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| Table 67 shows how to compute the analytic series for ''F'' = ‹''f'', ''g''› = ‹((''u'')(''v'')), ((''u'', ''v''))› in terms of coordinates, and Table 68 recaps these results in symbolic terms, agreeing with earlier derivations. | | Table 67 shows how to compute the analytic series for ''F'' = ‹''f'', ''g''› = ‹((''u'')(''v'')), ((''u'', ''v''))› in terms of coordinates, and Table 68 recaps these results in symbolic terms, agreeing with earlier derivations. |
| | | |
− | <pre>
| + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%" |
− | Table 67. Computation of an Analytic Series in Terms of Coordinates | + | |+ Table 67. Computation of an Analytic Series in Terms of Coordinates |
− | o--------o-------o-------o--------o-------o-------o-------o-------o
| + | | |
− | | u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg | | + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
− | o--------o-------o-------o--------o-------o-------o-------o-------o
| + | | |
− | | | | | | | | | | | + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%" |
− | | 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 | | + | | ''u'' |
− | | | | | | | | | | | + | | ''v'' |
− | | | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 | | + | |} |
− | | | | | | | | | | | + | | |
− | | | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 | | + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%" |
− | | | | | | | | | | | + | | d''u'' |
− | | | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 | | + | | d''v'' |
− | | | | | | | | | | | + | |} |
− | o--------o-------o-------o--------o-------o-------o-------o-------o
| + | | |
− | | | | | | | | | | | + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%" |
− | | 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 | | + | | ''u''<font face="courier new">’</font> |
− | | | | | | | | | | | + | | ''v''<font face="courier new">’</font> |
− | | | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 | | + | |} |
− | | | | | | | | | | | + | |- |
− | | | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 | | + | | valign="top" | |
− | | | | | | | | | | | + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
− | | | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 | | + | | 0 || 0 |
− | | | | | | | | | | | + | |} |
− | o--------o-------o-------o--------o-------o-------o-------o-------o
| + | | |
− | | | | | | | | | | | + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
− | | 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 | | + | | 0 || 0 |
− | | | | | | | | | | | + | |- |
− | | | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 | | + | | 0 || 1 |
− | | | | | | | | | | | + | |- |
− | | | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 | | + | | 1 || 0 |
− | | | | | | | | | | | + | |- |
− | | | 1 1 | 0 1 | | 1 0 | 0 0 | 1 0 | 1 0 | | + | | 1 || 1 |
− | | | | | | | | | | | + | |} |
− | o--------o-------o-------o--------o-------o-------o-------o-------o
| + | | |
− | | | | | | | | | | | + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
− | | 1 1 | 0 0 | 1 1 | 1 1 | 1 1 | 0 0 | 0 0 | 0 0 | | + | | 0 || 0 |
− | | | | | | | | | | | + | |- |
− | | | 0 1 | 1 0 | | 1 0 | 0 1 | 0 1 | 0 0 | | + | | 0 || 1 |
− | | | | | | | | | | | + | |- |
− | | | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 | | + | | 1 || 0 |
− | | | | | | | | | | | + | |- |
− | | | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 | | + | | 1 || 1 |
− | | | | | | | | | | | + | |} |
− | o--------o-------o-------o--------o-------o-------o-------o-------o
| + | |- |
− | </pre> | + | | valign="top" | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 1 |
| + | |} |
| + | | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | | 0 || 1 |
| + | |- |
| + | | 1 || 0 |
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| + | | 1 || 1 |
| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 1 |
| + | |- |
| + | | 0 || 0 |
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| + | | 1 || 1 |
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| + | | 1 || 0 |
| + | |} |
| + | |- |
| + | | valign="top" | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 1 || 0 |
| + | |} |
| + | | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
| + | |- |
| + | | 0 || 1 |
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| + | | 1 || 0 |
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| + | | 1 || 1 |
| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 1 || 0 |
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| + | | 1 || 1 |
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| + | | 0 || 0 |
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| + | | 0 || 1 |
| + | |} |
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| + | | valign="top" | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 1 || 1 |
| + | |} |
| + | | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 1 || 1 |
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| + | | 1 || 0 |
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| + | | 0 || 1 |
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| + | | 0 || 0 |
| + | |} |
| + | | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
| + | |- |
| + | | 0 || 1 |
| + | |- |
| + | | 1 || 0 |
| + | |- |
| + | | 1 || 1 |
| + | |} |
| + | |} |
| + | | |
| + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%" |
| + | | <math>\epsilon</math>''f'' |
| + | | <math>\epsilon</math>''g'' |
| + | |} |
| + | | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%" |
| + | | E''f'' |
| + | | E''g'' |
| + | |} |
| + | | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%" |
| + | | D''f'' |
| + | | D''g'' |
| + | |} |
| + | | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%" |
| + | | d''f'' |
| + | | d''g'' |
| + | |} |
| + | | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%" |
| + | | d<sup>2</sup>''f'' |
| + | | d<sup>2</sup>''g'' |
| + | |} |
| + | |- |
| + | | valign="top" | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 1 |
| + | |} |
| + | | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 1 |
| + | |- |
| + | | 1 || 0 |
| + | |- |
| + | | 1 || 0 |
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| + | | 1 || 1 |
| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | | 1 || 1 |
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| + | | 1 || 1 |
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| + | | 1 || 0 |
| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | | 0 || 0 |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 1 || 0 |
| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 1 || 0 |
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| + | | 0 || 1 |
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| + | | 1 || 1 |
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| + | | 1 || 0 |
| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | | 0 || 0 |
| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | | 1 || 1 |
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| + | | 0 || 1 |
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| + | | 1 || 0 |
| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | | valign="top" | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 1 || 0 |
| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 1 || 0 |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | |} |
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| + | | valign="top" | |
| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 1 || 1 |
| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 1 || 1 |
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| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | |} |
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| + | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%" |
| + | | 0 || 0 |
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| + | | 1 || 0 |
| + | |} |
| + | |} |
| + | |} |
| + | <br> |
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| {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%" | | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%" |