the idea is let us suppose i must solve(adsbygoogle = window.adsbygoogle || []).push({});

[tex] f(x)= 0 [/tex] (1)

let us suppose that f(x) have SEVERAL (perhaps infinite ) inverses, that is there is a finite or infinite solutions to the equation

[tex] f(x)= y [/tex] by [tex] g(y)= x [/tex] with [tex] f^{-1}(x)=g(x) [/tex]

then solution to equation (1) would be [tex] g(0)=x [/tex]

my problem is what would happen for multi-valued functions (example [tex] x^{2} [/tex] having several 'branches' (is this the correct word ?? )

Using Lagrange inversion theorem [tex] g(x) = a

+ \sum_{n=1}^{\infty}

\left(

\lim_{w \to a}\left(

\frac{\mathrm{d}^{\,n-1}}{\mathrm{d}w^{\,n-1}}

\left( \frac{w-a}{f(w) - b} \right)^n\right)

{\frac{(x - b)^n}{n!}}

\right).

[/tex]

then simply set x=0 but this would only give an UNIQUE solution to (1)

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# Solving equations by inversion formulae

Can you offer guidance or do you also need help?

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