# Differences

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quantum_integer [2014/12/10 12:02] nikolaj |
quantum_integer [2016/07/22 18:36] (current) nikolaj |
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The case $f=0$ is often considered. | The case $f=0$ is often considered. | ||

- | Quantum aspect: $f=n-1$ gives $[n]_{q^2}$=n+\mathcal{O}\left((q-1)^2\right)$. (The $q^2$ isn't necessary.) In the imaginary direction, $q\propto\mathrm{e}^{i\varphi}$, this corresponds to $\lim_{\varphi\to 0}\frac{\sin(n\varphi)}{\sin(\phi)}=n$. | + | Quantum aspect: $f=n-1$ gives |

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+ | $[n]_{q^2} = n + \mathcal{O}\left((q-1)^2\right)$. | ||

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+ | (The $q^2$ isn't necessary.) In the imaginary direction, $q\propto\mathrm{e}^{i\varphi}$, this corresponds to $\lim_{\varphi\to 0}\frac{\sin(n\varphi)}{\sin(\phi)}=n$. | ||

With $q=r\mathrm{e}^{i\varphi}$, along the positive real axis number $[n]_q$ is a valley with bottom at $q=1$, where $[n]_{1}=n$, and along $\varphi$ you have harmonic oscillations with period depending on $n$. | With $q=r\mathrm{e}^{i\varphi}$, along the positive real axis number $[n]_q$ is a valley with bottom at $q=1$, where $[n]_{1}=n$, and along $\varphi$ you have harmonic oscillations with period depending on $n$. | ||