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differentiable_function [2016/01/23 19:05] nikolaj |
differentiable_function [2016/01/23 19:07] nikolaj |
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| - | === Reference === | ||
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Differentiable_function|Differentiable function]] | ||
| - | |||
| === Theorems === | === Theorems === | ||
| - | Let $f(0)=0$, then | + | Let |
| - | $f(y\ f^{-1}(x))\ y^{-1}=x+(y-1)\cdot\dfrac{f''(0)}{f'(0)^2}\cdot\dfrac{1}{2}x^2+{\mathcal O}(x^3)$ | + | $f(0)=0\neq f'(0)$, |
| + | |||
| + | then | ||
| + | |||
| + | $\dfrac{ f(y\ f^{-1}(x)) }{y} =x+(y-1)\cdot\dfrac{f''(0)}{f'(0)^2}\cdot\dfrac{1}{2}x^2+{\mathcal O}(x^3)$ | ||
| + | |||
| + | === Reference === | ||
| + | Wikipedia: [[http://en.wikipedia.org/wiki/Differentiable_function|Differentiable function]] | ||
| ----- | ----- | ||
| === Context === | === Context === | ||
| [[Fréchet derivative]], [[Iterated function]] | [[Fréchet derivative]], [[Iterated function]] | ||
