# LaTeX2HTML 99.2beta6 (1.42)
# Associate images original text with physical files.


$key = q/-1<x<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$-1&lt;x&lt;1$">|; 

$key = q/displaystylevartheta_3^m(0,q);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$\displaystyle \vartheta_3^m(0,q)$">|; 

$key = q/{displaymath}frac{partial}{partialu}{{rm{sn}(u,k)={{rm{cn}(u,k);{{rm{dn}(u,k){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="\begin{displaymath}
\frac{\partial}{\partial u} {\rm sn}(u,k) = {\rm cn}(u,k) \; {\rm dn}(u,k)
\end{displaymath}">|; 

$key = q/{displaymath}(a^2+b^2+c^2+d^2)(w^2+x^2+y^2+z^2)=A^2+B^2+C^2+D^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="489" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="\begin{displaymath}
(a^2 + b^2 + c^2 + d^2)(w^2 + x^2 + y^2 + z^2)
= A^2 + B^2 + C^2 + D^2
\end{displaymath}">|; 

$key = q/(-2)^2+(-1)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$(-2)^2 + (-1)^2$">|; 

$key = q/2^2+(-1)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$2^2 + (-1)^2$">|; 

$key = q/{displaymath}frac{partial}{partialu}{{rm{dn}(u,k)=-k^2{{rm{sn}(u,k);{{rm{cn}(u,k){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="\begin{displaymath}
\frac{\partial}{\partial u} {\rm dn}(u,k) = -k^2 {\rm sn}(u,k) \; {\rm cn}(u,k)
\end{displaymath}">|; 

$key = q/m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$m$">|; 

$key = q/2^2+1^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$2^2 + 1^2$">|; 

$key = q/1^2+2^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$1^2 + 2^2$">|; 

$key = q/{displaymath}D=az-dw+by-cx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="\begin{displaymath}D = az - dw + by - cx. \end{displaymath}">|; 

$key = q/{displaymath}5=2^2+1^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="\begin{displaymath}
5 = 2^2 + 1^2
\end{displaymath}">|; 

$key = q/{displaymath}vartheta_3(0,q)=1+sum_{n=1}^infty2q^{n^2}=1+2q+2q^4+2q^9+2q^{16}+dots{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="453" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="\begin{displaymath}
\vartheta_3(0,q) = 1 + \sum_{n=1}^\infty 2 q^{n^2}
= 1 + 2q + 2q^4 +2q^9 + 2q^{16} + \dots
\end{displaymath}">|; 

$key = q/q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$q$">|; 

$key = q/{displaymath}100=5^2+5^2+5^2+5^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="\begin{displaymath}
100 = 5^2 + 5^2 + 5^2 + 5^2
\end{displaymath}">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$u$">|; 

$key = q/F(x|k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$F(x\vert k)$">|; 

$key = q/{displaymath}{{bf{K'}={{bf{K}'(k)={{bf{K}(k')=F(1|sqrt{1-k^2}){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="\begin{displaymath}
{\bf K'} = {\bf K}'(k) = {\bf K}(k') = F(1\vert\sqrt{1-k^2})
\end{displaymath}">|; 

$key = q/{displaymath}u=F(x|k){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="\begin{displaymath}
u = F(x\vert k)
\end{displaymath}">|; 

$key = q/y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$y$">|; 

$key = q/2^2=(-2)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$2^2=(-2)^2$">|; 

$key = q/{displaymath}sigma(n:4not|d){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="\begin{displaymath}
\sigma(n: 4 \not\vert d)
\end{displaymath}">|; 

$key = q/{displaymath}118=10^2+4^2+1^2+1^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="\begin{displaymath}
118 = 10^2 + 4^2 + 1^2 + 1^2
\end{displaymath}">|; 

$key = q/{displaymath}frac{2{{bf{K}}{pi}=1+4sum_{n=1}^{infty}(d_1(n)-d_3(n))q^n{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="\begin{displaymath}
\frac{2{\bf K}}{\pi} =
1 + 4 \sum_{n=1}^{\infty} (d_1(n)-d_3(n)) q^n
\end{displaymath}">|; 

$key = q/{displaymath}sin^{-1}x=int_0^xfrac{dx}{sqrt{1-x^2}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="\begin{displaymath}
\sin^{-1} x = \int_0^x \frac{dx}{\sqrt{1-x^2}}
\end{displaymath}">|; 

$key = q/{displaymath}5=2^2+1^2+0^2+0^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="\begin{displaymath}
5 = 2^2 + 1^2 + 0^2 + 0^2
\end{displaymath}">|; 

