https://mizugadro.mydns.jp/t/index.php?title=Abelpower&feed=atom&action=history
Abelpower - Revision history
2024-03-28T21:07:24Z
Revision history for this page on the wiki
MediaWiki 1.31.16
https://mizugadro.mydns.jp/t/index.php?title=Abelpower&diff=28405&oldid=prev
T: Text replacement - "\$([^\$]+)\$" to "\\(\1\\)"
2019-07-30T09:47:41Z
<p>Text replacement - "\$([^\$]+)\$" to "\\(\1\\)"</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 09:47, 30 July 2019</td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:AuPow2Plot.jpg|360px|thumb|<del class="diffchange diffchange-inline">$</del>y\!=<del class="diffchange diffchange-inline">$</del>[[AuPow]]<del class="diffchange diffchange-inline">$</del>_2(x)<del class="diffchange diffchange-inline">$</del>, blue; <del class="diffchange diffchange-inline">$</del>y\!=<del class="diffchange diffchange-inline">$</del>[[AdPow]]<del class="diffchange diffchange-inline">$</del>_2(x)<del class="diffchange diffchange-inline">$</del>, red]]</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:AuPow2Plot.jpg|360px|thumb|<ins class="diffchange diffchange-inline">\(</ins>y\!=<ins class="diffchange diffchange-inline">\)</ins>[[AuPow]]<ins class="diffchange diffchange-inline">\(</ins>_2(x)<ins class="diffchange diffchange-inline">\)</ins>, blue; <ins class="diffchange diffchange-inline">\(</ins>y\!=<ins class="diffchange diffchange-inline">\)</ins>[[AdPow]]<ins class="diffchange diffchange-inline">\(</ins>_2(x)<ins class="diffchange diffchange-inline">\)</ins>, red]]</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:Aupower2map.jpg|360px|thumb|<del class="diffchange diffchange-inline">$</del>u\!+\!\mathrm i v\!=\!\mathrm{AuPow}_2(x\!+\!\mathrm i y)<del class="diffchange diffchange-inline">$</del>]]</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:Aupower2map.jpg|360px|thumb|<ins class="diffchange diffchange-inline">\(</ins>u\!+\!\mathrm i v\!=\!\mathrm{AuPow}_2(x\!+\!\mathrm i y)<ins class="diffchange diffchange-inline">\)</ins>]]</div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:Adpow2map.jpg|360px|thumb|<del class="diffchange diffchange-inline">$</del>u\!+\!\mathrm i v\!=\!\mathrm{AdPow}_2(x\!+\!\mathrm i y)<del class="diffchange diffchange-inline">$</del>]]</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:Adpow2map.jpg|360px|thumb|<ins class="diffchange diffchange-inline">\(</ins>u\!+\!\mathrm i v\!=\!\mathrm{AdPow}_2(x\!+\!\mathrm i y)<ins class="diffchange diffchange-inline">\)</ins>]]</div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Abelpower]] appears [[inverse function]] of the [[superpower]] function.</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Abelpower]] appears [[inverse function]] of the [[superpower]] function.</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[Abelpower]] is [[Abel function]] for the [[power function]], id est, transfer function <del class="diffchange diffchange-inline">$</del>T(z)\!=\!z^a<del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[Abelpower]] is [[Abel function]] for the [[power function]], id est, transfer function <ins class="diffchange diffchange-inline">\(</ins>T(z)\!=\!z^a<ins class="diffchange diffchange-inline">\)</ins></div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <del class="diffchange diffchange-inline">$</del>a<del class="diffchange diffchange-inline">$</del> is parameter. Usually, it is assumed, that <del class="diffchange diffchange-inline">$</del>a\!>\!1<del class="diffchange diffchange-inline">$</del>.</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <ins class="diffchange diffchange-inline">\(</ins>a<ins class="diffchange diffchange-inline">\)</ins> is parameter. Usually, it is assumed, that <ins class="diffchange diffchange-inline">\(</ins>a\!>\!1<ins class="diffchange diffchange-inline">\)</ins>.</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Superpower]] and [[Abelpower]]==</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==[[Superpower]] and [[Abelpower]]==</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For the transfer function <del class="diffchange diffchange-inline">$</del>T(z)\!=\!z^a<del class="diffchange diffchange-inline">$</del>, the two superfunctions </div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For the transfer function <ins class="diffchange diffchange-inline">\(</ins>T(z)\!