I have a question if you don't mind. I have the following quotient operator: $$\frac{1}{e^{d/dx}(f(x))}$$ Where $f$ is a smooth function on $R$. I would like to get rid of the denominator. IS there any formula that i Can found in your papers or other references and use it for this case? Thanks and best regards.
1 Answer
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The exponentiated operator shifts $f$ by one unit in $x$, i.e., $$ \frac{1}{e^{d/dx} (f(x))} = \frac{1}{f(x+1)} $$ It's not quite clear what is desired by "getting rid of the denominator" - the result just happens to be the reciprocal of $f$. You could define $g=1/f$ and have an expression without a fraction.
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$\begingroup$ Where did you found this formula is this the developement of Taylor? $\endgroup$ Jun 22, 2021 at 14:03
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$\begingroup$ The shift operator is described here, en.wikipedia.org/wiki/Shift_operator - you can also find some references there if you want to cite something. $\endgroup$ Jun 22, 2021 at 15:45