Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision | Next revision Both sides next revision | ||
|
minus_twelve_._note [2019/09/09 22:45] nikolaj |
minus_twelve_._note [2019/09/21 00:34] nikolaj |
||
|---|---|---|---|
| Line 120: | Line 120: | ||
| The geometric series for z in (0,1) is | The geometric series for z in (0,1) is | ||
| - | [math] \sum_{n=0}^\infty z^n = \dfrac {1} {1-z} [/math] | + | $ \sum_{n=0}^\infty z^n = \dfrac {1} {1-z} $ |
| The smooth analogous is | The smooth analogous is | ||
| - | [math] \int_0^\infty z^n \, d n = \int_0^\infty e^{ n \, \log (z) } \ d n = \dfrac {1} {-\log(z)} [/math] | + | $ \int_0^\infty z^n \, d n = \int_0^\infty e^{ n \, \log (z) } \ d n = \dfrac {1} {-\log(z)} $ |
| - | Applying [math] z \dfrac {d} {dz} [/math] to the first yields | + | Applying $ z \dfrac {d} {dz} $ to the first yields |
| - | [math] z \dfrac {z} {dz} \dfrac {1} {1-z} = \sum_{n=0}^\infty n \, z^n [/math] | + | $ z \dfrac {z} {dz} \dfrac {1} {1-z} = \sum_{n=0}^\infty n \, z^n $ |
| and applying it to the second yields | and applying it to the second yields | ||
| - | [math] z \dfrac {z} {dz} \dfrac{1} {-\log(z)} = \dfrac {1} { \log(z)^2 } [/math] | + | $ z \dfrac {z} {dz} \dfrac{1} {-\log(z)} = \dfrac {1} { \log(z)^2 } $ |
| How do those two differ? | How do those two differ? | ||
| Line 138: | Line 138: | ||
| We have the Taylor expansion | We have the Taylor expansion | ||
| - | [math] \log(1+r) = \sum_{k=1}^{\infty} \dfrac {1} {k} (-r)^k = r - \dfrac {1} {2} r^2 + \dfrac {1} {3} r^3 + O (r^4) = r \left( 1 - \dfrac {1} {2} r + \dfrac {1} {3} r^2 + O (r^3) \right) [/math] | + | $ \log(1+r) = \sum_{k=1}^{\infty} \dfrac {1} {k} (-r)^k = r - \dfrac {1} {2} r^2 + \dfrac {1} {3} r^3 + O (r^4) = r \left( 1 - \dfrac {1} {2} r + \dfrac {1} {3} r^2 + O (r^3) \right) $ |
| Using the geometric series expansion, we get | Using the geometric series expansion, we get | ||
| - | [math] \dfrac {1} { \log (1+r) } = \dfrac {1} {r} \dfrac {1} {1 - \dfrac {1} {2} r + \dfrac {1} {3} r^2 + O (r^3) } = \dfrac {1} {r} \left( 1 + \dfrac {1} {2} r - \dfrac {1} { 2 \cdot 2 \cdot 3 } r^2 + O (r^3) \right) = \dfrac {1} {r} + \dfrac {1} {2} - \dfrac {1} {12} r + O (r^2) [/math] | + | $ \dfrac {1} { \log (1+r) } = \dfrac {1} {r} \dfrac {1} {1 - \dfrac {1} {2} r + \dfrac {1} {3} r^2 + O (r^3) } = \dfrac {1} {r} \left( 1 + \dfrac {1} {2} r - \dfrac {1} { 2 \cdot 2 \cdot 3 } r^2 + O (r^3) \right) = \dfrac {1} {r} + \dfrac {1} {2} - \dfrac {1} {12} r + O (r^2) $ |
| With r=z-1 we see | With r=z-1 we see | ||
| - | [math] \dfrac {1} { \log(z) } = - \dfrac {1} {1-z} + \dfrac {1} {2} - \dfrac {1} {12} (z-1) + O ((z-1)^2) [/math] | + | $ \dfrac {1} { \log(z) } = - \dfrac {1} {1-z} + \dfrac {1} {2} - \dfrac {1} {12} (z-1) + O ((z-1)^2) $ |
| and taking the derivative, we get | and taking the derivative, we get | ||
| - | [math] \sum_{n=0}^\infty n \, z^n - \dfrac {1} { \log(z)^2 } = - \dfrac {1} {12} + O ((z-1)) [/math] | + | $ \sum_{n=0}^\infty n \, z^n - \dfrac {1} { \log(z)^2 } = - \dfrac {1} {12} + O ((z-1)) $ |
| - | For z to 1 this says 1+2+3+4+... minus a 1/log^2 divergence is [math]- \dfrac {1} {12} [/math]. | + | For z to 1 this says 1+2+3+4+... minus a 1/log^2 divergence is $- \dfrac {1} {12} $. |
| - | For gathering the data bits [math]z^n[/math], that limit doesn’t exist for neither the operation [math]\sum_{n=0}^\infty[/math] nor [math]\int_0^\infty dn[/math], but it does for [math]\sum_{n=0}^\infty - \int_0^\infty dn[/math]. | + | For gathering the data bits $z^n$, that limit doesn’t exist for neither the operation $\sum_{n=0}^\infty$ nor $\int_0^\infty dn$, but it does for $\sum_{n=0}^\infty - \int_0^\infty dn$. |
| The space(time) your physical field theories are defined on generally fuck with you, but there are often such systematic renormalizations of your physical expressions, and for the addition of integers it relates to that relational. | The space(time) your physical field theories are defined on generally fuck with you, but there are often such systematic renormalizations of your physical expressions, and for the addition of integers it relates to that relational. | ||
