Logarithmic approximants: Difference between revisions
CompactStar (talk | contribs) |
Sintel bot (talk | contribs) Fix <math> environments for MathJax v4 (align->aligned, etc.) |
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:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
v &= \frac{n-d}{n+d} \\ | v &= \frac{n-d}{n+d} \\ | ||
&= \text{(frequency difference) / (frequency sum)} \\ | &= \text{(frequency difference) / (frequency sum)} \\ | ||
&= \frac{1}{2} \text{(frequency difference) / (mean frequency)} | &= \frac{1}{2} \text{(frequency difference) / (mean frequency)} | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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<math> | <math> | ||
\begin{ | \begin{aligned} | ||
v[-J] &= -v[J] \\ | v[-J] &= -v[J] \\ | ||
v[J_1 +J_2] &= \frac{v_1+v_2}{1+v_1 v_2} | v[J_1 +J_2] &= \frac{v_1+v_2}{1+v_1 v_2} | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
q(r) &= \frac{1}{2} \left( r^{\frac{1}{2}} - r^{-\frac{1}{2}} \right) \\ | q(r) &= \frac{1}{2} \left( r^{\frac{1}{2}} - r^{-\frac{1}{2}} \right) \\ | ||
&= \frac{1}{2} \left( e^{J} - e^{-J} \right) \\ | &= \frac{1}{2} \left( e^{J} - e^{-J} \right) \\ | ||
&= \sinh{J} \\ | &= \sinh{J} \\ | ||
&= J + \frac{1}{3!} J^3 + \frac{1}{5!} J^5 + \ldots | &= J + \frac{1}{3!} J^3 + \frac{1}{5!} J^5 + \ldots | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
v &= \tanh{J}, \; q = \sinh{J}, \; \frac{q}{v} = \cosh{J} \\ | v &= \tanh{J}, \; q = \sinh{J}, \; \frac{q}{v} = \cosh{J} \\ | ||
\sqrt{r} &= e^J = q(\frac{1}{v} + 1) \\ | \sqrt{r} &= e^J = q(\frac{1}{v} + 1) \\ | ||
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\frac {q[J_2 + J_1]}{q[J_2 - J_1]} &= \frac{v_2+v_1}{v_2-v_1} \\ | \frac {q[J_2 + J_1]}{q[J_2 - J_1]} &= \frac{v_2+v_1}{v_2-v_1} \\ | ||
q[J_2 + J_1] q[J_2 - J_1] &= q_2^2 - q_1^2 \\ | q[J_2 + J_1] q[J_2 - J_1] &= q_2^2 - q_1^2 \\ | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
\frac{\text{octave}}{\text{large tone}} &\approx \frac{q[F+f]}{q[F-f]} \\ | \frac{\text{octave}}{\text{large tone}} &\approx \frac{q[F+f]}{q[F-f]} \\ | ||
&= \frac{v[F] + v[f]}{v[F] - v[f]} \\ | &= \frac{v[F] + v[f]}{v[F] - v[f]} \\ | ||
&= \frac{1/5 + 1/7}{1/5 - 1/7} = 6 | &= \frac{1/5 + 1/7}{1/5 - 1/7} = 6 | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
b(J_1,J_2) &\approx \left( \frac{1}{3} \left(q_2^2 - q_1^2\right) - \frac{2}{15} \left( q_2^4 - q_1^4 \right) \right) b_m \\ | b(J_1,J_2) &\approx \left( \frac{1}{3} \left(q_2^2 - q_1^2\right) - \frac{2}{15} \left( q_2^4 - q_1^4 \right) \right) b_m \\ | ||
&= \frac{1}{3} \left( q_2^2 - q_1^2 \right) \left( 1 - \frac{2}{5} \left(q_1^2 + q_2^2\right) \right) b_m | &= \frac{1}{3} \left( q_2^2 - q_1^2 \right) \left( 1 - \frac{2}{5} \left(q_1^2 + q_2^2\right) \right) b_m | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||