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 — finite_exponential_power [2017/03/18 17:31] (current)nikolaj created 2017/03/18 17:31 nikolaj created 2017/03/18 17:31 nikolaj created Line 1: Line 1: + ===== 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]]