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<DIV dir=ltr align=left><SPAN class=295580208-10012021><FONT color=#0000ff
size=2 face=Arial>Yes, exactly. It's RECURSION, not
ITERATION.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=295580208-10012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=295580208-10012021><FONT color=#0000ff
size=2 face=Arial>Iteration implies that you have to converge to an answer by a
successive approximation. This process is exact and you know the
answer.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=295580208-10012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=295580208-10012021><FONT color=#0000ff
size=2 face=Arial>I did a bit more research into the Landen transformation and
discovered that, for each successive step, the argument of the function is
changing by a very simple formula. If the sn on the next step down the
table is sn(v), and the one above is sn(u), then u = (1+k)v. Hence, all
one needs to do is to calculate the argument for the top row of sn functions,
and then one can use a formula to find the actual function values. I'll
demonstrate this with a table: Note that u = (2n-1)K/32 or
(2n-1)(2K)/64. The sn(u) values at the bottom were calculated directly
from these arguments by the approximation formula given.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=295580208-10012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV><SPAN class=295580208-10012021>
<DIV dir=ltr align=left>
<TABLE style="WIDTH: 851pt; BORDER-COLLAPSE: collapse" border=0 cellSpacing=0
cellPadding=0 width=1134 x:str>
<COLGROUP>
<COL style="WIDTH: 65pt; mso-width-source: userset; mso-width-alt: 2446"
width=86>
<COL style="WIDTH: 36pt; mso-width-source: userset; mso-width-alt: 1365"
width=48>
<COL style="WIDTH: 75pt; mso-width-source: userset; mso-width-alt: 2844"
span=10 width=100>
<TBODY>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 65pt; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
height=21 width=86></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 36pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
width=48></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl29 width=100></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 width=100 x:num><FONT face=Arial>32</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 width=100 x:num><FONT face=Arial>1</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 width=100 x:num><FONT face=Arial>3</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 width=100 x:num><FONT face=Arial>5</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 width=100 x:num><FONT face=Arial>7</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 width=100 x:num><FONT face=Arial>9</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 width=100 x:num><FONT face=Arial>11</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 width=100 x:num><FONT face=Arial>13</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; WIDTH: 75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 width=100 x:num><FONT face=Arial>15</FONT></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30 height=21><FONT face=Arial>2K</FONT></TD>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30><FONT face=Arial>i</FONT></TD>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30><FONT face=Arial>k</FONT></TD>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30><FONT face=Arial>k'</FONT></TD>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30><FONT face=Arial>sn(K/32)</FONT></TD>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30><FONT face=Arial>sn(3K/32)</FONT></TD>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30><FONT face=Arial>sn(5K/32)</FONT></TD>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30><FONT face=Arial>sn(7K/32)</FONT></TD>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30><FONT face=Arial>sn(9K/32)</FONT></TD>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30><FONT face=Arial>sn(11K/32)</FONT></TD>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30><FONT face=Arial>sn(13K/32)</FONT></TD>
<TD
style="BORDER-BOTTOM: windowtext 0.5pt solid; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl30><FONT face=Arial>sn(15K/32)</FONT></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl31 height=21 x:num="21.19326946119158" x:fmla="=A4*(1+C3)"><FONT
face=Arial>21.1933</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 x:num><FONT face=Arial>0</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.99999999499999992" x:fmla="=SQRT(1-D3^2)"><FONT
