# LaTeX2HTML 2019.2 (Released June 5, 2019)
# Associate images original text with physical files.


$key = q/(s'=s,u'=sslashu);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img21.svg"
 ALT="$(s'=s,u'=s/u)$">|; 

$key = q/(s,u);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img20.svg"
 ALT="$(s,u)$">|; 

$key = q/0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.68ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img87.svg"
 ALT="$0.5$">|; 

$key = q/1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.68ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img86.svg"
 ALT="$1$">|; 

$key = q/1slashbar{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img79.svg"
 ALT="$1/\bar{s}$">|; 

$key = q/1slashbar{s}=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img88.svg"
 ALT="$1/\bar{s}= 4$">|; 

$key = q/2^u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.70ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img35.svg"
 ALT="$2^u$">|; 

$key = q/F(u,a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img29.svg"
 ALT="$F(u,a)$">|; 

$key = q/G(v,b);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img55.svg"
 ALT="$G(v, b)$">|; 

$key = q/L^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.03ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img30.svg"
 ALT="$L^2$">|; 

$key = q/a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.19ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img33.svg"
 ALT="$a$">|; 

$key = q/bar{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.50ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img77.svg"
 ALT="$\bar{s}$">|; 

$key = q/bar{s}=0.25;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.68ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img1.svg"
 ALT="$\bar {s}= 0.25$">|; 

$key = q/bar{s}slashh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img94.svg"
 ALT="$\bar{s}/ h$">|; 

$key = q/c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.19ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img16.svg"
 ALT="$c$">|; 

$key = q/displaystyle1;;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.13ex; vertical-align: -0.55ex; " SRC="|."$dir".q|img81.svg"
 ALT="$\displaystyle 1 ;$">|; 

$key = q/displaystyleC_w=int{1over|s|}tildew(s)ds.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.42ex; vertical-align: -2.27ex; " SRC="|."$dir".q|img10.svg"
 ALT="$\displaystyle C_w = \int { 1 \over \vert s\vert } \tilde w (s) ds .
$">|; 

$key = q/displaystyleC_w^{-1}intC_wdelta(t-t')f(t')dt';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img41.svg"
 ALT="$\displaystyle C_w^{-1} \int C_w \delta(t-t') f(t') dt'$">|; 

$key = q/displaystyleC_w^{-1}intleft{intw[u(t-t')]duright}f(t')dt';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.82ex; vertical-align: -1.84ex; " SRC="|."$dir".q|img40.svg"
 ALT="$\displaystyle C_w^{-1} \int \left \{ \int w[u(t - t' )] du \right \} f(t') dt'$">|; 

$key = q/displaystyleC_w^{-1}intleft{intw[u(t-t')]f(t')dt'right}du;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.82ex; vertical-align: -1.84ex; " SRC="|."$dir".q|img39.svg"
 ALT="$\displaystyle C_w^{-1} \int \left \{ \int w[u(t - t' )] f(t') dt' \right \} du$">|; 

$key = q/displaystyleC_wcdotdelta(t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img27.svg"
 ALT="$\displaystyle C_w \cdot \delta (t) .$">|; 

$key = q/displaystyleDelta^2s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.14ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img109.svg"
 ALT="$\displaystyle \Delta ^ 2 s$">|; 

$key = q/displaystyleDelta^2t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.14ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img105.svg"
 ALT="$\displaystyle \Delta ^ 2 t$">|; 

$key = q/displaystyleDeltas;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.77ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img111.svg"
 ALT="$\displaystyle \Delta s$">|; 

$key = q/displaystyleDeltasDeltat=frac{1}{4pi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.84ex; vertical-align: -1.69ex; " SRC="|."$dir".q|img113.svg"
 ALT="$\displaystyle \Delta s \Delta t = \frac{1}{4 \pi}.$">|; 

$key = q/displaystyleDeltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.77ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img107.svg"
 ALT="$\displaystyle \Delta t$">|; 

$key = q/displaystyleF(u,a)=intw[u(a-t)]f(t)dt.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img32.svg"
 ALT="$\displaystyle
F(u,a) = \int w[u(a - t)] f(t) dt .
$">|; 

$key = q/displaystyleG(v,b);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img65.svg"
 ALT="$\displaystyle G ( v, b)$">|; 

