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MyWikiBiz, Author Your Legacy — Thursday November 28, 2024
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{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:70%"
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|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math>
|+ Table 51.  Computation of an Analytic Series in Symbolic Terms
+
|- style="height:35px; background:ghostwhite; width:100%"
|
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| <math>u\!</math>
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
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| <math>v\!</math>
| ''u'' || ''v''
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| style="border-left:1px solid black" | <math>J\!</math>
|}
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| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math>
|
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| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math>
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
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| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math>
| ''J''
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| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math>
|}
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|
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{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
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| E''J''
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|
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{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
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| D''J''
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|
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{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
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| d''J''
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{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
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| d<sup>2</sup>''J''
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{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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| 0 || 1
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| 1 || 0
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| 1 || 1
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{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| 0
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| 0
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{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| &nbsp;d''u''&nbsp;&nbsp;d''v''&nbsp;
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|-
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| &nbsp;d''u''&nbsp;(d''v'')
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|-
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| (d''u'')&nbsp;d''v''&nbsp;
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|-
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| (d''u'')(d''v'')
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{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| &nbsp;&nbsp;d''u''&nbsp;&nbsp;d''v''&nbsp;&nbsp;
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|-
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| &nbsp;&nbsp;d''u''&nbsp;(d''v'')&nbsp;
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|-
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| &nbsp;(d''u'')&nbsp;d''v''&nbsp;&nbsp;
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|-
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| ((d''u'')(d''v''))
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|}
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{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| ()
   
|-
 
|-
| d''u''
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| style="border-top:1px solid black" |
|-
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<math>\begin{matrix}
| d''v''
+
0
|-
+
\\[4pt]
| (d''u'', d''v'')
+
0
|}
+
\\[4pt]
|
+
1
{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
\\[4pt]
| d''u'' d''v''
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1
|-
+
\end{matrix}</math>
| d''u'' d''v''
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| style="border-top:1px solid black" |
|-
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<math>\begin{matrix}
| d''u'' d''v''
+
0
|-
+
\\[4pt]
| d''u'' d''v''
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1
|}
+
\\[4pt]
 +
0
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\\[4pt]
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1
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\end{matrix}</math>
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| style="border-top:1px solid black; border-left:1px solid black" |
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<math>\begin{matrix}
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0
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\\[4pt]
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0
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\\[4pt]
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0
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\\[4pt]
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1
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\end{matrix}</math>
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| style="border-top:1px solid black; border-left:1px solid black" |
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<math>\begin{matrix}
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\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}
 +
\\[4pt]
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\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
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\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
\\[4pt]
 +
\texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
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<math>\begin{matrix}
 +
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}
 +
\\[4pt]
 +
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
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\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
\\[4pt]
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\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
\end{matrix}</math>
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| style="border-top:1px solid black; border-left:1px solid black" |
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<math>\begin{matrix}
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0
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\\[4pt]
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\mathrm{d}u
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\\[4pt]
 +
\mathrm{d}v
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\\[4pt]
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\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
\end{matrix}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
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\mathrm{d}u \cdot \mathrm{d}v
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\\[4pt]
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\mathrm{d}u \cdot \mathrm{d}v
 +
\\[4pt]
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\mathrm{d}u \cdot \mathrm{d}v
 +
\\[4pt]
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\mathrm{d}u \cdot \mathrm{d}v
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\end{matrix}</math>
 
|}
 
|}
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