMF2#F?F4*&*&F(F.),&F,F-F(F?F-F.
F.-F06#FUF2FQ*&*&F(F.),(F,F)F(F?F?F)F-F.F.-F06#FenF2#F?FC*&*&F(F.),(F(
F)F)F)F,F)F-F.F.-F06#F\\oF2Fhn*&F(F.-F06#F(F)#F?F-F)-%$sinG6#%%phi|irG
F)F)*&,4F&FQF5FQF;FhnFDFBFJFQFRFQFXFhnFinFBF_o#F)F-F)-%$cosGFeoF)F)" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "L[1] := location(%,coeff(%,sin
(phi),1));" "6#>&%\"LG6#\"\"\"-%)locationG6$%\"%G-%&coeffG6%F+-%$sinG6
#%$phiG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"LG6#\"\"\"7$F'F'
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "L[2] := location(%%,coeff(%
%,cos(phi),1));" "6#>&%\"LG6#\"\"#-%)locationG6$%#%%G-%&coeffG6%F+-%$c
osG6#%$phiG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"LG6#\"\"#7$F
'\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subsop(L[1] = map(pr
oc (x) options operator, arrow; simplify(factor(x)) end,coeff(%%%,sin(
phi),1)),L[2] = map(proc (x) options operator, arrow; simplify(factor(
x)) end,coeff(%%%,cos(phi),1)),%%%);" "6#-%'subsopG6%/&%\"LG6#\"\"\"-%
$mapG6$R6#%\"xG7\"6$%)operatorG%&arrowG6\"-%)simplifyG6#-%'factorG6#F0
F5F5F5-%&coeffG6%%$%%%G-%$sinG6#%$phiG\"\"\"/&F(6#\"\"#-F,6$R6#F07\"6$
F3F4F5-F76#-F:6#F0F5F5F5-F=6%F?-%$cosG6#FC\"\"\"F?" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#,&*&,4*&%#f|irG\"\"\"-%$absG6#,&F'F(%#g|irG\"\"#F(#F(
\"\")*&F'\"\"\"-F*6#,&F'F(F.F(F(F/*&F'F2-F*6#,(F-F(F'F(!\"\"F(F(#F(\"
\"%*&F'F2-F*6#,(F-F(F'F:F(F(F(F;*&F'F2-F*6#,&!\"#F(F'F(F(#F:F0*&-F*6#,
&F-F.F'F:F(F'F2FF*&-F*6#,(F-F(F'F:F:F(F(F'F2#F:F<*&F'F2-F*6#,(F'F(F(F(
F-F(F(FO*&F'F2-F*6#F'F(#F:F.F(-%$sinG6#%%phi|irGF(F(*&,4F&FFF1FFF6FOF=
F;FAFFFGFFFKFOFPF;FT#F(F.F(-%$cosGFZF(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "('Iv') = Pi*f*S^2*collect(8*%/f,[sin, cos, f, g])/8;" "
6#/.%#IvG*,%#PiG\"\"\"%\"fGF(%\"SG\"\"#-%(collectG6$*(\"\")F(%\"%GF(F)
!\"\"7&%$sinG%$cosGF)%\"gGF(\"\")F2" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#/%#IvG,$**%#PiG\"\"\"%#f|irGF()%#S|irG\"\"#\"\"\",&*&,4-%$absG6#,&F)F
(%#g|irGF,F(-F26#,&F)F(F,F(F(-F26#,(F5F(F)F(!\"\"F(F,-F26#,(F5F(F)F<F(
F(F,-F26#,&!\"#F(F)F(F<-F26#,&F5F,F)F<F<-F26#,(F5F(F)F<F<F(FC-F26#,(F)
F(F(F(F5F(FC-F26#F)!\"%F(-%$sinG6#%%phi|irGF(F(*&,4F1F<F6F<F9FCF=F,F@F
<FDF<FGFCFJF,FM\"\"%F(-%$cosGFRF(F(F(#F(\"\")" }}}{EXCHG {PARA 0 "" 0 
"" {XPPEDIT 19 1 "simplify(rhs(%)-convert(Iv2,abs));" "6#-%)simplifyG6
#,&-%$rhsG6#%\"%G\"\"\"-%(convertG6$%$Iv2G%$absG!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Iv
