The Exponential Function

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 10 September 2010

The exponential function

Let $k\in {\mathbb{Z}}_{\ge 0}$. Define $k$ factorial by $$0!=1\text{and}k!=k·\left(k-1\right)\cdots 3·2·1,\text{if }k\in {\mathbb{Z}}_{>0}\text{.}$$ Let $n,k\in {\mathbb{Z}}_{\ge 0}$ with $k\le n$. Define $$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!\left(n-k\right)!}\text{.}$$

Let $n,k\in {\mathbb{Z}}_{\ge 0}$ with $k\le n$.

1. Let $S$ be a set of cardinality $n$. Then ${\displaystyle \left(\genfrac{}{}{0ex}{}{n}{k}\right)}$ is the number of subsets of $S$ with cardinality $k$.
2. ${\displaystyle \left(\genfrac{}{}{0ex}{}{n}{k}\right)}$ is the coefficient of $x^k{y}^{n-k}$ in ${\left(x+y\right)}^n$.
3. ${\displaystyle \left(\genfrac{}{}{0ex}{}{n}{n}\right)=1}$, ${\displaystyle \left(\genfrac{}{}{0ex}{}{n}{0}\right)=1}$ and if $1\le k\le n-1$ then $$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)+\left(\genfrac{}{}{0ex}{}{n-1}{k}\right).$$

The exponential function is the element $e^x$ of $\mathbb{Q}[[x]]$ given by $$e^x=\sum _{k\in {\mathbb{Z}}_{\ge 0}}\frac{x^k}{k!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots \text{.}$$

As an element of $\mathbb{Q}[[x,y]]$, $$e^{x+y}=e^x{e}^y\text{.}$$

HW: Show that $e^0=1$.

HW: Show that $e^{-x}=\frac{1}{e^x}$.

The logarithm is $$\log (1+x)=\sum _{k\in {\mathbb{Z}}_{>0}}{(-1)}^{k-1}\frac{x^k}{k}\text{.}$$

Let $$G=\left\{p\left(x\right)\in 𝔽[[x]]\left|p\right(0)=1\right\}\text{and}𝔤=\left\{p\left(x\right)\in 𝔽[[x]]\left|p\right(0)=0\right\}\text{.}$$

1. $\log \left(1+\left(e^x-1\right)\right)=e^{\log \left(1+x\right)}-1=x$.
2. $G$ is an abelian group under multiplication, $𝔤$ is a commutative group under addition and $$\begin{array}{rcl}G& ⟶& 𝔤\\ p& ⟼& e^p-1\end{array}$$ is an isomorphism of groups.

References

[Bou] N. Bourbaki FIX THIS, Mason, Paris, ????

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