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Let r[y] \ be a branch of the inverse function of q in a small neighborhood about y = 0, \ meaning that q[r[y]] = 0 for any sufficient small y. Then r[0] is clearly a \ root of q. In order to find r[0] as a function of a, b and c, we will compute \ the formal power series of r[y] in y at zero; for this it suffices to invert \ q[x] as a formal power series:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "s = InverseSeries[Series[a x^2 + b x + c, {x, 0, 9}], y]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ SeriesData[y, c, {b^(-1), -(a/b^3), (2*a^2)/b^5}, 1, 4, 1]\ \>", "\<\ 2 2 3 -c + y a (-c + y) 2 a (-c + y) 4 ------ - ----------- + -------------- + O[-c + y] b 3 5 b b\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "This gives nine terms of the series, which we evaluate at zero to get \ r[0]:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Normal[s] /. y -> 0"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ -(c/b) - (a*c^2)/b^3 - (2*a^2*c^3)/b^5\ \>", "\<\ 2 2 3 c a c 2 a c -(-) - ---- - ------- b 3 5 b b\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["The general term appears to be: "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "tau = -c^(k + 1) / b^(2k + 1) a^k Binomial[2 k, k]/(k + 1);"], "Input", AspectRatioFixed->True], Cell[TextData[ "We can sum this series to get a closed-form solution for one of the roots of \ the original quadratic equation:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["<True], Cell[CellGroupData[{Cell[TextData["SymbolicSum[tau, {k, 0, Infinity}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ (b*(-1 + (1 - (4*a*c)/b^2)^(1/2)))/(2*a)\ \>", "\<\ 4 a c b (-1 + Sqrt[1 - -----]) 2 b ------------------------ 2 a\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Quintic"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Now we show how this method works for a quintic of the form: "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["t^5 - t - rho"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ -rho - t + t^5\ \>", "\<\ 5 -rho - t + t\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["Proceeding as above, we compute:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "s = InverseSeries[Series[t^5 - t - rho, {t, 0, 29}], y];\nNormal[s] /. y -> \ 0"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ -rho - rho^5\ \>", "\<\ 5 -rho - rho\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ " In version 2.3 this gives the first eight terms in the series for one of \ the roots. The general term of this series can be written as:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["tau = - rho^(4k + 1) Binomial[5k, k] / (4k + 1);"], "Input", AspectRatioFixed->True], Cell[TextData[ "This series can be summed in hypergeometric functions:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["SymbolicSum[tau, {k, 0, Infinity}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ -(rho*HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5}, {1/2, 3/4, 5/4}, (3125*rho^4)/256])\ \>", "\<\ 1 2 3 4 1 3 5 -(rho HypergeometricPFQ[{-, -, -, -}, {-, -, -}, 5 5 5 5 2 4 4 4 3125 rho ---------]) 256\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "It is possible to get the other four roots by building a Riemann surface for \ this function. In principle this method will work for a polynomial of any \ degree. "], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 640}, {0, 460}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{520, 365}, WindowMargins->{{12, Automatic}, {30, Automatic}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, MacintoshSystemPageSetup->"\<\ AVU/IFiQKFD000000V:^/09R]g0000000OVaH097bCP0AP1Y06`0I@1^0642HZj` 0V:gT0000001nK500TO9>000000000000000009R[[0000000000000000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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