2 := rhs(%%):" "6#>%$Iv2G-%$rhsG6#%#%%G" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 105 "This is not the same as in section 1 \227 how dissapoint
ing.  At least it does not contain those suspicious " }{TEXT 280 2 "ln
" }{TEXT -1 8 " terms. " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Delt
a = collect(simplify(Iv2-Iv),[S, Pi, sin, cos, ln],factor);" "6#/%&Del
taG-%(collectG6%-%)simplifyG6#,&%$Iv2G\"\"\"%#IvG!\"\"7'%\"SG%#PiG%$si
nG%$cosG%#lnG%'factorG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%&DeltaG*&,
&*&,&*(%#f|irG\"\"\",*-%$absG6#,(F*F+F+F+%#g|irGF+\"\"#-F.6#,&F*F+F2F+
!\"\"-F.6#,&F*F+F1F2F6-F.6#F*\"\"%F+-%$sinG6#%%phi|irGF+#F6\"\")*(F*\"
\"\",,F*!\"%F-F2F3F6F7F6F:F<F+-%$cosGF?F+#F+FBF+%#PiGF+F+*&,6*(F*FD,(F
1F+F*F6F+F+F+-%#lnG6#,(F1F6F*F+F6F+F+#F+F<*(F*FD,(F1F+F*F+F6F+F+-FP6#,
(F1F6F*F6F+F+F+FS*(F*FD,&F1F2F*F6F+-FP6#,&F1!\"#F*F+F+FA*(F*FD,(F1F+F*
F6F6F+F+-FP6#,(F1F6F*F+F+F+F+#F6F<*(F*FD,&FhnF+F*F+F+-FP6#F`oF+FI*(F*F
DF`oFD-FP6#,&F2F+F*F6F+FA*(F*FDFUFD-FP6#FUF+F^o*(F*FDFjnFD-FP6#FjnF+FS
*(F*FDFZFD-FP6#FZF+FI*(F*FDFNFD-FP6#FNF+F^oF+F=FDF+F+)%#S|irGF2FD" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 289 3 "sin" }{TEXT -1 
5 " and " }{TEXT 290 3 "cos" }{TEXT -1 10 " parts are" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "Iv2_sin := coeff( Iv2, sin(phi) );" "6#
>%(Iv2_sinG-%&coeffG6$%$Iv2G-%$sinG6#%$phiG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%(Iv2_sinG,$**%#PiG\"\"\"%#f|irGF()%#S|irG\"\"#\"\"\",
4-%$absG6#,&F)F(%#g|irGF,F(-F06#,&F)F(F,F(F(-F06#,(F3F(F)F(!\"\"F(F,-F
06#,(F3F(F)F:F(F(F,-F06#,&!\"#F(F)F(F:-F06#,&F3F,F)F:F:-F06#,(F3F(F)F:
F:F(FA-F06#,(F)F(F(F(F3F(FA-F06#F)!\"%F(#F(\"\")" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "Iv2_cos := coeff( Iv2, cos(phi) );" "6#>%(Iv2_co
sG-%&coeffG6$%$Iv2G-%$cosG6#%$phiG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
>%(Iv2_cosG,$**%#PiG\"\"\"%#f|irGF()%#S|irG\"\"#\"\"\",4-%$absG6#,&F)F
(%#g|irGF,!\"\"-F06#,&F)F(F,F(F4-F06#,(F3F(F)F(F4F(!\"#-F06#,(F3F(F)F4
F(F(F,-F06#,&F;F(F)F(F4-F06#,&F3F,F)F4F4-F06#,(F3F(F)F4F4F(F;-F06#,(F)
F(F(F(F3F(F,-F06#F)\"\"%F(#F(\"\")" }}}}{SECT 1 {PARA 4 "" 0 "3.3.  Ch
aracterization of the sin and cos Terms." {TEXT -1 30 "3.3.  Character
ization of the " }{TEXT 285 3 "sin" }{TEXT -1 5 " and " }{TEXT 286 3 "
cos" }{TEXT -1 7 " Terms." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "As be
fore, define the coefficients of " }{XPPEDIT 18 0 "Pi/8*S^2*sin(phi)" 
"6#**%#PiG\"\"\"\"\")!\"\"%\"SG\"\"#-%$sinG6#%$phiGF%" }{TEXT -1 5 " a
