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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 25 "Heaviside's step function" }}
{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}
{PARA 0 "" 0 "" {TEXT -1 19 "Version:  27.3.2007" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 73 "The integral of the Dirac delta function and the Heavis
ide step function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Int(delta(x),x = -
epsilon .. epsilon) = 1;" "6#/-%$IntG6$-%&deltaG6#%\"xG/F*;,$%(epsilon
G!\"\"F.\"\"\"" }{TEXT -1 39 ", for any (small) positive real number \+
" }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 16 ",  the functio
n " }{XPPEDIT 18 0 "Int(delta(t),t = -infinity .. x)" "6#-%$IntG6$-%&d
eltaG6#%\"tG/F);,$%)infinityG!\"\"%\"xG" }{TEXT -1 9 "  is the " }
{TEXT 261 18 "Heaviside function" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "H(x
)=PIECEWISE([1, 0 < x],[0, x < 0])" "6#/-%\"HG6#%\"xG-%*PIECEWISEG6$7$
\"\"\"2\"\"!F'7$F.2F'F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 22 "No value is given for " }{XPPEDIT 
18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 76 ", but it can be assigned an
y finite value without affecting the discussion. " }}{PARA 0 "" 0 "" 
{TEXT -1 27 "A specific choice of value " }{XPPEDIT 18 0 "H(0)=1/2" "6
#/-%\"HG6#\"\"!*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 18 "  could be used.  \+
" }{XPPEDIT 18 0 "H(0)=0" "6#/-%\"HG6#\"\"!F'" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "H(0)=1" "6#/-%\"HG6#\"\"!\"\"\"" }{TEXT -1 30 " are oth
er convenient choices." }}{PARA 0 "" 0 "" {TEXT -1 33 "Whatever choice
 is made, we have " }{XPPEDIT 18 0 "Limit(H(x),x=0,left)=0" "6#/-%&Lim
itG6%-%\"HG6#%\"xG/F*\"\"!%%leftGF," }{TEXT -1 6 " and  " }{XPPEDIT 
18 0 "Limit(H(x),x=0,right)=1" "6#/-%&LimitG6%-%\"HG6#%\"xG/F*\"\"!%&r
ightG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "alias(delta=Dirac): alias(H=
Heaviside):\nInt(delta(t),t=-infinity..x);\nvalue(%);\n\n" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%$IntG6$-%&deltaG6#%\"tG/F);,$%)infinityG!\"\"
%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"HG6#%\"xG" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(H(
x),x=-2..2,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 523 168 168 
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-1 6 "Since " }{XPPEDIT 18 0 "Int(delta(t),t = -infinity .. x)=H(x)" "
6#/-%$IntG6$-%&deltaG6#%\"tG/F*;,$%)infinityG!\"\"%\"xG-%\"HG6#F0" }
{TEXT -1 27 ", it makes sense to regard " }{XPPEDIT 18 0 "delta(x)" "6
#-%&deltaG6#%\"xG" }{TEXT -1 28 " as being the derivative of " }
{XPPEDIT 18 0 "H(x)" "6#-%\"HG6#%\"xG" }{TEXT -1 64 ". Thus we think o
f H(x) as having infinite gradient where x = 0." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Diff(H(x),x)
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" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Although " }
{XPPEDIT 18 0 "H(x)" "6#-%\"HG6#%\"xG" }{TEXT -1 121 " is clearly disc
ontinuous at 0 it can be regarded as being the limit of a family of co
ntinuous functions in various ways." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 19 "1st description of " }{XPPEDIT 18 0 "H(x)" "6#-%\"HG6#%
\"xG" }{TEXT -1 35 " as a limit of continuous functions" }}{EXCHG 
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{MPLTEXT 1 0 180 "s := (n,x) -> piecewise(x<=0,exp(n*x)/2,x>0,1-exp(-n
