## Implicit function: Taylor, Puiseux and Newton's diagram

### Implicit function: Taylor, Puiseux and Newton's diagram

Asymptotic expansion of implicit functions.
(Evgeniy Maevskiy: emaevskiy at e-math dot ru)

=== imp_taylor ===

X - list of variables
F - one expression (d=1) or a list of expressions of the variables X, d:length(F)
A - a fixed point, the value of X, F(A)=0
n - the order of the expansion

Let m:length(X). List X splits into two parts: X[1],...,X[m-d] - the independent variables, X[m-d+1],...,X[m] - the dependent variables. If F at point A satisfies the implicit function theorem, the imp_taylor(F,X,A,n) returns a single expansion or a list of expansions for X[m-d+1],...,X[m] in powers of the differences (X[1]-A[1]), ..., (X[m-d]-A[m-d]) up to degree n inclusive.

Description of the method:
Let F: R^{m-d} * R^d -> R^d, F(A)=0. Let D is the Jacobian (matrix) of F at A. Then the only solution y(x) to the equation F(x,y)=0 can be found by the method of successive approximations: y_{k+1}(x)=y_k(x)-D^{-1}F(x,y_k(x))

Examples:
Code: Select all
`imp_taylor(x-asin(y),[x,y],[%pi/6,1/2],5);imp_taylor(x+s*sin(x+t)-x*exp(x-s),[t,s,x],[%pi,1,0],3);imp_taylor([x[1]-log(y[1]),x[2]-exp(y[2])],[x[1],x[2],y[1],y[2]],[0,1,1,0],5);`

=== imp_puiseux ===

X - list of 2 variables
F - an expression of the variables X
A - a fixed point, the value of X, F(A)=0
n - the order of the Puiseux expansion

Let F: R * R -> R satisfies the Taylor's theorem of n-th order. Let for simplicity X=[x,y]. Further suppose that:

at(F,[x=a,y=b])=0
at(diff(F,y,i),y=b)=0 for i=1,...,p-1
D: 1/p!* at(diff(F,y,p),[x=a,y=b])#0

Then the Puiseux's expansions of y(x) in a neighborhood of x=a can be found by the method of successive approximations: y_{k+1}^p(x)=y_k^p(x)-D^{-1}F(x,y_k(x))

Examples:
Code: Select all
`imp_puiseux(x-y*exp(y),[x,y],[-%e^-1,-1],5);imp_puiseux(y-2*sin(y/2)-sin(x)*(y^4+y^5)+cos(x)*(y^3-y^4),[x,y],[0,0],7);`

=== imp_newton ===

X - list of 2 variables
F - an irreducible polynomial of the variables X
A - a fixed point, the value of X, F(A)=0
n - the number of terms in the Puiseux expansion

Newton's diagram method for Puiseux's expansions of all branches of the algebraic function y(x)

Examples:
Code: Select all
`imp_newton(x^2+y^2-1,[x,y],[1/2,sqrt(3)/2],5);imp_newton(2*x^7-x^8-x^3*y+(4*x^2+x^3)*y^2+(x^3-x^4)*y^3-4*x*y^4+7*x^5*y^5+(1-x^2)*y^6+5*x^6*y^7+x^3*y^8,[x,y],[0,0],3);imp_newton(ratdisrep(taylor(x-asin(y),y,0,9)),[x,y],[0,0],5);`
Attachments
implicit.mac