nd " }{XPPEDIT 18 0 "Pi/8*S^2*cos(phi)" "6#**%#PiG\"\"\"\"\")!\"\"%\"S
G\"\"#-%$cosG6#%$phiGF%" }{TEXT -1 17 " as functions of " }{TEXT 279 
1 "g" }{TEXT -1 5 " and " }{TEXT 278 1 "f" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "G2[s] := fn(collect(8*Iv2_sin/(Pi*S^2),
[f, g]),g,f):" "6#>&%#G2G6#%\"sG-%#fnG6%-%(collectG6$*(\"\")\"\"\"%(Iv
2_sinGF0*&%#PiGF0*$%\"SG\"\"#F0!\"\"7$%\"fG%\"gGF:F9" }}}{EXCHG {PARA 
0 "" 0 "" {XPPEDIT 19 1 "G2[c] := fn(collect(8*Iv2_cos/(Pi*S^2),[f, g]
),g,f):" "6#>&%#G2G6#%\"cG-%#fnG6%-%(collectG6$*(\"\")\"\"\"%(Iv2_cosG
F0*&%#PiGF0*$%\"SG\"\"#F0!\"\"7$%\"fG%\"gGF:F9" }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 19 1 "G2[c](g,f);" "6#-&%#G2G6#%\"cG6$%\"gG%\"fG" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#*&%#f|irG\"\"\",4-%$absG6#,&F$F%%#g|ir
G\"\"#!\"\"-F(6#,&F$F%F,F%F--F(6#,(F+F%F$F%F-F%!\"#-F(6#,(F+F%F$F-F%F%
F,-F(6#,&F4F%F$F%F--F(6#,&F+F,F$F-F--F(6#,(F+F%F$F-F-F%F4-F(6#,(F$F%F%
F%F+F%F,-F(6#F$\"\"%F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "G2[s]
(g,f);" "6#-&%#G2G6#%\"sG6$%\"gG%\"fG" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#*&%#f|irG\"\"\",4-%$absG6#,&F$F%%#g|irG\"\"#F%-F(6#,&F$F%F,F%F%-F(6
#,(F+F%F$F%!\"\"F%F,-F(6#,(F+F%F$F3F%F%F,-F(6#,&!\"#F%F$F%F3-F(6#,&F+F
,F$F3F3-F(6#,(F+F%F$F3F3F%F:-F(6#,(F$F%F%F%F+F%F:-F(6#F$!\"%F%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "g = 0" "6#/%
\"gG\"\"!" }{TEXT -1 58 " these of course reduce to much simpler expre
ssions.  The " }{TEXT 281 3 "cos" }{TEXT -1 13 " term becomes" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "G2[c](0,f);" "6#-&%#G2G6#%\"cG6$
\"\"!%\"fG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#f|irG\"\"\",(-%$absG
6#F$\"\"#-F(6#,&F$F%F*F%!\"\"-F(6#,&!\"#F%F$F%F.F%" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 47 "This is identical to eq. (11) in TM97-01.  For " 
}{XPPEDIT 18 0 "abs(f)>2" "6#2\"\"#-%$absG6#%\"fG" }{TEXT -1 43 " this
 expression is identically zero.  The " }{TEXT 282 3 "sin" }{TEXT -1 
13 " term becomes" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "G2[s](0,f)
;" "6#-&%#G2G6#%\"sG6$\"\"!%\"fG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&
%#f|irG\"\"\",,-%$absG6#F$!\"%-F(6#,&F$F%\"\"#F%F%-F(6#,&F$F%!\"\"F%\"
\"%-F(6#,&!\"#F%F$F%F2-F(6#,&F$F%F%F%F*F%" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "Let's look at some plots.   " }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 443 "Gplot := proc( F, vals::list, frange::range )\n  loc