=\!z^a<ins class="diffchange diffchange-inline">\)</ins>, the two superfunctions </div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$</del>F(z)=\exp(a^z)<del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\(</ins>F(z)=\exp(a^z)<ins class="diffchange diffchange-inline">\)</ins></div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>and </div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>and </div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$</del>F(z)=\exp(-a^z)<del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\(</ins>F(z)=\exp(-a^z)<ins class="diffchange diffchange-inline">\)</ins></div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>are denoted as [[SuPow]] and [[SdPow]].</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>are denoted as [[SuPow]] and [[SdPow]].</div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For [[superpower]] function <del class="diffchange diffchange-inline">$</del>F<del class="diffchange diffchange-inline">$</del>, the inverse function <del class="diffchange diffchange-inline">$</del>G\!=\!F^{-1}<del class="diffchange diffchange-inline">$</del> is solution of the Abel equation,</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For [[superpower]] function <ins class="diffchange diffchange-inline">\(</ins>F<ins class="diffchange diffchange-inline">\)</ins>, the inverse function <ins class="diffchange diffchange-inline">\(</ins>G\!=\!F^{-1}<ins class="diffchange diffchange-inline">\)</ins> is solution of the Abel equation,</div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$</del>G(T(z))=G(z)+1<del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\(</ins>G(T(z))=G(z)+1<ins class="diffchange diffchange-inline">\)</ins></div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The two solutions (corresponding to the two superfunctions <del class="diffchange diffchange-inline">$</del>F<del class="diffchange diffchange-inline">$</del> above) are:</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The two solutions (corresponding to the two superfunctions <ins class="diffchange diffchange-inline">\(</ins>F<ins class="diffchange diffchange-inline">\)</ins> above) are:</div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$</del>G(z)=\mathrm{AuPow}_a^{-1}(z)=\mathrm{AuPow}_a(z)<del class="diffchange diffchange-inline">$</del> <del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\(</ins>G(z)=\mathrm{AuPow}_a^{-1}(z)=\mathrm{AuPow}_a(z)<ins class="diffchange diffchange-inline">\)</ins> <ins class="diffchange diffchange-inline">\(</ins></div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>=\log_a\!\big(\ln(z)\big)<del class="diffchange diffchange-inline">$</del> <del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>=\log_a\!\big(\ln(z)\big)<ins class="diffchange diffchange-inline">\)</ins> <ins class="diffchange diffchange-inline">\(</ins></div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>=\ln\!\big(\ln(z)\big)/\ln(a)<del class="diffchange diffchange-inline">$</del> <del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>=\ln\!\big(\ln(z)\big)/\ln(a)<ins class="diffchange diffchange-inline">\)</ins> <ins class="diffchange diffchange-inline">\(</ins></div></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>=\ln^2(z)/\ln(a) <del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>=\ln^2(z)/\ln(a) <ins class="diffchange diffchange-inline">\)</ins></div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>and</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>and</div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$</del>G(z)=\mathrm{AdPow}_a^{-1}(z)=\mathrm{AdPow}_a(z)<del class="diffchange diffchange-inline">$</del> <del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\(</ins>G(z)=\mathrm{AdPow}_a^{-1}(z)=\mathrm{AdPow}_a(z)<ins class="diffchange diffchange-inline">\)</ins> <ins class="diffchange diffchange-inline">\(</ins></div></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>=\log_a\!\big(\ln(1/z)\big)<del class="diffchange diffchange-inline">$</del> <del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>=\log_a\!\big(\ln(1/z)\big)<ins class="diffchange diffchange-inline">\)</ins> <ins class="diffchange diffchange-inline">\(</ins></div></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>=\ln\!