algeqtoseries - Maple Help

gfun

 algeqtoseries
 Puiseux expansions of algebraic functions

 Calling Sequence algeqtoseries(pol, x, y, order, pos_slopes)

Parameters

 pol - polynomial equation in two variables x and y x - name; variable name y - name; variable name order - positive integer order of the expansions pos_slopes - (optional) only branches tending to 0 are computed

Description

 • The algeqtoseries(pol, x, y, order) command computes an expansion of all the branches at the origin.  The equation $\mathrm{pol}\left(x,y\right)=0$ defines a multivalued function y(x).
 • If pos_slopes is specified, only those branches tending to 0 are computed. Note that this function is not designed to compute expansions to a large order. In this case, the differential equation should be used.  For more information, see algeqtodiffeq.

Examples

 > $\mathrm{with}\left(\mathrm{gfun}\right):$
 > $P≔y-{x}^{2}-{x}^{3}{y}^{2}+{x}^{6}{y}^{5}$
 ${P}{≔}{{x}}^{{6}}{}{{y}}^{{5}}{-}{{x}}^{{3}}{}{{y}}^{{2}}{-}{{x}}^{{2}}{+}{y}$ (1)
 > $\mathrm{algeqtoseries}\left(P,x,y,4\right)$
 $\left[{{x}}^{{2}}{+}{O}{}\left({{x}}^{{6}}\right){,}\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{4}}{+}{1}\right)}{{{x}}^{{3}}{{2}}}}{-}\frac{{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{4}}{+}{1}\right)}^{{2}}}{{4}}{+}{\mathrm{O}}{}\left(\sqrt{{x}}\right)\right]$ (2)
 > $\mathrm{algeqtoseries}\left(P,x,y,10,\mathrm{true}\right)$
 $\left[{{x}}^{{2}}{+}{{x}}^{{7}}{+}{O}{}\left({{x}}^{{12}}\right)\right]$ (3)