Lonely

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15 years, 13 days

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These are questions asked by Lonely

how can i form a taylor series of:

F(x,y)

here F(x,y) = [f1(x,y); f2(x,y)]

about a point X = [alpha, beta] such that F(alpha, beta) = 0

 

Suppose we have:-
 
x = a * e  + (2*b - 3*a^2) * e^2

Now let us find functions:

F(x) = 1 + a * e + (2 * b - a^2) * e^2 + ......

we are just interested in the first three terms.

Now two such functions can be:

F1(x) =  1 + x + 2 * x^2

and

F2(x) = (1-x)/(1-2*x)

How can we find all such functions F(x)?

we have:

x = c[2]*e/c[1]+(2*c[3]*c[1]-3*c[2]^2)*e^2/c[1]^2+(3*c[4]*c[1]^2-10*c[2]*c[3]*c[1]+8*c[2]^3)*e^3/c[1]^3

how can we find a function "F(x)" the first three terms of whose power series around "e=0"  is given as:

1+c[2]*e/c[1]+(2*c[3]*c[1]-c[2]^2)*e^2/c[1]^2

 

we have the following:

  CM := MultiSeries:-series((c[1]*e+c[2]*e^2+c[3]*e^3+c[4]*e^4+O(e^5))^m, e)

we have the following:

CM := MultiSeries:-series((c[1]*e+c[2]*e^2+c[3]*e^3+c[4]*e^4+O(e^5))^m, e)

CM := c[1]^m*e^m+c[2]*m*c[1]^m*e^(1+m)/c[1]+(1/2)*m*(2*c[3]*c[1]+c[2]^2*m-c[2]^2)*c[1]^m*e^(2+m)/c[1]^2+(1/6)*m*(6*c[4]*c[1]^2+6*c[3]*c[2]*c[1]*m-6*c[3]*c[2]*c[1]+c[2]^3*m^2-3*c[2]^3*m+2*c[2]^3)*c[1]^m*e^(3+m)/c[1]^3+O(e^(4+m))

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