$key = q/displaystyleG(v,b)=intw[v(b-tau)]g(tau)dtau~~~~{{rm{and};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img53.svg"
 ALT="$\displaystyle G(v, b ) = \int w [ v ( b - \tau ) ] g (\tau) d\tau&nbsp;&nbsp;&nbsp;&nbsp; {\rm and}
$">|; 

$key = q/displaystyleG(v,b)approxleft|{dtoverdtau}(b)right|^{-1}~Fleft{u=vcdotleft[{dtoverdtau}(b)right]^{-1},~a=t(b)right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 7.14ex; vertical-align: -3.00ex; " SRC="|."$dir".q|img56.svg"
 ALT="$\displaystyle
G(v,b) \approx
\left \vert {dt \over d\tau} ( b ) \right \vert^{...
...v \cdot
\left [ {dt \over d\tau} ( b ) \right ]^{-1} ,&nbsp; a = t(b) \right \} .
$">|; 

$key = q/displaystyleb+[t'-t(b)]cdotleft[{dtoverdtau}(b)right]^{-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 6.36ex; vertical-align: -2.42ex; " SRC="|."$dir".q|img64.svg"
 ALT="$\displaystyle b + [ t' - t(b) ]
\cdot \left [ {dt \over d\tau} ( b ) \right ]^{-1} .$">|; 

$key = q/displaystylecos(2pibar{s}t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img74.svg"
 ALT="$\displaystyle \cos ( 2 \pi \bar{s}t) ,$">|; 

$key = q/displaystylee^{-pi(bar{s}tslashh)^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.44ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img72.svg"
 ALT="$\displaystyle e^{ - \pi (\bar{s}t / h )^2 }$">|; 

$key = q/displaystylee^{-pi(bar{s}tslashh)^2}cos(2pibar{s}t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 3.02ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img76.svg"
 ALT="$\displaystyle e^{ - \pi (\bar{s}t / h )^2 } \cos ( 2 \pi \bar{s}t) .$">|; 

$key = q/displaystyleexp(-pislash4)approx0.456.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img84.svg"
 ALT="$\displaystyle \exp (- \pi / 4) \approx 0.456 .$">|; 

$key = q/displaystylef(t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img42.svg"
 ALT="$\displaystyle f(t).$">|; 

$key = q/displaystylef(t)=C_w^{-1}intF(u,a=t)du.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img37.svg"
 ALT="$\displaystyle
f(t) = C_w^{-1} \int F(u, a=t) du .
$">|; 

$key = q/displaystylefrac{1}{2sqrt{pi}}frac{bar{s}}{h}~.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.40ex; vertical-align: -2.26ex; " SRC="|."$dir".q|img112.svg"
 ALT="$\displaystyle \frac{1}{2 \sqrt{\pi}}\frac{\bar{s}}{h}&nbsp;.$">|; 

$key = q/displaystylefrac{1}{2sqrt{pi}}frac{h}{bar{s}}~.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.52ex; vertical-align: -2.26ex; " SRC="|."$dir".q|img108.svg"
 ALT="$\displaystyle \frac{1}{2 \sqrt{\pi}} \frac{h}{\bar{s}}&nbsp;.$">|; 

$key = q/displaystylefrac{1}{2}delta(s+bar{s})+frac{1}{2}delta(s-bar{s}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.84ex; vertical-align: -1.69ex; " SRC="|."$dir".q|img98.svg"
 ALT="$\displaystyle \frac{1}{2}\delta (s+\bar{s}) + \frac{1}{2}\delta(s - \bar{s}).$">|; 

$key = q/displaystylefrac{h}{2bar{s}}e^{-pi[h(s+bar{s})slashbar{s}]^2}+frac{h}{2bar{s}}e^{-pi[h(s-bar{s})slashbar{s}]^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="|."$dir".q|img102.svg"
 ALT="$\displaystyle \frac{h}{2 \bar{s}} e^{- \pi [ h ( s + \bar{s}) /\bar{s}] ^2 } +
\frac{h}{2 \bar{s}} e^{- \pi [ h ( s - \bar{s}) /\bar{s}] ^2 }.$">|; 

