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quantum_integer [2014/12/10 12:02]
nikolaj
quantum_integer [2016/07/22 18:36]
nikolaj
Line 15: Line 15:
 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 $n$.
  
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