face=Arial>1.000000</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl26 x:num="1E-4"><FONT color=#ff0000
face=Arial><STRONG>0.000100</STRONG></FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl27 x:num="0.31954908778216407"
x:fmla="=(1+$C4)*E4/(1+$C4*E4^2)"><STRONG><FONT color=#3366ff
face=Arial>0.319549</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl27 x:num="0.75882300219892307"
x:fmla="=(1+$C4)*F4/(1+$C4*F4^2)"><STRONG><FONT color=#3366ff
face=Arial>0.758823</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl27 x:num="0.92963895262695051"
x:fmla="=(1+$C4)*G4/(1+$C4*G4^2)"><STRONG><FONT color=#3366ff
face=Arial>0.929639</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl27 x:num="0.98079395245268941"
x:fmla="=(1+$C4)*H4/(1+$C4*H4^2)"><STRONG><FONT color=#3366ff
face=Arial>0.980794</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl27 x:num="0.99485657133820127"
x:fmla="=(1+$C4)*I4/(1+$C4*I4^2)"><STRONG><FONT color=#3366ff
face=Arial>0.994857</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl27 x:num="0.99862970112246652"
x:fmla="=(1+$C4)*J4/(1+$C4*J4^2)"><STRONG><FONT color=#3366ff
face=Arial>0.998630</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl27 x:num="0.99963543572806379"
x:fmla="=(1+$C4)*K4/(1+$C4*K4^2)"><STRONG><FONT color=#3366ff
face=Arial>0.999635</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl27 x:num="0.99990304582766598"
x:fmla="=(1+$C4)*L4/(1+$C4*L4^2)"><STRONG><FONT color=#3366ff
face=Arial>0.999903</FONT></STRONG></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl31 height=21 x:num="10.596634757087378" x:fmla="=A5*(1+C4)"><FONT
face=Arial>10.5966</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 x:num><FONT face=Arial>1</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.99980001999800017" x:fmla="=(1-D3)/(1+D3)"><FONT
face=Arial>0.999800</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="1.9998000199982782E-2" x:fmla="=SQRT(1-C4^2)"><FONT
face=Arial>0.019998</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.16409222159323647"
x:fmla="=(1+$C5)*E5/(1+$C5*E5^2)"><FONT face=Arial>0.164092</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.45957748887243821"
x:fmla="=(1+$C5)*F5/(1+$C5*F5^2)"><FONT face=Arial>0.459577</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.67939430999979955"
x:fmla="=(1+$C5)*G5/(1+$C5*G5^2)"><FONT face=Arial>0.679394</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.82079804538860868"
x:fmla="=(1+$C5)*H5/(1+$C5*H5^2)"><FONT face=Arial>0.820798</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.9034429805416675"
x:fmla="=(1+$C5)*I5/(1+$C5*I5^2)"><FONT face=Arial>0.903443</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.94906239839059414"
x:fmla="=(1+$C5)*J5/(1+$C5*J5^2)"><FONT face=Arial>0.949062</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.97345209132416455"
x:fmla="=(1+$C5)*K5/(1+$C5*K5^2)"><FONT face=Arial>0.973452</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.98626912113215703"
x:fmla="=(1+$C5)*L5/(1+$C5*L5^2)"><FONT face=Arial>0.986269</FONT></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl31 height=21 x:num="5.2988472102815436" x:fmla="=A6*(1+C5)"><FONT
face=Arial>5.2988</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 x:num><FONT face=Arial>2</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.96078815802371775" x:fmla="=(1-D4)/(1+D4)"><FONT
face=Arial>0.960788</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.27728345677553784" x:fmla="=SQRT(1-C5^2)"><FONT
face=Arial>0.277283</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="8.4257695419860446E-2"
x:fmla="=(1+$C6)*E6/(1+$C6*E6^2)"><FONT face=Arial>0.084258</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.24826384223891873"
x:fmla="=(1+$C6)*F6/(1+$C6*F6^2)"><FONT face=Arial>0.248264</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.39966617248610581"
x:fmla="=(1+$C6)*G6/(1+$C6*G6^2)"><FONT face=Arial>0.399666</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.5327627875975175"
x:fmla="=(1+$C6)*H6/(1+$C6*H6^2)"><FONT face=Arial>0.532763</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.64482349359736113"
x:fmla="=(1+$C6)*I6/(1+$C6*I6^2)"><FONT face=Arial>0.644823</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.73578453382704834"
x:fmla="=(1+$C6)*J6/(1+$C6*J6^2)"><FONT face=Arial>0.735785</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.80743701351050501"