$key = q/N=(-n)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$N=(-n)^2$">|; 

$key = q/{displaymath}{{rm{cn}(0,k)={{rm{dn}(0,k)=1{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="\begin{displaymath}
{\rm cn}(0,k) = {\rm dn}(0,k) = 1
\end{displaymath}">|; 

$key = q/d_1(5)=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$d_1(5)=2$">|; 

$key = q/(-1)^2+(-2)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$(-1)^2 + (-2)^2$">|; 

$key = q/(-1)^2+2^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$(-1)^2 + 2^2$">|; 

$key = q/{displaymath}1+x^2+y^2=mp.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="\begin{displaymath}
1 + x^2 + y^2 = mp.
\end{displaymath}">|; 

$key = q/p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$p$">|; 

$key = q/{displaymath}{{rm{cn}^2(u,k)=1-{{rm{sn}^2(u,k){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="\begin{displaymath}
{\rm cn}^2(u,k) = 1-{\rm sn}^2(u,k)
\end{displaymath}">|; 

$key = q/2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$2$">|; 

$key = q/d_1(n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$d_1(n)$">|; 

$key = q/{displaymath}intR(sqrt{P(x)},x);dx{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="\begin{displaymath}
\int R(\sqrt{P(x)}, x) \; dx
\end{displaymath}">|; 

$key = q/d_3(n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$d_3(n)$">|; 

$key = q/{displaymath}frac{2{{bf{K}}{pi}=1+4sumfrac{q^n}{1+q^{2n}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="\begin{displaymath}
\frac{2{\bf K}}{\pi} =
1 + 4 \sum \frac{q^n}{1+q^{2n}}
\end{displaymath}">|; 

$key = q/x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$x$">|; 

$key = q/N=0=0^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$N=0=0^2$">|; 

$key = q/P(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$P(x)$">|; 

$key = q/q^N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$q^N$">|; 

$key = q/sinu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$\sin u$">|; 

$key = q/{displaymath}left(frac{2{{bf{K}}{pi}right)^2=1+8sumfrac{q^n}{left(1+(-q)^nright)^2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="\begin{displaymath}
\left(\frac{2{\bf K}}{\pi}\right)^2 =
1 + 8 \sum \frac{q^n}{\left(1+(-q)^n\right)^2}
\end{displaymath}">|; 

$key = q/{displaymath}8(1+2+5+10+25+50)=8(93)=744{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="\begin{displaymath}
8(1+2+5+10+25+50) = 8(93) = 744
\end{displaymath}">|; 

$key = q/q^5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$q^5$">|; 

$key = q/{displaymath}x=sinu{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="\begin{displaymath}
x = \sin u
\end{displaymath}">|; 

$key = q/displaystylevartheta_3^2(0,q);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$\displaystyle \vartheta_3^2(0,q)$">|; 

$key = q/{displaymath}F(x|k)=int_0^xfrac{dx}{sqrt{(1-x^2)(1-k^2x^2)}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="\begin{displaymath}
F(x\vert k) = \int_0^x \frac{dx}{\sqrt{(1-x^2)(1-k^2 x^2)}}
\end{displaymath}">|; 

$key = q/N=n^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$N=n^2$">|; 

$key = q/k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$k$">|; 

$key = q/{displaymath}B=ax-bw+cz-dy,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="\begin{displaymath}B = ax - bw + cz - dy, \end{displaymath}">|; 

$key = q/1^2+(-2)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$1^2 + (-2)^2$">|; 

$key = q/(-2)^2+1^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$(-2)^2 + 1^2$">|; 

$key = q/24=16+8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$24=16+8$">|; 

$key = q/{displaymath}{{rm{dn}^2(u,k)=1-k^2{{rm{sn}^2(u,k){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="\begin{displaymath}
{\rm dn}^2(u,k) = 1-k^2{\rm sn}^2(u,k)
\end{displaymath}">|; 

$key = q/N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$N$">|; 

$key = q/{displaymath}x={{rm{sn}(u,k){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="\begin{displaymath}
x = {\rm sn}(u,k)
\end{displaymath}">|; 

$key = q/1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$1$">|; 

$key = q/{displaymath}{{bf{K}={{bf{K}(k)=F(1|k){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="\begin{displaymath}
{\bf K} = {\bf K}(k) = F(1\vert k)
\end{displaymath}">|; 

$key = q/displaystylefrac{2{{bf{K}x}{pi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$\displaystyle \frac{2{\bf K}x}{\pi}$">|; 