$key = q/displaystylefrac{h}{bar{s}}e^{-pi(hsslashbar{s})^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="|."$dir".q|img93.svg"
 ALT="$\displaystyle \frac{h}{\bar{s}} e^{ - \pi ( h s /\bar{s})^2 } .$">|; 

$key = q/displaystyleg(tau)=C_w^{-1}intG(v,b=tau)dv.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img54.svg"
 ALT="$\displaystyle g(\tau) = C_w^{-1} \int G(v, b = \tau ) dv.
$">|; 

$key = q/displaystyleg(tau)=f[t(tau)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img43.svg"
 ALT="$\displaystyle g(\tau) = f[t(\tau)].
$">|; 

$key = q/displaystyleinte^{-i2pist}m(t)dt=inte^{-i2pist}m_g(t)m_c(t)dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img100.svg"
 ALT="$\displaystyle \int e^{-i 2 \pi s t } m (t) dt =
\int e^{-i 2 \pi s t } m_g (t) m_c (t) dt$">|; 

$key = q/displaystyleinte^{-i2pist}m_c(t)dt=inte^{-i2pist}cos(2pibar{s}t)dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img97.svg"
 ALT="$\displaystyle \int e^{-i 2 \pi s t } m_c(t) dt
= \int e^{-i 2 \pi s t } \cos (2 \pi \bar{s}t) dt$">|; 

$key = q/displaystyleinte^{-i2pist}m_g(t)dt=inte^{-i2pist}e^{-pi(bar{s}tslashh)^2}dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img92.svg"
 ALT="$\displaystyle \int e^{-i 2 \pi s t } m_g(t) dt =
\int e^{-i 2 \pi s t } e^{ - \pi (\bar{s}t / h )^2 } dt$">|; 

$key = q/displaystyleinte^{-i2pist}w(t)dt{{rm{~and};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img7.svg"
 ALT="$\displaystyle \int e^{-i 2 \pi s t } w(t) dt {\rm&nbsp;and}$">|; 

$key = q/displaystyleinte^{i2pist}tildew(s)ds.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img9.svg"
 ALT="$\displaystyle \int e^{i 2 \pi s t } \tilde w (s) ds .$">|; 

$key = q/displaystyleintintleft|{u'overs'}right|tildew(u')e^{i2pis't}left|{s'overu'^2}right|ds'du';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.98ex; vertical-align: -2.42ex; " SRC="|."$dir".q|img24.svg"
 ALT="$\displaystyle \int \int \left \vert {u' \over s'} \right \vert \tilde w ( u' ) e^{i 2 \pi s' t }
\left \vert { s' \over u'^2 } \right \vert ds' du'$">|; 

$key = q/displaystyleintint{1over|u'|}tildew(u')e^{i2pis't}ds'du';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.42ex; vertical-align: -2.27ex; " SRC="|."$dir".q|img25.svg"
 ALT="$\displaystyle \int \int {1 \over \vert u'\vert} \tilde w ( u' ) e^{i 2 \pi s' t } ds' du'$">|; 

$key = q/displaystyleintm_g(t)dt=inte^{-pi(bar{s}tslashh)^2}dt=frac{h}{bar{s}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="|."$dir".q|img85.svg"
 ALT="$\displaystyle \int m_g(t) dt = \int e^{- \pi (\bar{s}t / h )^2 } dt = \frac{h}{\bar{s}} ,$">|; 

$key = q/displaystyleintw(ut)du;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img23.svg"
 ALT="$\displaystyle \int w(ut) du$">|; 

$key = q/displaystyleintw(ut)du=C_wcdotdelta(t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img17.svg"
 ALT="$\displaystyle
\int w(ut) du = C_w \cdot \delta (t) .
$">|; 

$key = q/displaystyleintw[v(b-tau)]cdotf[t(tau)]dtau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img58.svg"
 ALT="$\displaystyle \int w[v(b-\tau)] \cdot f[t(\tau)] d \tau$">|; 

$key = q/displaystyleintw[v(b-tau)]cdotfleft[t(b)+(tau-b)cdot{dtoverdtau}(b)right]dtauldots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.98ex; vertical-align: -2.42ex; " SRC="|."$dir".q|img59.svg"
 ALT="$\displaystyle \int w [ v(b-\tau)] \cdot f \left [ t(b) + (\tau-b)
\cdot {dt \over d\tau} ( b ) \right ] d\tau \ldots$">|; 