x:fmla="=(1+$C6)*K6/(1+$C6*K6^2)"><FONT face=Arial>0.807437</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.86253585063777793"
x:fmla="=(1+$C6)*L6/(1+$C6*L6^2)"><FONT face=Arial>0.862536</FONT></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl31 height=21 x:num="2.7024067789262163" x:fmla="=A7*(1+C6)"><FONT
face=Arial>2.7024</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 x:num><FONT face=Arial>3</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.56582314551300739" x:fmla="=(1-D5)/(1+D5)"><FONT
face=Arial>0.565823</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.82452663268190807" x:fmla="=SQRT(1-C6^2)"><FONT
face=Arial>0.824527</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="5.3898932282505044E-2"
x:fmla="=(1+$C7)*E7/(1+$C7*E7^2)"><FONT face=Arial>0.053899</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.16087342405339869"
x:fmla="=(1+$C7)*F7/(1+$C7*F7^2)"><FONT face=Arial>0.160873</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.2654175649715656"
x:fmla="=(1+$C7)*G7/(1+$C7*G7^2)"><FONT face=Arial>0.265418</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.36603901998352861"
x:fmla="=(1+$C7)*H7/(1+$C7*H7^2)"><FONT face=Arial>0.366039</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.46142187386290923"
x:fmla="=(1+$C7)*I7/(1+$C7*I7^2)"><FONT face=Arial>0.461422</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.55046926314605304"
x:fmla="=(1+$C7)*J7/(1+$C7*J7^2)"><FONT face=Arial>0.550469</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.63232407619966602"
x:fmla="=(1+$C7)*K7/(1+$C7*K7^2)"><FONT face=Arial>0.632324</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.7063682545863379"
x:fmla="=(1+$C7)*L7/(1+$C7*L7^2)"><FONT face=Arial>0.706368</FONT></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl31 height=21 x:num="1.7258697361002622" x:fmla="=A8*(1+C7)"><FONT
face=Arial>1.7259</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 x:num><FONT face=Arial>4</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="9.6174736052035664E-2" x:fmla="=(1-D6)/(1+D6)"><FONT
face=Arial>0.096175</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.99536446598485784" x:fmla="=SQRT(1-C7^2)"><FONT
face=Arial>0.995364</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="4.9181457081762414E-2"
x:fmla="=(1+$C8)*E8/(1+$C8*E8^2)"><FONT face=Arial>0.049181</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.14706419000911147"
x:fmla="=(1+$C8)*F8/(1+$C8*F8^2)"><FONT face=Arial>0.147064</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.24351156938729249"
x:fmla="=(1+$C8)*G8/(1+$C8*G8^2)"><FONT face=Arial>0.243512</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.33758389401240496"
x:fmla="=(1+$C8)*H8/(1+$C8*H8^2)"><FONT face=Arial>0.337584</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.42836692199998322"
x:fmla="=(1+$C8)*I8/(1+$C8*I8^2)"><FONT face=Arial>0.428367</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.51498138851900621"
x:fmla="=(1+$C8)*J8/(1+$C8*J8^2)"><FONT face=Arial>0.514981</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.59659189985388728"
x:fmla="=(1+$C8)*K8/(1+$C8*K8^2)"><FONT face=Arial>0.596592</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.67241508194302824"
x:fmla="=(1+$C8)*L8/(1+$C8*L8^2)"><FONT face=Arial>0.672415</FONT></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl31 height=21 x:num="1.5744476490273123" x:fmla="=A9*(1+C8)"><FONT
face=Arial>1.5744</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 x:num><FONT face=Arial>5</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="2.3231515315444832E-3" x:fmla="=(1-D7)/(1+D7)"><FONT
face=Arial>0.002323</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.99999730147983978" x:fmla="=SQRT(1-C8^2)"><FONT
face=Arial>0.999997</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="4.9067740373185675E-2"
x:fmla="=(1+$C9)*E9/(1+$C9*E9^2)"><FONT face=Arial>0.049068</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.14673066817083066"
x:fmla="=(1+$C9)*F9/(1+$C9*F9^2)"><FONT face=Arial>0.146731</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.24298048839154618"
x:fmla="=(1+$C9)*G9/(1+$C9*G9^2)"><FONT face=Arial>0.242980</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.33689025635558889"
x:fmla="=(1+$C9)*H9/(1+$C9*H9^2)"><FONT face=Arial>0.336890</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.42755556485782747"
x:fmla="=(1+$C9)*I9/(1+$C9*I9^2)"><FONT face=Arial>0.427556</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.51410325451718708"