\big(\ln(1/z)\big)/\ln(a)<del class="diffchange diffchange-inline">$</del> <del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>=\ln\!\big(\ln(1/z)\big)/\ln(a)<ins class="diffchange diffchange-inline">\)</ins> <ins class="diffchange diffchange-inline">\(</ins></div></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>=\ln^2(1/z)/\ln(a)<del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>=\ln^2(1/z)/\ln(a)<ins class="diffchange diffchange-inline">\)</ins></div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>These functions are called [[Abelpower]] functions.</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>These functions are called [[Abelpower]] functions.</div></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For <del class="diffchange diffchange-inline">$</del>a\!=\!2<del class="diffchange diffchange-inline">$</del>, the explicit plots of these two functions are shown in figure at top.</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For <ins class="diffchange diffchange-inline">\(</ins>a\!=\!2<ins class="diffchange diffchange-inline">\)</ins>, the explicit plots of these two functions are shown in figure at top.</div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The [[complex map]]s of them are shown below with levels of constant real part and levels of constant imaginary part;</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The [[complex map]]s of them are shown below with levels of constant real part and levels of constant imaginary part;</div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$</del>u\!+\!\mathrm i v=G(x\!+\!\mathrm i y)<del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\(</ins>u\!+\!\mathrm i v=G(x\!+\!\mathrm i y)<ins class="diffchange diffchange-inline">\)</ins></div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:Apow2ma4.jpg|600px|thumb|<del class="diffchange diffchange-inline">$</del>u\!+\!\mathrm i v=\mathrm{AuPow}_2(x\!+\!\mathrm i y)~ ~<del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:Apow2ma4.jpg|600px|thumb|<ins class="diffchange diffchange-inline">\(</ins>u\!+\!\mathrm i v=\mathrm{AuPow}_2(x\!+\!\mathrm i y)~ ~<ins class="diffchange diffchange-inline">\)</ins></div></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">$</del>u\!+\!\mathrm i v=\mathrm{AdPow}_2(x\!+\!\mathrm i y)<del class="diffchange diffchange-inline">$</del>]]</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">\(</ins>u\!+\!\mathrm i v=\mathrm{AdPow}_2(x\!+\!\mathrm i y)<ins class="diffchange diffchange-inline">\)</ins>]]</div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In the most of the complex plane, values of functions </div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In the most of the complex plane, values of functions </div></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[AuPow]]<del class="diffchange diffchange-inline">$</del>_a<del class="diffchange diffchange-inline">$</del> and</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[AuPow]]<ins class="diffchange diffchange-inline">\(</ins>_a<ins class="diffchange diffchange-inline">\)</ins> and</div></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[AdPow]]<del class="diffchange diffchange-inline">$</del>_a<del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[AdPow]]<ins class="diffchange diffchange-inline">\(</ins>_a<ins class="diffchange diffchange-inline">\)</ins></div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>differ for a constant; however, they have different cut lines. </div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>differ for a constant; however, they have different cut lines. </div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In order to compare the systems of cuts for these two functions, the zooming of the central parts of maps </div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In order to compare the systems of cuts for these two functions, the zooming of the central parts of maps </div></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>for [[AuPow]]<del class="diffchange diffchange-inline">$</del>_2<del class="diffchange diffchange-inline">$</del> and</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>for [[AuPow]]<ins class="diffchange diffchange-inline">\(</ins>_2<ins class="diffchange diffchange-inline">\)</ins> and</div></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[AdPow]]<del class="diffchange diffchange-inline">$</del>_2<del class="diffchange diffchange-inline">$</del> are shown together in figure</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[AdPow]]<ins class="diffchange diffchange-inline">\(</ins>_2<ins class="diffchange diffchange-inline">\)</ins> are shown together in figure</div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Application==</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Application==</div></td>
