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taylor_s_formula [2013/09/16 21:36] nikolaj |
taylor_s_formula [2014/03/21 11:11] (current) |
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| $f\in C^k(\mathbb R^n,\mathbb R)$ | | | $f\in C^k(\mathbb R^n,\mathbb R)$ | | ||
- | | @#55EE55: $f(x) = \sum_{|\alpha|\le k} \frac{1}{\alpha !} f^{(\alpha)}(0)\ x^\alpha + R_k(x) $ | | + | | @#55EE55: postulate | @#55EE55: $f(x) = \sum_{|\alpha|\le k} \frac{1}{\alpha !} f^{(\alpha)}(0)\ x^\alpha + R_k(x) $ | |
with | with | ||
- | | @#55EE55: $ R_k(x) = \sum_{|\alpha|=k+1} \frac{k+1}{\alpha !} \left( \int_0^1\ (1-s)^k\ F^{(\alpha)}(s\ x)\ \mathrm ds \right)\ x^\alpha $ | | + | | @#55EE55: postulate | @#55EE55: $ R_k(x) = \sum_{|\alpha|=k+1} \frac{k+1}{\alpha !} \left( \int_0^1\ (1-s)^k\ F^{(\alpha)}(s\ x)\ \mathrm ds \right)\ x^\alpha $ | |
where we use multi-index notation for $\alpha \in \mathrm{FinSequence}(\mathbb N)$, see [[Multi-index power]]. | where we use multi-index notation for $\alpha \in \mathrm{FinSequence}(\mathbb N)$, see [[Multi-index power]]. | ||
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=== Reference === | === Reference === | ||
==== Parents ==== | ==== Parents ==== | ||
- | === Requirements === | + | === Context === |
[[Fréchet derivative]], [[Function integral]] | [[Fréchet derivative]], [[Function integral]] |