$key = q/{displaymath}frac{2{{bf{K}}{pi};{{rm{sn}(frac{2{{bf{K}x}{pi},k)=frac{4}{k}sum_{nfty}frac{q^{(2n+1)slash2}}{1-q^{2n+1}}sin(2n+1)x{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="\begin{displaymath}
\frac{2{\bf K}}{\pi} \; {\rm sn}(\frac{2{\bf K}x}{\pi}, k) =...
..._{n=0}^{\infty}
\frac{q^{(2n+1)/2}}{1-q^{2n+1}}
\sin (2n+1)x
\end{displaymath}">|; 

$key = q/1^2=(-1)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$1^2 = (-1)^2$">|; 

$key = q/R(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$R(x,y)$">|; 

$key = q/{displaymath}frac{2{{bf{K}}{pi};{{rm{dn}(frac{2{{bf{K}x}{pi},k)=1+4sum_{n=1}^{infty}frac{q^{n}}{1+q^{2n}}cos(2n)x{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="\begin{displaymath}
\frac{2{\bf K}}{\pi} \; {\rm dn}(\frac{2{\bf K}x}{\pi}, k) =
1 + 4 \sum_{n=1}^{\infty}
\frac{q^{n}}{1+q^{2n}}
\cos (2n)x
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{sn}(0,k)=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="\begin{displaymath}
{\rm sn}(0,k) = 0
\end{displaymath}">|; 

$key = q/{displaymath}frac{2{{bf{K}}{pi};{{rm{cn}(frac{2{{bf{K}x}{pi},k)=frac{4}{k}sum_{nfty}frac{q^{(2n+1)slash2}}{1+q^{2n+1}}cos(2n+1)x{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="\begin{displaymath}
\frac{2{\bf K}}{\pi} \; {\rm cn}(\frac{2{\bf K}x}{\pi}, k) =...
..._{n=0}^{\infty}
\frac{q^{(2n+1)/2}}{1+q^{2n+1}}
\cos (2n+1)x
\end{displaymath}">|; 

$key = q/{displaymath}4=2^2+0^2+0^2+0^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="\begin{displaymath}
4 = 2^2 + 0^2 + 0^2 + 0^2
\end{displaymath}">|; 

$key = q/0<m<p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$0&lt;m&lt;p$">|; 

$key = q/{displaymath}4=1^2+1^2+1^2+1^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="\begin{displaymath}
4 = 1^2 + 1^2 + 1^2 + 1^2
\end{displaymath}">|; 

$key = q/z=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$z=0$">|; 

$key = q/{displaymath}A=aw+bx+cy+dz,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="\begin{displaymath}A = aw + bx + cy + dz, \end{displaymath}">|; 

$key = q/16=2^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$16=2^4$">|; 

$key = q/d_3(5)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$d_3(5)=0$">|; 

$key = q/{displaymath}C=ay-cw+dx-bz,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="\begin{displaymath}C = ay - cw + dx - bz, \end{displaymath}">|; 

$key = q/n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$n$">|; 

$key = q/{displaymath}q=exp-pifrac{{{bf{K}'}{{bf{K}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="\begin{displaymath}
q = \exp -\pi\frac{{\bf K}'}{\bf K}
\end{displaymath}">|; 

$key = q/d_1(75)=d_3(75)=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$d_1(75)=d_3(75)=3$">|; 

$key = q/{displaymath}frac{partial}{partialu}{{rm{cn}(u,k)=-{{rm{sn}(u,k);{{rm{dn}(u,k){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="\begin{displaymath}
\frac{\partial}{\partial u} {\rm cn}(u,k) = - {\rm sn}(u,k) \; {\rm dn}(u,k)
\end{displaymath}">|; 

$key = q/{displaymath}vartheta_3(0,q)=sqrt{frac{2{{bf{K}}{pi}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="\begin{displaymath}
\vartheta_3(0,q) = \sqrt{\frac{2{\bf K}}{\pi}}
\end{displaymath}">|; 

$key = q/displaystylevartheta_3(0,q);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$\displaystyle \vartheta_3(0,q)$">|; 

$key = q/{displaymath}left(frac{2{{bf{K}}{pi}right)^2=1+8sum_{n=1}^{infty}sigma(n:4not|d)q^n{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="\begin{displaymath}
\left( \frac{2{\bf K}}{\pi} \right)^2 =
1 + 8 \sum_{n=1}^{\infty}
\sigma(n:4 \not\vert d) q^n
\end{displaymath}">|; 

$key = q/z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$z$">|; 

1;