$key = q/displaystyleintwleft{vcdot[t(b)-t']cdotleft[{dtoverdtau}(b)right]^{-1}right}cdotf(t')cdotleft|{dtoverdtau}(b)right|^{-1}~dt';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 7.14ex; vertical-align: -3.00ex; " SRC="|."$dir".q|img66.svg"
 ALT="$\displaystyle \int w \left \{ v \cdot [t(b) - t'] \cdot
\left [ {dt \over d\tau...
...\} \cdot f(t')
\cdot \left \vert {dt \over d\tau} ( b ) \right \vert^{-1}&nbsp; dt'$">|; 

$key = q/displaystyleint|m_g(t)|^2dt=inte^{-2pi(bar{s}tslashh)^2}dt=frac{1}{sqrt{2}}frac{h}{bar{s}}~,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.46ex; vertical-align: -2.20ex; " SRC="|."$dir".q|img103.svg"
 ALT="$\displaystyle \int \vert m_g(t) \vert^2 dt =
\int e^{- 2 \pi (\bar{s}t / h )^2 } dt = \frac{1}{\sqrt{2}} \frac{h}{\bar{s}}&nbsp;,$">|; 

$key = q/displaystyleint|m_g(t)|^2t^2dt=inte^{-2pi(bar{s}tslashh)^2}t^2dt=frac{1}{4sqrt{2}pi}frac{h^3}{bar{s}^3}~.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.69ex; vertical-align: -2.20ex; " SRC="|."$dir".q|img104.svg"
 ALT="$\displaystyle \int \vert m_g(t) \vert^2 t^2 dt =
\int e^ { - 2 \pi (\bar{s}t / h )^2 } t^2 dt = \frac{1}{4 \sqrt{2} \pi}\frac{h^3}{\bar{s}^3}&nbsp;.$">|; 

$key = q/displaystyleleft{int{1over|u'|}tildew(u')du'right}cdotleft{inte^{i2pis't}ds'right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.98ex; vertical-align: -2.42ex; " SRC="|."$dir".q|img26.svg"
 ALT="$\displaystyle \left \{ \int {1 \over \vert u'\vert} \tilde w ( u' ) du' \right \} \cdot
\left \{ \int e^{i 2 \pi s' t } ds' \right \}$">|; 

$key = q/displaystyleleft|{dtoverdtau}(b)right|^{-1}intwleft{vleft[{dtoverdtau}(b)right]^{-1}cdot[t(b)-t']right}cdotf(t')dt';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 7.14ex; vertical-align: -3.00ex; " SRC="|."$dir".q|img67.svg"
 ALT="$\displaystyle \left \vert {dt \over d\tau} ( b ) \right \vert^{-1}
\int w \left...
...t \over d\tau} ( b ) \right ]^{-1}
\cdot [t(b) - t']
\right \} \cdot f(t') d t'$">|; 

$key = q/displaystyleleft|{dtoverdtau}(b)right|^{-1}~Fleft{u=vcdotleft[{dtoverdtau}(b)right]^{-1},~a=t(b)right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 7.14ex; vertical-align: -3.00ex; " SRC="|."$dir".q|img68.svg"
 ALT="$\displaystyle \left \vert {dt \over d\tau} ( b ) \right \vert^{-1}
&nbsp; F \left \{ u=v \cdot
\left [ {dt \over d\tau} ( b ) \right ]^{-1} ,&nbsp; a = t(b) \right \} .$">|; 

$key = q/displaystylem(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img69.svg"
 ALT="$\displaystyle m(t)$">|; 

$key = q/displaystylem_g(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.58ex; vertical-align: -0.75ex; " SRC="|."$dir".q|img80.svg"
 ALT="$\displaystyle m_g(0)$">|; 

$key = q/displaystylem_g(t)m_c(t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.58ex; vertical-align: -0.75ex; " SRC="|."$dir".q|img70.svg"
 ALT="$\displaystyle m_g(t) m_c(t) ,$">|; 

$key = q/displaystylem_g[-hslash(2bar{s})];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.58ex; vertical-align: -0.75ex; " SRC="|."$dir".q|img83.svg"
 ALT="$\displaystyle m_g[-h/(2 \bar{s})]$">|; 

