From here:

http://i48.tinypic.com/2n7304h.png

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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- Nov 26th 2009, 07:46 AMMrWaeseLSimplification of Spherical Bessel functions
From here:

http://i48.tinypic.com/2n7304h.png

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

Thanks in advance for any help. - Nov 26th 2009, 10:32 AMJester
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}.

$ - Nov 26th 2009, 11:00 AMMrWaeseL
I see...so it would be wrong to state that $\displaystyle \left(\frac{1}{x} \frac{d}{dx} \right)^n = (\frac{1}{x})^n (\frac{d}{dx})^n$?

- Nov 26th 2009, 12:22 PMJester