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Thread: Name of a rule

  1. #1
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    Name of a rule

    What is the name of the rule which states:

    Let $\displaystyle \alpha(x)$ be a function which $\displaystyle x->\alpha(x)$ such that $\displaystyle \alpha^n(x)=x$ for some $\displaystyle n$. Given an equation:

    $\displaystyle A(x)f(x) + B(x)f(\alpha(x)) = C(x)$

    by repeatedly apply $\displaystyle \alpha$ to $\displaystyle x$ we get a series of equation:

    $\displaystyle A(x)f(x) + B(x)f(\alpha(x)) = C(x)$
    $\displaystyle A(\alpha(x))f(\alpha(x)) + B(\alpha(x))f(\alpha^2(x)) = C(\alpha(x))$
    $\displaystyle A(\alpha^2(x))f(\alpha^2(x)) + B(\alpha^2(x))f(\alpha^3(x)) = C(\alpha^2(x))$
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  2. #2
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    I'm not aware of the rule you are referring to.

    The rule of substitution?

    For a general equality with free variable $\displaystyle x$,
    $\displaystyle F(x) = G(x)$ means it is true for all values $\displaystyle x$.
    For convenience, since $\displaystyle x$ is a free variable, we can rename it to $\displaystyle t$.
    $\displaystyle F(t) = G(t)$, now substitute $\displaystyle \alpha(x)$ for $\displaystyle t$.
    $\displaystyle F(\alpha(x)) = G(\alpha(x))$

    You can repeat this to show:
    $\displaystyle F(\alpha^n(x)) = G(\alpha^n(x))$
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  3. #3
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    Quote Originally Posted by TheKovachki View Post
    $\displaystyle x->\alpha(x)$ such that $\displaystyle \alpha^n(x)=x$ for some $\displaystyle n$.
    Why is this fact important? Where is it being used?
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  4. #4
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    Never mind, I guess there is no real rule to it, just continually plunging in $\displaystyle \alpha(x)$ to get a series of equations.
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