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****F17$Fct$\"0*f&p*********F17$Fht$\"0t&)***********F17$F]u$\"0#*****
*********F17$FbuFe_o7$FguFe_o7$F\\vFe_o7$FavFe_o7$FfvFe_o7$F[wFe_oFd_o
Fg_oFh_oFi_oFj_oF[`oF\\`oF]`oF^`oF_`oF``oFa`o-F\\[l6&F^[lF*F_[lF_[l-%+
AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu
rve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT 
-1 69 "The derivatives of this sequence of functions gives exactly seq
uence " }{XPPEDIT 18 0 "p[n](x) = n*exp(-n*abs(x))/2" "6#/-&%\"pG6#%\"
nG6#%\"xG*(F(\"\"\"-%$expG6#,$*&F(F,-%$absG6#F*F,!\"\"F,\"\"#F5" }
{TEXT -1 51 ", whose limit provides one possible realisation of " }
{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 35 "The step function as a distribution" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Th
e " }{TEXT 261 18 "Heaviside function" }{TEXT -1 1 " " }{XPPEDIT 18 0 
"H(x)" "6#-%\"HG6#%\"xG" }{TEXT -1 11 ", given by " }{XPPEDIT 18 0 "H(
x)=PIECEWISE([1, 0 < x],[0, x < 0])" "6#/-%\"HG6#%\"xG-%*PIECEWISEG6$7
$\"\"\"2\"\"!F'7$F.2F'F." }{TEXT -1 37 ", can be realised as a distrib
ution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 
"Let " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 51 " be an in
finitely differentiable function such that" }}{PARA 0 "" 0 "" {TEXT 
-1 5 " (i) " }{XPPEDIT 18 0 "Limit(h(x),x=infinity)=1" "6#/-%&LimitG6$
-%\"hG6#%\"xG/F*%)infinityG\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 6 " (ii) " }{XPPEDIT 18 0 "Limit(h(x),x=-infinity)=0" "6#/-%&
LimitG6$-%\"hG6#%\"xG/F*,$%)infinityG!\"\"\"\"!" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }
{XPPEDIT 18 0 "H(x)" "6#-%\"HG6#%\"xG" }{TEXT -1 93 " can be represent
ed as a distribution by the sequence of infinitely differentiable func
tions " }{XPPEDIT 18 0 "h[n](x) = h(n*x)" "6#/-&%\"hG6#%\"nG6#%\"xG-F&
6#*&F(\"\"\"F*F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 76 "1st representation of H(x) by a sequence infinitely dif
ferentiable functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }
}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "H(x)" "6#-%\"HG6#%\"x
G" }{TEXT -1 92 " can be represented as a distribution by the sequence
 of infinitely differentiable functions" }}{PARA 256 "" 0 "" {TEXT -1 
2 "  " }{XPPEDIT 18 0 "h[n](x) = 1/2+1/Pi;" "6#/-&%\"hG6#%\"nG6#%\"xG,
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}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
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jrzFM$\"1NGm))H^R**Fhp7$$\"1&\\(=n3T)z)FM$\"1E7;qvFk**Fhp7$Fd^m$\"1&yd
(RX#*y**Fhp7$$\"1+v=nY(y7\"Fhp$\"1a#)e5cn#***Fhp7$Fds$\"1@0ZHoX(***Fhp
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{SECT 1 {PARA 4 "" 0 "" {TEXT -1 76 "3rd representation of H(x) by a s
equence infinitely differentiable functions" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"H(x)" "6#-%\"HG6#%\"xG" }{TEXT -1 93 " can be represented as a distri
bution by the sequence of infinitely differentiable functions " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h[n](x) = PIECEWISE([
0, x <= -1/n],[(1+tanh(n*x/(1-n^2*x^2)))/2, -1/n < x and x < 1/n],[1, \+
1/n <= x]);" "6#/-&%\"hG6#%\"nG6#%\"xG-%*PIECEWISEG6%7$\"\"!1F*,$*&\"
\"\"F3F(!\"\"F47$*&,&F3F3-%%tanhG6#*(F(F3F*F3,&F3F3*&F(\"\"#F*F>F4F4F3
F3F>F432,$*&F3F3F(F4F4F*2F**&F3F3F(F47$F31*&F3F3F(F4F*" }{TEXT -1 2 ".
