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The composition of formal power series $g \circ f$ is well defined if f has vanishing constant term. My question is how one can generalize composition of power series to several variables? If we substitute $g_1(y_1,...,y_m), ..., g_n(y_1,...,y_m)$ into the series $f(x_1,...,x_n)$, which conditions do we have to give on the series $g_i$ to get a well definied series $f(g_1,...,g_n)$?

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In Electricl/Elctronics Engg. studies ,we usually restrict ourselves product of two functions at a time.If the problem can be solved , then this product is represented as a new function.Then we can consider the product of this function with another function. Is there any real life situation , where you come across functions like these ? –  B.Sahu Nov 28 '12 at 5:11

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Similarly to the one-variable case, if each $g_k(y_1,...,y_m)$ has no constant term you can substitute them into the power series for $f(x_1,...,x_n)$ and get another power series. The basic idea is that if $f(x_1,...,x_n) = \sum_{\alpha} f_{\alpha}x^{\alpha}$, then if one substitutes the $g_k(y_1,...,y_m)$ into this power series, given any $\beta$ a term of the form $cy^{\beta}$ can appear only finitely many times; it must come from a $f_{\alpha}x^{\alpha}$ with $|\alpha| \leq |\beta|$, and for a given $\alpha$ such a term can only come from terms of degree $\leq |\beta|$ from the power series of the various $g_k(y_1,...,y_n)$.

If these power series are convergent in some neighborhood of the origin, so will be the power series of the composition.

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