</tr>
</table>
T
https://mizugadro.mydns.jp/t/index.php?title=Abelpower&diff=14345&oldid=prev
Maintenance script at 21:57, 30 November 2018
2018-11-30T21:57:33Z
<p></p>
<p><b>New page</b></p><div>[[File:AuPow2Plot.jpg|360px|thumb|$y\!=$[[AuPow]]$_2(x)$, blue; $y\!=$[[AdPow]]$_2(x)$, red]]<br />
<br />
[[File:Aupower2map.jpg|360px|thumb|$u\!+\!\mathrm i v\!=\!\mathrm{AuPow}_2(x\!+\!\mathrm i y)$]]<br />
[[File:Adpow2map.jpg|360px|thumb|$u\!+\!\mathrm i v\!=\!\mathrm{AdPow}_2(x\!+\!\mathrm i y)$]]<br />
<br />
[[Abelpower]] appears [[inverse function]] of the [[superpower]] function.<br />
<br />
[[Abelpower]] is [[Abel function]] for the [[power function]], id est, transfer function $T(z)\!=\!z^a$<br />
where $a$ is parameter. Usually, it is assumed, that $a\!>\!1$.<br />
<br />
==[[Superpower]] and [[Abelpower]]==<br />
<br />
For the transfer function $T(z)\!=\!z^a$, the two superfunctions <br />
<br />
$F(z)=\exp(a^z)$<br />
<br />
and <br />
<br />
$F(z)=\exp(-a^z)$<br />
<br />
are denoted as [[SuPow]] and [[SdPow]].<br />
For [[superpower]] function $F$, the inverse function $G\!=\!F^{-1}$ is solution of the Abel equation,<br />
<br />
$G(T(z))=G(z)+1$<br />
<br />
The two solutions (corresponding to the two superfunctions $F$ above) are:<br />
<br />
$G(z)=\mathrm{AuPow}_a^{-1}(z)=\mathrm{AuPow}_a(z)$ $<br />
=\log_a\!\big(\ln(z)\big)$ $<br />
=\ln\!\big(\ln(z)\big)/\ln(a)$ $<br />
=\ln^2(z)/\ln(a) $<br />
<br />
and<br />
<br />
$G(z)=\mathrm{AdPow}_a^{-1}(z)=\mathrm{AdPow}_a(z)$ $<br />
=\log_a\!\big(\ln(1/z)\big)$ $<br />
=\ln\!\big(\ln(1/z)\big)/\ln(a)$ $<br />
=\ln^2(1/z)/\ln(a)$<br />
<br />
These functions are called [[Abelpower]] functions.<br />
For $a\!=\!2$, the explicit plots of these two functions are shown in figure at top.<br />
The [[complex map]]s of them are shown below with levels of constant real part and levels of constant imaginary part;<br />
<br />
$u\!+\!\mathrm i v=G(x\!+\!\mathrm i y)$<br />
<br />
[[File:Apow2ma4.jpg|600px|thumb|$u\!+\!\mathrm i v=\mathrm{AuPow}_2(x\!+\!\mathrm i y)~ ~$<br />
$u\!+\!\mathrm i v=\mathrm{AdPow}_2(x\!+\!\mathrm i y)$]]<br />
In the most of the complex plane, values of functions <br />
[[AuPow]]$_a$ and<br />
[[AdPow]]$_a$<br />
differ for a constant; however, they have different cut lines. <br />
In order to compare the systems of cuts for these two functions, the zooming of the central parts of maps <br />
for [[AuPow]]$_2$ and<br />
[[AdPow]]$_2$ are shown together in figure<br />
<br />
==Application==<br />
<br />
[[Abelpower]] function shows behaviour, similar to those for other [[abelfunction]]s, (see [[Table of superfunctions]]), that cannot be expressed as elementary functions.<br />
<br />
So, the primary application of the [[superpower]] and [[abelpower]] functions is testing, tracing of general algorithms for evaluation of [[superfunctions]] and [[abelfunction]]s. Superpower and abelpower are easy to evaluate through their explicit representations; so, the precision of some general method of evaluation is easy to estimate.<br />
<br />
Articles about [[Abelpower]]<br />
function are loaded in [[TORI]] as draft of for the special chapter of the Book [[Superfunctions]].<br />
Such a chapter is expected to be especially useful for the colleagues who complain that "The formalism of superfunctions is too complicated"<br />
<br />
==References==<br />
<references/><br />
<br />
==Keywords==<br />
[[Abel function]],<br />
[[AdPow]],<br />
[[AuPow]],<br />
[[Power function]],<br />
[[Superfunction]],<br />
[[Superfunctions]],<br />
[[SdPow]],<br />
[[SuPow]],<br />
<br />
[[Category:Abel equation]]<br />
[[Category:Abel function]]<br />
[[Category:Book]]<br />
[[Category:Power function]]<br />
[[Category:Superfunction]]<br />
[[Category:Superfunctions]]</div>
Maintenance script