 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 208 "h := (n,x) ->\n piecewise(x<-1/n,0,abs(x)<1/n,(1+tan
h(n*x/(1-n^2*x^2)))/2,1):\n'h(n,x)'=h(n,x);\nplot([h(1,x),h(2,x),h(4,x
),h(8,x),h(16,x),h(32,x)],x=-2.5..2.5,\n       color=[red,green,blue,m
agenta,coral,cyan]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6$%\"nG
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\"1'p>(fy#e`)FY7$$\"1gRA'zUY;#F]q$\"1bCsT;@[$*FY7$Fbgl$\"1%RT=RHA*)*FY
7$$\"17F>k\\i`GF]q$\"1W*4ywJ)****FY7$FhcmFjw7$$\"1k9;KrgUNF]qFjw7$Fd^m
Fjw7$F`dmFjw7$F`_lFjwFhdmFjdmF[`mF\\`mF^[mF`[mFfblFgblFhblFiblFjblFgwF
\\xF_xFbxFexFhxF[yF^yFayFdyFgyFjyF]zF`zFcz-Fgz6&FizF+FjzFjz-%+AXESLABE
LSG6$Q\"x6\"%!G-%%VIEWG6$;$!#D!\"\"$\"#DFb[n%(DEFAULTG" 1 2 0 1 10 0 
2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "The int
egral property of the Heaviside step function" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 22 "For any test f
unction " }{XPPEDIT 18 0 "xi(x);" "6#-%#xiG6#%\"xG" }{TEXT -1 1 " " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(H(x)*xi(x),x = -i
nfinity .. infinity) = Int(xi(x),x = 0 .. infinity);" "6#/-%$IntG6$*&-
%\"HG6#%\"xG\"\"\"-%#xiG6#F+F,/F+;,$%)infinityG!\"\"F3-F%6$-F.6#F+/F+;
\"\"!F3" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 
262 17 "_________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "h(x)" "6#-%\"
hG6#%\"xG" }{TEXT -1 51 " be an infinitely differentiable function suc
h that" }}{PARA 0 "" 0 "" {TEXT -1 5 " (i) " }{XPPEDIT 18 0 "h(x) >=0
" "6#1\"\"!-%\"hG6#%\"xG" }{TEXT -1 9 " for all " }{TEXT 263 1 "x" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 " (ii) If " }{XPPEDIT 18 
0 "x[1]<x[2]" "6#2&%\"xG6#\"\"\"&F%6#\"\"#" }{TEXT -1 6 ", and " }
{XPPEDIT 18 0 "h(x[1])<>0" "6#0-%\"hG6#&%\"xG6#\"\"\"\"\"!" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "h(x[2])<>1" "6#0-%\"hG6#&%\"xG6#\"\"#\"\"\"
" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "h(x[1])<h(x[2])" "6#2-%\"hG6#&
%\"xG6#\"\"\"-F%6#&F(6#\"\"#" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 
0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 56 " is strictly increasing, unle
ss it has the value 0 or 1." }}{PARA 0 "" 0 "" {TEXT -1 7 " (iii) " }
{XPPEDIT 18 0 "h(-x)=1-h(x)" "6#/-%\"hG6#,$%\"xG!\"\",&\"\"\"F+-F%6#F(
F)" }{TEXT -1 9 " for all " }{TEXT 265 1 "x" }{TEXT -1 21 ", so, in pa
rticular, " }{XPPEDIT 18 0 "h(0)=1/2" "6#/-%\"hG6#\"\"!*&\"\"\"F)\"\"#
!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 " (iv) " }
{XPPEDIT 18 0 "Limit(h(x),x=infinity)=1" "6#/-%&LimitG6$-%\"hG6#%\"xG/
F*%)infinityG\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 " (v
) " }{XPPEDIT 18 0 "Limit(h(x),x=-infinity)=0" "6#/-%&LimitG6$-%\"hG6#
%\"xG/F*,$%)infinityG!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 42 "Note that (v) follows from (iii) and (iv)." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 
