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,y - variable names
order - positive integer order of the expansions
pos_slopes - (optional) only branches tending to 0 are computed
Description
- The equation
pol(x,y)=0defines a multivalued functiony(x),algeqtoseriescomputes an expansion of all the branches at the origin. - When a fifth argument is given, 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 (see algeqtodiffeq).
Examples
> with(gfun):
P:=y-x^2-x^3*y^2+x^6*y^5;\[x^{6} y^{5}-x^{3} y^{2}-x^{2}+y\]
> algeqtoseries(P,x,y,4);\[\left[x^{2}+\mathrm{O}\! \left(x^{6}\right), \frac{\textrm{RootOf}\left(\_Z^{4}+1\right)}{x^{{3}/{2}}}-\frac{\textrm{RootOf}\left(\_Z^{4}+1\right)^{2}}{4}+O\! \left(\sqrt{x}\right)\right]\]
> algeqtoseries(P,x,y,10,true);\[[x^{2}+x^{7}+\mathrm{O}\! \left(x^{12}\right)]\]