x:fmla="=(1+$C9)*J9/(1+$C9*J9^2)"><FONT face=Arial>0.514103</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.59569982302833779"
x:fmla="=(1+$C9)*K9/(1+$C9*K9^2)"><FONT face=Arial>0.595700</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.67155945230844161"
x:fmla="=(1+$C9)*L9/(1+$C9*L9^2)"><FONT face=Arial>0.671559</FONT></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl31 height=21 x:num="1.570798446211249" x:fmla="=A10*(1+C9)"><FONT
face=Arial>1.5708</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 x:num><FONT face=Arial>6</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="1.3492619006131591E-6" x:fmla="=(1-D8)/(1+D8)"><FONT
face=Arial>0.000001</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.99999999999908973" x:fmla="=SQRT(1-C9^2)"><FONT
face=Arial>1.000000</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="4.9067674327440296E-2"
x:fmla="=(1+$C10)*E10/(1+$C10*E10^2)"><FONT face=Arial>0.049068</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.14673047445542708"
x:fmla="=(1+$C10)*F10/(1+$C10*F10^2)"><FONT face=Arial>0.146730</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.24298017990336795"
x:fmla="=(1+$C10)*G10/(1+$C10*G10^2)"><FONT face=Arial>0.242980</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.33688985339235594"
x:fmla="=(1+$C10)*H10/(1+$C10*H10^2)"><FONT face=Arial>0.336890</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.42755509343044112"
x:fmla="=(1+$C10)*I10/(1+$C10*I10^2)"><FONT face=Arial>0.427555</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.51410274419339386"
x:fmla="=(1+$C10)*J10/(1+$C10*J10^2)"><FONT face=Arial>0.514103</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.59569930449260833"
x:fmla="=(1+$C10)*K10/(1+$C10*K10^2)"><FONT face=Arial>0.595699</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.6715589548471862"
x:fmla="=(1+$C10)*L10/(1+$C10*L10^2)"><FONT
face=Arial>0.671559</FONT></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl31 height=21 x:num="1.5707963267956115"
x:fmla="=A11*(1+C10)"><FONT face=Arial>1.5708</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 x:num><FONT face=Arial>7</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="4.5513592894529012E-13" x:fmla="=(1-D9)/(1+D9)"><FONT
face=Arial>0.000000</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num x:fmla="=SQRT(1-C10^2)"><FONT
face=Arial>1.000000</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="4.9067674327418015E-2"
x:fmla="=(1+$C11)*E11/(1+$C11*E11^2)"><FONT face=Arial>0.049068</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.14673047445536175"
x:fmla="=(1+$C11)*F11/(1+$C11*F11^2)"><FONT face=Arial>0.146730</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.24298017990326387"
x:fmla="=(1+$C11)*G11/(1+$C11*G11^2)"><FONT face=Arial>0.242980</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.33688985339222005"
x:fmla="=(1+$C11)*H11/(1+$C11*H11^2)"><FONT face=Arial>0.336890</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.42755509343028208"
x:fmla="=(1+$C11)*I11/(1+$C11*I11^2)"><FONT face=Arial>0.427555</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.51410274419322166"
x:fmla="=(1+$C11)*J11/(1+$C11*J11^2)"><FONT face=Arial>0.514103</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.59569930449243336"
x:fmla="=(1+$C11)*K11/(1+$C11*K11^2)"><FONT face=Arial>0.595699</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num="0.67155895484701833"
x:fmla="=(1+$C11)*L11/(1+$C11*L11^2)"><FONT
face=Arial>0.671559</FONT></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl31 height=21 x:num="1.5707963267948966" x:fmla="=PI()/2"><FONT
face=Arial>1.5708</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24 x:num><FONT face=Arial>8</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num x:fmla="=(1-D10)/(1+D10)"><FONT
face=Arial>0.000000</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25 x:num x:fmla="=SQRT(1-C11^2)"><FONT
face=Arial>1.000000</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28 x:num="4.9067674327418015E-2"
x:fmla="=SIN((PI()/2)*E1/$D$1)"><STRONG><FONT
face=Arial>0.049068</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28 x:num="0.14673047445536175"
x:fmla="=SIN((PI()/2)*F1/$D$1)"><STRONG><FONT
face=Arial>0.146730</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28 x:num="0.24298017990326387"
x:fmla="=SIN((PI()/2)*G1/$D$1)"><STRONG><FONT
face=Arial>0.242980</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28 x:num="0.33688985339222005"
x:fmla="=SIN((PI()/2)*H1/$D$1)"><STRONG><FONT