0 "H(x)" "6#-%\"HG6#%\"xG" }{TEXT -1 93 " can be represented as a dist
ribution by the sequence of infinitely differentiable functions " }
{XPPEDIT 18 0 "h[n](x) = h(n*x)" "6#/-&%\"hG6#%\"nG6#%\"xG-F&6#*&F(\"
\"\"F*F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }
{XPPEDIT 18 0 "xi(x)" "6#-%#xiG6#%\"xG" }{TEXT -1 26 " be a test funct
ion. Then " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Int(H(x)*xi(x),x = -inf
inity .. infinity) = Limit(Int(h[n](x)*xi(x),x = -infinity .. infinity
),n = infinity);" "6#/-%$IntG6$*&-%\"HG6#%\"xG\"\"\"-%#xiG6#F+F,/F+;,$
%)infinityG!\"\"F3-%&LimitG6$-F%6$*&-&%\"hG6#%\"nG6#F+F,-F.6#F+F,/F+;,
$F3F4F3/F?F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(h[n](x)*xi(x),x =
 -infinity .. infinity) = Int(h[n](x)*xi(x),x = -infinity .. -1/sqrt(n
))+Int(h[n](x)*xi(x),x = -1/sqrt(n) .. 1/sqrt(n))+Int(h[n](x)*xi(x),x \+
= 1/sqrt(n) .. infinity);" "6#/-%$IntG6$*&-&%\"hG6#%\"nG6#%\"xG\"\"\"-
%#xiG6#F.F//F.;,$%)infinityG!\"\"F6,(-F%6$*&-&F*6#F,6#F.F/-F16#F.F//F.
;,$F6F7,$*&F/F/-%%sqrtG6#F,F7F7F/-F%6$*&-&F*6#F,6#F.F/-F16#F.F//F.;,$*
&F/F/-FH6#F,F7F7*&F/F/-FH6#F,F7F/-F%6$*&-&F*6#F,6#F.F/-F16#F.F//F.;*&F
/F/-FH6#F,F7F6F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "The \+
first two integrals converge to zero." }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "abs(Int(h[n](x)*xi(x),x = -infinity .. -1/sqrt(n))
);" "6#-%$absG6#-%$IntG6$*&-&%\"hG6#%\"nG6#%\"xG\"\"\"-%#xiG6#F0F1/F0;
,$%)infinityG!\"\",$*&F1F1-%%sqrtG6#F.F9F9" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= Int(h[n](x)*abs(xi(x)
),x = -infinity .. -1/sqrt(n));" "6#1%!G-%$IntG6$*&-&%\"hG6#%\"nG6#%\"
xG\"\"\"-%$absG6#-%#xiG6#F/F0/F/;,$%)infinityG!\"\",$*&F0F0-%%sqrtG6#F
-F;F;" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "`` <= h(-sqrt(n))*Int(abs(xi(x)),x = -infinity .. infinity);" "6
#1%!G*&-%\"hG6#,$-%%sqrtG6#%\"nG!\"\"\"\"\"-%$IntG6$-%$absG6#-%#xiG6#%
\"xG/F9;,$%)infinityGF.F=F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT 
-1 20 "which tends to 0 as " }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"
nG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }
{XPPEDIT 18 0 "abs(xi(x)) <= B;" "6#1-%$absG6#-%#xiG6#%\"xG%\"BG" }
{TEXT -1 9 " for all " }{TEXT 264 1 "x" }{TEXT -1 6 ", then" }}{PARA 
256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "abs(Int(h[n](x)*xi(x),x = \+
-1/sqrt(n) .. 1/sqrt(n)));" "6#-%$absG6#-%$IntG6$*&-&%\"hG6#%\"nG6#%\"
xG\"\"\"-%#xiG6#F0F1/F0;,$*&F1F1-%%sqrtG6#F.!\"\"F<*&F1F1-F:6#F.F<" }
{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+
<= Int(h[n](x)*abs(xi(x)),x = -1/sqrt(n) .. 1/sqrt(n));" "6#1%!G-%$Int
G6$*&-&%\"hG6#%\"nG6#%\"xG\"\"\"-%$absG6#-%#xiG6#F/F0/F/;,$*&F0F0-%%sq
rtG6#F-!\"\"F>*&F0F0-F<6#F-F>" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= B*Int(h[n](x),x = -1/sqrt(n) .. 1
