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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]]