face=Arial>0.336890</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28 x:num="0.42755509343028208"
x:fmla="=SIN((PI()/2)*I1/$D$1)"><STRONG><FONT
face=Arial>0.427555</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28 x:num="0.51410274419322166"
x:fmla="=SIN((PI()/2)*J1/$D$1)"><STRONG><FONT
face=Arial>0.514103</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28 x:num="0.59569930449243336"
x:fmla="=SIN((PI()/2)*K1/$D$1)"><STRONG><FONT
face=Arial>0.595699</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28 x:num="0.67155895484701833"
x:fmla="=SIN((PI()/2)*L1/$D$1)"><STRONG><FONT
face=Arial>0.671559</FONT></STRONG></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl31 height=21><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl28><STRONG><FONT face=Arial></FONT></STRONG></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl31 height=21><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25><STRONG><FONT face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl34><FONT face=Arial>u</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl26 x:num="0.33114483533111844"
x:fmla="=$A$3*E1/$D$1/2"><STRONG><FONT color=#ff0000
face=Arial>0.331145</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl26 x:num="0.99343450599335537"
x:fmla="=$A$3*F1/$D$1/2"><STRONG><FONT color=#ff0000
face=Arial>0.993435</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl26 x:num="1.6557241766555921"
x:fmla="=$A$3*G1/$D$1/2"><STRONG><FONT color=#ff0000
face=Arial>1.655724</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl26 x:num="2.3180138473178289"
x:fmla="=$A$3*H1/$D$1/2"><STRONG><FONT color=#ff0000
face=Arial>2.318014</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl26 x:num="2.9803035179800661"
x:fmla="=$A$3*I1/$D$1/2"><STRONG><FONT color=#ff0000
face=Arial>2.980304</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl26 x:num="3.6425931886423029"
x:fmla="=$A$3*J1/$D$1/2"><STRONG><FONT color=#ff0000
face=Arial>3.642593</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl26 x:num="4.3048828593045396"
x:fmla="=$A$3*K1/$D$1/2"><STRONG><FONT color=#ff0000
face=Arial>4.304883</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl26 x:num="4.9671725299667768"
x:fmla="=$A$3*L1/$D$1/2"><STRONG><FONT color=#ff0000
face=Arial>4.967173</FONT></STRONG></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
height=21><STRONG><FONT color=#ff0000 face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24><STRONG><FONT color=#ff0000 face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl25><STRONG><FONT color=#ff0000 face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl34><FONT face=Arial>sn(u)</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl32 x:num="0.31954908703883617"
x:fmla="=TANH(E13)+0.25*$D$3^2*(SINH(E13)*COSH(E13)-E13)/((COSH(E13))^2)"><STRONG><FONT
color=#0000ff face=Arial>0.319549</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl32 x:num="0.75882300114541623"
x:fmla="=TANH(F13)+0.25*$D$3^2*(SINH(F13)*COSH(F13)-F13)/((COSH(F13))^2)"><STRONG><FONT
color=#0000ff face=Arial>0.758823</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl32 x:num="0.92963895206495017"
x:fmla="=TANH(G13)+0.25*$D$3^2*(SINH(G13)*COSH(G13)-G13)/((COSH(G13))^2)"><STRONG><FONT
color=#0000ff face=Arial>0.929639</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl32 x:num="0.98079395223222776"
x:fmla="=TANH(H13)+0.25*$D$3^2*(SINH(H13)*COSH(H13)-H13)/((COSH(H13))^2)"><STRONG><FONT
color=#0000ff face=Arial>0.980794</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl32 x:num="0.99485657126175375"
x:fmla="=TANH(I13)+0.25*$D$3^2*(SINH(I13)*COSH(I13)-I13)/((COSH(I13))^2)"><STRONG><FONT
color=#0000ff face=Arial>0.994857</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl32 x:num="0.99862970109752747"
x:fmla="=TANH(J13)+0.25*$D$3^2*(SINH(J13)*COSH(J13)-J13)/((COSH(J13))^2)"><STRONG><FONT
color=#0000ff face=Arial>0.998630</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl32 x:num="0.99963543572022229"
x:fmla="=TANH(K13)+0.25*$D$3^2*(SINH(K13)*COSH(K13)-K13)/((COSH(K13))^2)"><STRONG><FONT
color=#0000ff face=Arial>0.999635</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl32 x:num="0.99990304582527423"
x:fmla="=TANH(L13)+0.25*$D$3^2*(SINH(L13)*COSH(L13)-L13)/((COSH(L13))^2)"><STRONG><FONT
color=#0000ff face=Arial>0.999903</FONT></STRONG></TD></TR>
<TR style="HEIGHT: 15.75pt" height=21>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; HEIGHT: 15.75pt; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
