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This is a specific idea I have to rewriting $x$ as $x y$, which I recently asked about in this question.

Suppose we have a power series $f(x) = \sum_{i=0}^\infty{c_i x^i} = c_0 x^0 + c_1 x^1 + c_2 x^2 + \dots$

We'd like $f(x y) = \sum_{i=0}^\infty{c_i (x y)^i} = c_0 (x y)^0 + c_1 (x y)^1 + c_2 (x y)^2 + \dots$

MY IDEA/METHOD

We first take $I = \int_0^{x y}{f(x)dx}$ $= C + c_0 (x y)^1 + \frac{1}{2}c_1 (x y)^2 + \frac{1}{3}c_2 (x y)^3+ \dots$

We next eliminate the constant of integration $C$. We do this as well as eliminating the fractions and correcting the powers of $x$ and $y$ by taking the derivative with respect to $(x y)$ to get:$c_0 (x y)^0 + c_1 (x y)^1 + c_2 (x y)^2+ \dots$ We have then rewritten $f(x)$ as $f(x y)$.

I'm wondering first if this is correct, and second if there are any restrictions, limitations, or problems associated with it. I realize it may seem a bit roundabout and inefficient, but I still would like to try and make it work.

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    @GerryMyerson:Thanks!2012-07-05

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