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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]]
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