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Thread: Simplification of Spherical Bessel functions

  1. #1
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    Simplification of Spherical Bessel functions

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    Why isn't the $\displaystyle (-x)^n (1/x)^n$ product simplified to $\displaystyle (-1)^n$?

    Thanks in advance for any help.
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  2. #2
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    Quote Originally Posted by MrWaeseL View Post
    From here:



    Why isn't the $\displaystyle (-x)^n (1/x)^n$ product simplified to $\displaystyle (-1)^n$?

    Thanks in advance for any help.
    Because it's an operator not just an algebraic expression. To demonstrate let $\displaystyle n = 2$ (and $\displaystyle f(x) = \frac{\sin x}{x}$ ) so

    $\displaystyle
    (-x)^2 \left(\frac{1}{x} \frac{d}{dx} \right)^2 f(x) = x^2 \frac{1}{x} \frac{d}{dx} \left(\frac{1}{x} \frac{df}{dx} \right) = \frac{d^2f}{d x^2} - \frac{1}{x} \frac{df}{dx}.
    $
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    I see...so it would be wrong to state that $\displaystyle \left(\frac{1}{x} \frac{d}{dx} \right)^n = (\frac{1}{x})^n (\frac{d}{dx})^n$?
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  4. #4
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    Quote Originally Posted by MrWaeseL View Post
    I see...so it would be wrong to state that $\displaystyle \left(\frac{1}{x} \frac{d}{dx} \right)^n = (\frac{1}{x})^n (\frac{d}{dx})^n$?
    Absolutely. It only applies when $\displaystyle n = 1$.
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