/sqrt(n))" "6#1%!G*&%\"BG\"\"\"-%$IntG6$-&%\"hG6#%\"nG6#%\"xG/F1;,$*&F
'F'-%%sqrtG6#F/!\"\"F9*&F'F'-F76#F/F9F'" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "`` <= B*h[n](1/sqrt(n));" "6#1%!G*&%\"BG\"\"\"-&%\"hG6#%\"nG6#*&F'F
'-%%sqrtG6#F,!\"\"F'" }{XPPEDIT 18 0 "``=B*h(sqrt(n))*``(2/sqrt(n)" "6
#/%!G*(%\"BG\"\"\"-%\"hG6#-%%sqrtG6#%\"nGF'-F$6#*&\"\"#F'-F,6#F.!\"\"F
'" }{XPPEDIT 18 0 "``<=2*B/sqrt(n)" "6#1%!G*(\"\"#\"\"\"%\"BGF'-%%sqrt
G6#%\"nG!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 20 "which t
ends to 0 as " }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7\"6$%)opera
torG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We now turn our attention
 to the last integral. " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 
18 0 "abs(xi(x)) <= B;" "6#1-%$absG6#-%#xiG6#%\"xG%\"BG" }{TEXT -1 9 "
 for all " }{TEXT 266 1 "x" }{TEXT -1 6 ", then" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "abs(Int(xi(x),x = 1/sqrt(n) .. infinit
y)-Int(h[n](x)*xi(x),x = 1/sqrt(n) .. infinity));" "6#-%$absG6#,&-%$In
tG6$-%#xiG6#%\"xG/F-;*&\"\"\"F1-%%sqrtG6#%\"nG!\"\"%)infinityGF1-F(6$*
&-&%\"hG6#F56#F-F1-F+6#F-F1/F-;*&F1F1-F36#F5F6F7F6" }{TEXT -1 1 " " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= Int((1-h[n](x))
*abs(xi(x)),x = 1/sqrt(n) .. infinity);" "6#1%!G-%$IntG6$*&,&\"\"\"F*-
&%\"hG6#%\"nG6#%\"xG!\"\"F*-%$absG6#-%#xiG6#F1F*/F1;*&F*F*-%%sqrtG6#F/
F2%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= B*Int(``(1-h[n]
(x)),x = 1/sqrt(n) .. infinity);" "6#1%!G*&%\"BG\"\"\"-%$IntG6$-F$6#,&
F'F'-&%\"hG6#%\"nG6#%\"xG!\"\"/F4;*&F'F'-%%sqrtG6#F2F5%)infinityGF'" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``<=
``" "6#1%!GF$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-h(sqrt(n)))*Int(abs(
xi(x)),x = -infinity .. infinity);" "6#*&,&\"\"\"F%-%\"hG6#-%%sqrtG6#%
\"nG!\"\"F%-%$IntG6$-%$absG6#-%#xiG6#%\"xG/F7;,$%)infinityGF-F;F%" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 20 "which tends to 0 as " }
{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%
)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "It fo
llows that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(H(x
)*xi(x),x = -infinity .. infinity);" "6#-%$IntG6$*&-%\"HG6#%\"xG\"\"\"
-%#xiG6#F*F+/F*;,$%)infinityG!\"\"F2" }{TEXT -1 2 "  " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(Int(h[n](x)*xi(x),x = 1
/sqrt(n) .. infinity),n = infinity);" "6#/%!G-%&LimitG6$-%$IntG6$*&-&%
\"hG6#%\"nG6#%\"xG\"\"\"-%#xiG6#F2F3/F2;*&F3F3-%%sqrtG6#F0!\"\"%)infin
ityG/F0F>" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = Int(xi(x),x = 0 .. infinity);" "6#/%!G-%$IntG6$-%#
xiG6#%\"xG/F+;\"\"!%)infinityG" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 \+
0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }