Differences
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| Both sides previous revision Previous revision | Next revision Both sides next revision | ||
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minus_twelve_._note [2016/05/31 20:38] nikolaj |
minus_twelve_._note [2016/11/11 21:48] nikolaj |
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| $\dfrac {1} { \log(1+r)} = \dfrac {1} {r} \dfrac {1} {1 - \dfrac{1}{2}r + \dfrac{1}{3}r^2 + {\mathcal O}(r^3) } = \dfrac {1} {r} \left( 1 + \dfrac{1}{2} r - \dfrac{1}{2\cdot 2\cdot 3} r^2 + {\mathcal O}(r^3) \right) = \dfrac {1} {r} + \dfrac{1}{2} - \dfrac{1}{12} r + {\mathcal O}(r^2)$ | $\dfrac {1} { \log(1+r)} = \dfrac {1} {r} \dfrac {1} {1 - \dfrac{1}{2}r + \dfrac{1}{3}r^2 + {\mathcal O}(r^3) } = \dfrac {1} {r} \left( 1 + \dfrac{1}{2} r - \dfrac{1}{2\cdot 2\cdot 3} r^2 + {\mathcal O}(r^3) \right) = \dfrac {1} {r} + \dfrac{1}{2} - \dfrac{1}{12} r + {\mathcal O}(r^2)$ | ||
| + | |||
| + | <code> | ||
| + | Plot[{ | ||
| + | Log[1 + r] | ||
| + | , 1/Log[1 + r] - 1/r | ||
| + | , 1/Log[1 + r] | ||
| + | , 1/2 + (-1/12) r | ||
| + | }, {r, -0.5, 1.5}, PlotRange -> {-0.5, 1.5}] | ||
| + | </code> | ||
| With $r=z-1$ we see | With $r=z-1$ we see | ||