height=21><STRONG><FONT color=#0000ff face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl24><STRONG><FONT color=#0000ff face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"><STRONG><FONT
color=#0000ff face=Arial></FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl29><FONT face=Arial>tanh(u)</FONT></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl33 x:num="0.31954908698329121" x:fmla="=TANH(E13)"><STRONG><FONT
color=#993366 face=Arial>0.319549</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl33 x:num="0.75882300030186534" x:fmla="=TANH(F13)"><STRONG><FONT
color=#993366 face=Arial>0.758823</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl33 x:num="0.92963895030285282" x:fmla="=TANH(G13)"><STRONG><FONT
color=#993366 face=Arial>0.929639</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl33 x:num="0.98079395000070468" x:fmla="=TANH(H13)"><STRONG><FONT
color=#993366 face=Arial>0.980794</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl33 x:num="0.99485656885106011" x:fmla="=TANH(I13)"><STRONG><FONT
color=#993366 face=Arial>0.994857</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl33 x:num="0.99862969862589335" x:fmla="=TANH(J13)"><STRONG><FONT
color=#993366 face=Arial>0.998630</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl33 x:num="0.99963543322897941" x:fmla="=TANH(K13)"><STRONG><FONT
color=#993366 face=Arial>0.999635</FONT></STRONG></TD>
<TD
style="BORDER-BOTTOM: #ece9d8; BORDER-LEFT: #ece9d8; BACKGROUND-COLOR: transparent; BORDER-TOP: #ece9d8; BORDER-RIGHT: #ece9d8"
class=xl33 x:num="0.99990304332792446" x:fmla="=TANH(L13)"><STRONG><FONT
color=#993366
face=Arial>0.999903</FONT></STRONG></TD></TR></TBODY></TABLE></DIV>
<DIV dir=ltr align=left><FONT color=#0000ff size=2
face=Arial></FONT> </DIV>
<DIV dir=ltr align=left><SPAN class=295580208-10012021><FONT color=#0000ff
size=2 face=Arial>Since k' is so small, the approximation formula for the
elliptic sine may be used:</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=295580208-10012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=295580208-10012021><IMG border=0 hspace=0
alt="" align=baseline src="cid:295580208@10012021-0571"></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=295580208-10012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=295580208-10012021><FONT color=#0000ff
size=2 face=Arial>Of course, since (k')^2 is only 1E-8, sn(u) ~ tanh(u).
In fact, this is true to six decimal places until k' is larger than about
0.001. Above k' = about 0.005, the approximation starts to become
inaccurate at the sixth decimal place. That means that, for all of our PDN
networks (where the bandwidth is always at least 3 decades), the elliptic sine
can be taken as the hyperbolic tangent. One only needs the Landen
transformation to find the actual quarter-period of the tanh. That's not
iterative. That's recursive. In any case, the Landen transformation
technique is nearly as easy and more
accurate.</FONT></SPAN></DIV></SPAN><BR>
<DIV dir=ltr lang=en-us class=OutlookMessageHeader align=left>
<HR tabIndex=-1>
<FONT size=2 face=Tahoma><B>From:</B> Ian Fritz [mailto:ijfritz@comcast.net]
<BR><B>Sent:</B> Saturday, January 09, 2021 11:42 PM<BR><B>To:</B> Bernard
Arthur Hutchins, Jr<BR><B>Cc:</B> synth-diy@synth-diy.org; David G Dixon; Brian
Willoughby<BR><B>Subject:</B> Re: [sdiy] 90-degree phase displacement network
calculations<BR></FONT><BR></DIV>
<DIV></DIV><SPAN
style="BACKGROUND-COLOR: #ffecb3; COLOR: #000000; FONT-SIZE: 12px"><SPAN
style="BACKGROUND-COLOR: #ffecb3; COLOR: #000000; FONT-SIZE: 12px"><SPAN
style="PADDING-BOTTOM: 3px; LINE-HEIGHT: 1.6; BACKGROUND-COLOR: #ffecb3; FONT-STYLE: normal; PADDING-LEFT: 3px; PADDING-RIGHT: 3px; COLOR: #000000; FONT-SIZE: 12px; FONT-WEIGHT: normal; PADDING-TOP: 3px">[<STRONG>CAUTION:</STRONG>
Non-UBC Email]</SPAN></SPAN></SPAN>
<DIV><SPAN></SPAN></DIV>
<DIV>
<DIV><SPAN></SPAN></DIV>
<DIV>
<DIV><SPAN></SPAN></DIV>
<DIV>
<DIV><SPAN></SPAN></DIV>
<DIV>
<DIV><SPAN></SPAN></DIV>
<DIV>
<DIV></DIV>
<DIV>Hmmm... If you substitute the first equation into the second, then you have
a recursive definition of the components of the vector k'. I would vote for
calling this closed form, since there is an explicit formula for calculating
each component.</DIV>
<DIV><BR></DIV>
<DIV>As another example, consider, say, the function e^x. This may be
defined recursively by its Taylor expansion, but I think most folks would
consider e^x a closed form expression.</DIV>
<DIV><BR></DIV>
<DIV>Ian</DIV></DIV></DIV></DIV></DIV></DIV></BODY></HTML>