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Math Help - Name of a rule

  1. #1
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    Name of a rule

    What is the name of the rule which states:

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

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

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

    A(x)f(x) + B(x)f(\alpha(x)) = C(x)
    A(\alpha(x))f(\alpha(x)) + B(\alpha(x))f(\alpha^2(x)) = C(\alpha(x))
    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 x,
    F(x) = G(x) means it is true for all values x.
    For convenience, since x is a free variable, we can rename it to t.
    F(t) = G(t), now substitute \alpha(x) for t.
    F(\alpha(x)) = G(\alpha(x))

    You can repeat this to show:
    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
    x->\alpha(x) such that \alpha^n(x)=x for some 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 \alpha(x) to get a series of equations.
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