$key = q/displaystylem_g[hslash(2bar{s})];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.58ex; vertical-align: -0.75ex; " SRC="|."$dir".q|img82.svg"
 ALT="$\displaystyle m_g[h/(2 \bar{s})]$">|; 

$key = q/displaystylembox{and}m_c(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img73.svg"
 ALT="$\displaystyle \mbox{ and }
m_c(t)$">|; 

$key = q/displaystylembox{so}m(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img75.svg"
 ALT="$\displaystyle \mbox{ so }
m(t)$">|; 

$key = q/displaystylembox{where}m_g(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.58ex; vertical-align: -0.75ex; " SRC="|."$dir".q|img71.svg"
 ALT="$\displaystyle \mbox{ where }
m_g(t)$">|; 

$key = q/displaystylet';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.01ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img60.svg"
 ALT="$\displaystyle t'$">|; 

$key = q/displaystylet(b)+(tau-b)cdot{dtoverdtau}(b)~~~~{{rm{and};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="|."$dir".q|img62.svg"
 ALT="$\displaystyle t(b) + (\tau-b) \cdot {dt \over d\tau} ( b )
&nbsp;&nbsp;&nbsp;&nbsp; {\rm and}$">|; 

$key = q/displaystylet(tau);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img47.svg"
 ALT="$\displaystyle t(\tau)$">|; 

$key = q/displaystylet(tau_0)+(tau-tau_0)cdot{dtoverdtau}(tau_0)~~~~{{rm{and};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="|."$dir".q|img49.svg"
 ALT="$\displaystyle t(\tau_0 ) + (\tau - \tau_0 )
\cdot {dt \over d\tau} ( \tau_0 )&nbsp;&nbsp;&nbsp;&nbsp;{\rm and }$">|; 

$key = q/displaystyletau(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img50.svg"
 ALT="$\displaystyle \tau(t)$">|; 

$key = q/displaystyletau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.19ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img63.svg"
 ALT="$\displaystyle \tau$">|; 

$key = q/displaystyletau_0+[t-t(tau_0)]cdotleft[{dtoverdtau}(tau_0)right]^{-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 6.36ex; vertical-align: -2.42ex; " SRC="|."$dir".q|img51.svg"
 ALT="$\displaystyle \tau_0 + [ t - t (\tau_0 ) ]
\cdot \left [ {dt \over d\tau} ( \tau_0 ) \right ]^{-1} .$">|; 

$key = q/displaystyletildem(s);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img99.svg"
 ALT="$\displaystyle \tilde m(s)$">|; 

$key = q/displaystyletildem_c(s);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img96.svg"
 ALT="$\displaystyle \tilde m_c(s)$">|; 

$key = q/displaystyletildem_g(s)*tildem_c(s)=inttildem_g(s')tildem_c(s-s')ds';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img101.svg"
 ALT="$\displaystyle \tilde m_g(s) * \tilde m_c(s) = \int \tilde m_g(s') \tilde m_c(s -s') d s'$">|; 

$key = q/displaystyletildem_g(s);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.58ex; vertical-align: -0.75ex; " SRC="|."$dir".q|img91.svg"
 ALT="$\displaystyle \tilde m_g(s)$">|; 

$key = q/displaystyletildew(s);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img5.svg"
 ALT="$\displaystyle \tilde w (s)$">|; 

$key = q/displaystylew(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img8.svg"
 ALT="$\displaystyle w(t)$">|; 

$key = q/displaystylew(t)=w(-t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img28.svg"
 ALT="$\displaystyle w(t) = w(-t) .
$">|; 

$key = q/displaystyle{int|m_g(t)|^2t^2dtoverint|m_g(t)|^2dt}=frac{1}{4pi}frac{h^2}{bar{s}^2};;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.83ex; vertical-align: -2.34ex; " SRC="|."$dir".q|img106.svg"
 ALT="$\displaystyle { \int \vert m_g(t) \vert^2 t^2 dt \over
\int \vert m_g(t) \vert^2 dt }
= \frac{1}{4 \pi} \frac{h^2}{\bar{s}^2} ;$">|; 

