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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 02:36, 9 September 2013

146 bytes removed , 02:36, 9 September 2013

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Line 5,191: Line 5,191:

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{| align="center" cellpadding="1" 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%"

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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%"

|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math>

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| ~~width~~="~~50%~~" |

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|- style="height:40px; background:ghostwhite"

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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:100%~~"

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| style="width:8%; border-bottom:1px solid black" | <math>u\!</math>

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|- style="~~height~~:~~35px~~; ~~background~~:~~ghostwhite~~"

+

| style="width:8%; border-bottom:1px solid black" | <math>v\!</math>

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| style="width:16%; border-bottom:1px solid black" | <math>u\!</math>

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| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math>

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| style="width:16%; border-bottom:1px solid black" | <math>v\!</math>

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| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math>

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| style="width:16%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math>

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| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math>

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| style="width:16%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math>

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| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math>

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| style="width:16%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math>

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| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math>

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| style="width:16%; border-bottom:1px solid black" | <math>v'\!</math>

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| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math>

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| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math>

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| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math>

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| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math>

|-

| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>

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| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>

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| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>

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| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>

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| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>

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| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>

|-

| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>

| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>

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| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math>

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| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}1\\0\\1\\0\end{matrix}~~</math>~~

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<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math>

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|-

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~~| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>~~

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~~| style="vertical-align:top; border-top:1px solid black" | <math>0~~\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}~~</math>~~

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<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math>

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~~| style="vertical-align:top; border-top:1px solid black" |~~

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~~<math>~~\~~begin{matrix}0\\1\\0\\1\end{matrix}~~</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}1\\1\\0\\0\end{matrix}</math>

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>

|-

| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>

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| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>

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| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}1\\1\\0\\0\end{matrix}</math>

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<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}1\\0\\1~~\\0~~\end{matrix}~~</math>~~

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>

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|}

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| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math>

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~~| width="50%" |~~

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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:100%"

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~~|- style="height:35px; background:ghostwhite"~~

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~~| style="width:20%; border-bottom:1px solid black" | <math>\boldsymbol\varepsilon J~~\!</math>

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| style="~~width:20%; border~~-~~bottom~~:~~1px solid black" | <math>\mathrm{E}J\!</math>~~

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~~| style="width:20%~~; border-~~bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math>~~

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~~| style="width:20%; border-bottom~~:1px solid black; border-left:~~1px solid black" | <math>\mathrm{d}J\!</math>~~

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~~| style="width:20%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math>~~

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|-

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~~| style="vertical-align:top; border-top:1px solid~~ black" | <math>0\!</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}</math>

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<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}</math>

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<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}</math>

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>

|-

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| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>

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| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>

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| style="vertical-align:top; border-top:1px solid black" |

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| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>

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<math>~~\begin{matrix}0\\0\\~~1\~~\0\end{matrix}~~</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\0\\1\\0\end{matrix}~~</math>~~

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<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>

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~~| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |~~

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~~<math>~~\~~begin{matrix}0\\0\\1\\1\end{matrix}~~</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}~~</math>~~

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<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>

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|-

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~~| style="vertical-align:top; border-top:1px solid black" | <math>0~~\!~~</math>~~

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~~| style="vertical-align:top; border-top:1px solid black" |~~

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~~<math>\begin{matrix}0\\1\\0\\0\end{matrix}~~</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\1\\0\\0\end{matrix}~~</math>~~

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<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math>

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~~| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |~~

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~~<math>\begin{matrix}0\\1\\0\\1~~\~~end{matrix}~~</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}</math>

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<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math>

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|-

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| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math>

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| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}1\\0\\0\\0\end{matrix}</math>

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<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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<math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>

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<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>

| style="vertical-align:top; border-top:1px solid black" |

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}</math>

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<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>

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|}

Line 6,934: Line 6,923:

|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math>

|- style="height:40px; background:ghostwhite"

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| style="border-bottom:1px solid black" | <math>u\!</math>

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| style="width:6%; border-bottom:1px solid black" | <math>u\!</math>

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| style="border-bottom:1px solid black" | <math>v\!</math>

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| style="width:6%; border-bottom:1px solid black" | <math>v\!</math>

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| style="border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math>

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| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math>

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| style="border-bottom:1px solid black" | <math>\mathrm{d}v\!</math>

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| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math>

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| style="border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math>

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| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math>

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| style="border-bottom:1px solid black" | <math>v'\!</math>

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| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math>

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| style="border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math>

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| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math>

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| style="border-bottom:1px solid black" | <math>g\!</math>

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| style="width:6%; border-bottom:1px solid black" | <math>g\!</math>

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| style="border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math>

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| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math>

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| style="border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math>

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| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math>

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| style="border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math>

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| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math>

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| style="border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math>

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| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math>

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| style="border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math>

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| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math>

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| style="border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math>

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| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math>

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| style="border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math>

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| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math>

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| style="border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math>

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| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math>

|-

| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>

Line 6,961: Line 6,950:

| style="vertical-align:top; border-top:1px solid black" |

<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>

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| style="vertical-align:top; border-top:1px solid black; border-left: 4px double black" | <math>0\!</math>

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| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math>

| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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| style="vertical-align:top; border-top:1px solid black" |

<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math>

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| style="vertical-align:top; border-top:1px solid black; border-left: 4px double black" | <math>1\!</math>

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| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math>

| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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| style="vertical-align:top; border-top:1px solid black" |

<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>

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| style="vertical-align:top; border-top:1px solid black; border-left: 4px double black" | <math>1\!</math>

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| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math>

| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

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| style="vertical-align:top; border-top:1px solid black" |

<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math>

−

| style="vertical-align:top; border-top:1px solid black; border-left: 4px double black" | <math>1\!</math>

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| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math>

| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>

| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |

Jon Awbrey

12,080

edits