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/* find the taylor series of a function of x which is defined implicitly
by a polynomial equation in x and y, if there is a unique branch at a particular
pt.
implicit_taylor(f,pt,deg,init_a0) where F is the implicit defining equation,
pt is the value of x around which we wish to form the expansion, deg is the
highest degree of x to appear in the expansion, and init_a0 is the initial value
of y at point, for the branch for which we will form the expansion.
(C2) implicit_taylor(y^3+x*y^2+x*y+y-2,0,10,1);
10 9 8 7 6 5 4 3
83787 X 2847 X 5853 X 189 X 21 X 23 X 15 X X
(D2) - --------- + ------- + ------- - ------ - ----- + ----- - ----- - --
134217728 8388608 4194304 65536 32768 2048 1024 32
2
3 X X
+ ---- - - + 1
16 2
*/
implicit_taylor(f,pt,deg,init_a0):=
block([n:deg,ratwtlvl:deg,ratweights:[],expansion,res,ans:[],eqns],
ratweight(x,1),
expansion:sum((x-pt)^i*concat(a,i),i,0,n),
res:sublis([y=expansion],f),
res:ratsimp(res),
eqns:create_list(coeff(res,x,i),i,1,n),
ans:[a0=init_a0],
eqns:sublis(ans,eqns),
for i:1 thru n do (ans:append(ans,solve(eqns[i],concat(a,i))),
eqns: sublis(ans,eqns)),
sublis(ans,expansion));
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