$key = q/displaystyle{int|tildem_g(s)|^2s^2dsoverint|tildem_g(s)|^2ds}=frac{1}{4pi}frac{bar{s}^2}{h^2};;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.83ex; vertical-align: -2.34ex; " SRC="|."$dir".q|img110.svg"
 ALT="$\displaystyle {\int \vert \tilde m_g(s) \vert^2 s^2 ds \over
\int \vert \tilde m_g(s) \vert^2 ds } =
\frac{1}{4 \pi}\frac{\bar{s}^2}{h^2} ;$">|; 

$key = q/displaystyle{{bf{Proof:~}C_w^{-1}intF(u,a=t)du;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.48ex; vertical-align: -1.67ex; " SRC="|."$dir".q|img38.svg"
 ALT="$\displaystyle {\bf Proof:&nbsp;}
C_w^{-1} \int F(u,a=t) du$">|; 

$key = q/displaystyle{{bf{Proof:~}G(v,b);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img57.svg"
 ALT="$\displaystyle {\bf Proof:&nbsp;}
G ( v, b)$">|; 

$key = q/displaystyle{{bf{Proof:~}intw(ut)du=intleft{int{1over|u|}tildew({soveru})e^{i2pist}dsright}duldots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.98ex; vertical-align: -2.42ex; " SRC="|."$dir".q|img19.svg"
 ALT="$\displaystyle {\bf Proof:&nbsp;} \int w(ut) du
= \int \left \{ \int {1 \over \vert u\vert} \tilde w ( { s \over u } )
e^{i 2 \pi s t } ds \right \} du \ldots
$">|; 

$key = q/epsilon>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.75ex; vertical-align: -0.16ex; " SRC="|."$dir".q|img14.svg"
 ALT="$\epsilon &gt; 0 $">|; 

$key = q/f(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img31.svg"
 ALT="$f(t)$">|; 

$key = q/g(tau);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img52.svg"
 ALT="$g(\tau)$">|; 

$key = q/h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.80ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img78.svg"
 ALT="$h$">|; 

$key = q/h=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.80ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img2.svg"
 ALT="$h=4$">|; 

$key = q/hslashbar{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img95.svg"
 ALT="$h/\bar{s}$">|; 

$key = q/hslashbar{s}=4slash0.25=16;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img90.svg"
 ALT="$h/\bar{s}= 4/0.25 = 16$">|; 

$key = q/partial(s,u)slashpartial(s',u')=|s'slashu'^2|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.61ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img22.svg"
 ALT="$\partial (s,u) / \partial (s',u') = \vert s'/u'^2\vert$">|; 

$key = q/rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.68ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img12.svg"
 ALT="$\rightarrow 0 $">|; 

$key = q/srightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.68ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img13.svg"
 ALT="$s \rightarrow 0$">|; 

$key = q/t(tau);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img44.svg"
 ALT="$t(\tau)$">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.62ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img45.svg"
 ALT="$t$">|; 

$key = q/t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.68ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img18.svg"
 ALT="$t=0$">|; 

$key = q/t=pm8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.88ex; vertical-align: -0.29ex; " SRC="|."$dir".q|img89.svg"
 ALT="$t=\pm 8$">|; 

$key = q/tau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.19ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img46.svg"
 ALT="$\tau$">|; 

$key = q/textstyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.03ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img6.svg"
 ALT="$\textstyle =$">|; 

$key = q/textstyleapprox;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.31ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img48.svg"
 ALT="$\textstyle \approx$">|; 

$key = q/textstyleequiv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.27ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img61.svg"
 ALT="$\textstyle \equiv$">|; 

$key = q/tildew(s);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img4.svg"
 ALT="$\tilde w (s)$">|; 

$key = q/tildew(s)slash|s|^epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img11.svg"
 ALT="$\tilde w (s) / \vert s\vert^\epsilon $">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.19ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img34.svg"
 ALT="$u$">|; 

$key = q/u^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.03ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img36.svg"
 ALT="$u^{-1}$">|; 

$key = q/w(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img3.svg"
 ALT="$w(t)$">|; 

$key = q/w(t)={{rm{cos}(2pit)(e^{-pit^2slashc^2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.90ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img15.svg"
 ALT="$w(t) = {\rm cos}(2\pi t) (e^{- \pi t^2 / c^2} )$">|; 

1;