al p, k, Grange;\n  if nargs=3 then\n    Grange := args[3];\n  else\n \+
   Grange := -2.5..2.5;\n  fi;\n  p := [];\n  for k from 1 to nops(gva
ls) do\n    p := [ op(p), plot( F(vals[k],f), f=frange, \n            \+
            color=mycolors[(k-1 mod 10)+1], \n                        \+
view=[frange,Grange], numpoints=200, \n                        thickne
ss=2 ) ];\n  od;\n  plots[display](p,axes=normal);\nend:" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "gvals := [seq(0.1*i,i=0..10)]:" "6#>%&g
valsG7#-%$seqG6$*&$\"\"\"!\"\"\"\"\"%\"iGF-/F.;\"\"!\"#5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 9 "Plot the " }{TEXT 283 3 "cos" }{TEXT -1 
11 " term only:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Gplot( G2[c]
, gvals, -2.5..2.5 );" "6#-%&GplotG6%&%#G2G6#%\"cG%&gvalsG;,$$\"#D!\"
\"!\"\"$\"#D!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 671 340 340 {PLOTDATA 
2 "64-%'CURVESG6%7iw7$$!1+++++++D!#:\"\"!7$$!1sV([(>rtCF*F+7$$!1pQx/*Q
3X#F*F+7$$!1IhAXd6DCF*F+7$$!1(Rze2A#*R#F*F+7$$!1;Jiu9XtBF*F+7$$!1g>RG)
e&\\BF*F+7$$!1yc8x$>[K#F*F+7$$!1\"Gc7!QB*H#F*F+7$$!1;Kky-ttAF*F+7$$!1r
U&3x'\\ZAF*F+7$$!1tX\"HG!RCAF*F+7$$!1c7D]yP)>#F*F+7$$!1,-/3'eA<#F*F+7$
$!1g>Ryy3Z@F*F+7$$!1iBZW/BC@F*F+7$$!1=OsW20(4#F*F+7$$!1Z$pQ(f-u?F*F+7$
$!1V&3<pUs/#F*F+7$$!1JhA?))QN?F*F+7$$!1=Pu[\\`B?F*F+7$$!1]**)HPH0,#F*F
+7$$!1\"=OszBv*>F*$\"1g<-&*ya#*)*!#=7$$!1Ryc8%R^)>F*$\"1QF/nr1+f!#<7$$
!1(\\**)H]vs>F*$\"1gX5OI&\\2\"!#;7$$!1!*zf>9\"p%>F*$\"1'R)po_<n?Ffp7$$
!1'>Ry\"*yJ#>F*$\"1F]98P\"[&HFfp7$$!19Fa3-e(*=F*$\"1n$[YQ&*p)QFfp7$$!1
\\)pRM!*4(=F*$\"1ncA9k^F[Ffp7$$!1nMp)yVy%=F*$\"1M\"y%*>=Ki&Ffp7$$!1sV(
[([%G#=F*$\"1*ffyBE&ekFfp7$$!1pPv]&=qz\"F*$\"1Ts3&fG_H(Ffp7$$!1zd:JDvr
<F*$\"1yH481'z3)Ffp7$$!1,-/ejIZ<F*$\"1;QlWTmI))Ffp7$$!1pPv]I;?<F*$\"1$
[!e1OIF'*Ffp7$$!1[&4>etdp\"F*$\"1;NVX#)zJ5F*7$$!1_/4=Etp;F*$\"1=9`Sk\"
H5\"F*7$$!16@UMc8Y;F*$\"1t)fQ[<];\"F*7$$!1iC\\)*yL?;F*$\"16zQ$yh.B\"F*
7$$!1OsWRU1'f\"F*$\"1xV!\\%[T*G\"F*7$$!1S!3;F\"pq:F*$\"1'z?]%Hi[8F*7$$
!1X*)ydX)ea\"F*$\"1O^8*4?SS\"F*7$$!1NpQFU\"*>:F*$\"1%H@BRy$f9F*7$$!19G
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{TEXT -1 79 "These curves appear the same as those plotted in TM97-01.
  Hence, the function " }{XPPEDIT 18 0 "G[c]" "6#&%\"GG6#%\"cG" }
{TEXT -1 30 " is probably identical to the " }{TEXT 287 1 "G" }{TEXT 
-1 50 " function in Appendix A of TM97-01.  Now plot the " }{TEXT 284 
3 "sin" }{TEXT -1 11 " term only:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
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