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


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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I don't know what $Id$ is (we're not doing Freudian psychology here, are we?). I suppose you mean you're integrating $f$ from 0 to $xy$ and then differentiating with respect to $x$. I'd rather see it written as $\int_0^{xy}f(t)\,dt$. – Gerry Myerson Jul 5 '12 at 0:45
What's the point? What's wrong with Leonid Kovalev's answer to your previous question? What you're doing here looks a bit like writing $2+2=4$ as $\lim_{n\to\infty}(1+\log 2/n)^n+\lim_{n\to\infty}(1+\log 2/n)^n=\lim_{n\to\infty}(1+2\log 2/n)^n$. – tomasz Jul 5 '12 at 1:31
@GerryMyerson: Sorry for the poor description - it's due to my lack of mathematical knowledge. I'm differentiating with respect to $x$ and $y$ both, and I tried to make the correction in the question. I hope this isn't too hard for a good analysis. I've been attempting to find information on operator calculus to learn about the potential of various methods, especially this one, to perform the task. I'm a little lost as to how the integration and differentiation interact as well. – Matt Groff Jul 5 '12 at 1:32
@tomasz: I realize that. I'm just trying to explore all my options to finding a way to represent this function composition as a single function if possible. I guess I've just ended up confusing everyone, including me. I was really hoping that I could simplify the composition as much as possible, and I reacted to the book I mention in the previous question by trying to adapt their methods to fit the problem. – Matt Groff Jul 5 '12 at 1:36
OK, it looks like what you are doing is ${d\over dQ}\int_0^Qf(t)\,dt=f(Q)$. There's no problem with that. – Gerry Myerson Jul 5 '12 at 4:10

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