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finite_exponential_power [2017/03/18 17:31] (current) nikolaj created |
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+ | ===== Finite exponential power ===== | ||
+ | ==== Function ==== | ||
+ | | @#55CCEE: context | @#55CCEE: $ m\in{\mathbb N} | | ||
+ | | @#FF9944: definition | @#FF9944: ${\rm pexp}_n: \mathbb C\to\mathbb C$ | | ||
+ | | @#FF9944: definition | @#FF9944: ${\rm pexp}_n(z) := \left(1 + \dfrac {x} {n} \right)^n $ | | ||
+ | ----- | ||
+ | |||
+ | ${\rm pexp}_n(x) = \sum_{k=0}^n a_k(n)\dfrac {1} {k!} x^k $ | ||
+ | |||
+ | with | ||
+ | |||
+ | $a_k(n)=\prod_{j=1}^{k-1}\left(1-\dfrac{k-j}{n}\right)\le 1$ | ||
+ | |||
+ | **Elaboration** | ||
+ | |||
+ | $\left(x+y\right)^m=\sum_{k=0}^m \dfrac{n!}{k!\,(m-k)!} x^k y^{m-k}$ | ||
+ | |||
+ | so | ||
+ | |||
+ | $\left(1 + b(n)\,x \right)^n = \sum_{k=0}^n \left( b(n)^{-k}\dfrac {n!} {(n-k)!} \right) \dfrac {x^k} {k!}$ | ||
+ | |||
+ | (Note that here the summands depend on the upper sum bound $n$, this sum doesn't make for an infinite sum of partial sums - the to be partial sums are all different) | ||
+ | |||
+ | The above sum follows. Also, | ||
+ | |||
+ | $= \sum_{k=0}^n \left(\prod_{j=1}^{k}\left(\dfrac{1}{j}-\dfrac{1}{n}\left(\frac{k}{j}-1\right)\right) x \right)$ | ||
+ | |||
+ | **Derivative** | ||
+ | |||
+ | $\frac{\mathrm d}{\mathrm d z}{\rm pexp}_n(z)=\dfrac{1}{1+z/n}{\rm pexp}_n(z)$ | ||
+ | |||
+ | $\frac{\mathrm d}{\mathrm d n}{\rm pexp}_n(z)=-\left(\dfrac{1}{1+(z/n)^{-1}}+\log\left(\dfrac{1}{1+z/n}\right)\right){\rm pexp}_n(z)$ | ||
+ | |||
+ | |||
+ | === References === | ||
+ | Wikipedia: | ||
+ | [[http://en.wikipedia.org/wiki/Exponential_function|Exponential function]], | ||
+ | [[http://en.wikipedia.org/wiki/Matrix_exponential|Matrix exponential]], | ||
+ | [[http://en.wikipedia.org/wiki/Exponential_map|Exponential map]] | ||
+ | |||
+ | ----- | ||
+ | === Related === | ||
+ | [[Finite exponential series]] |