The binomial theorema and the exponential function

The binomial theorem and the exponential function

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

Last updates: 20 May 2012

The binomial theorem

Let $k \in ℤ_{\geq 0}$ . Define $k$ factorial by $0! = 1 and k! = k \cdot (k - 1) \dots 3 \cdot 2 \cdot 1, if k \in ℤ_{> 0} .$ Let $n, k \in ℤ_{\geq 0}$ with $k \leq n$ . Define $(\binom{n}{k}) = \frac{n!}{k! (n - k)!} .$

(Binomial theorem) Let $n, k \in ℤ_{\geq 0}$ with $k \leq n$ .

Let $S$ be a set of cardinality $n$ . Then $(\binom{n}{k})$ is the number of subsets of $S$ with cardinality $k$ .
$(\binom{n}{k})$ is the coefficient of $x^{k} y^{n - k}$ in ${(x + y)}^{n}$ .
$(\binom{n}{n}) = 1$ , $(\binom{n}{0}) = 1$ , and if $1 \leq k \leq n - 1$ then $(\binom{n}{k}) = (\binom{n - 1}{k - 1}) + (\binom{n - 1}{k}) .$

This theorem says that the table of numbers

\begin{matrix} (\binom{0}{0}) \\ (\binom{1}{0}) & (\binom{1}{1}) \\ (\binom{2}{0}) & (\binom{2}{1}) & (\binom{2}{2}) \\ (\binom{3}{0}) & (\binom{3}{1}) & (\binom{3}{2}) & (\binom{3}{3}) \\ (\binom{4}{0}) & (\binom{4}{1}) & (\binom{4}{2}) & (\binom{4}{3}) & (\binom{4}{4}) \\ (\binom{5}{0}) & (\binom{5}{1}) & (\binom{5}{2}) & (\binom{5}{3}) & (\binom{5}{4}) & (\binom{5}{5}) \\ ⋰ & ⋮ & ⋱ \end{matrix}

are the numbers in Pascal's triangle

\begin{matrix} 1 \\ 1 & 1 \\ 1 & 2 & 1 \\ 1 & 3 & 3 & 1 \\ 1 & 4 & 6 & 4 & 1 \\ 1 & 5 & 10 & 10 & 5 & 1 \\ ⋰ & ⋮ & ⋱ \end{matrix}

and that

\begin{matrix} {(x + y)}^{0} & = & 1, \\ {(x + y)}^{1} & = & x + y, \\ {(x + y)}^{2} & = & x^{2} + x y + y^{2}, \\ {(x + y)}^{3} & = & x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}, \\ {(x + y)}^{4} & = & x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}, \\ {(x + y)}^{5} & = & x^{5} + 5 x^{4} y + 10 x^{3} y^{2} + 10 x^{2} y^{3} + 5 x y^{4} + y^{5}, \\ ⋮ & ⋮ \end{matrix}

The exponential function

The exponential function is the element $e^{x}$ of $ℚ [[x]]$ given by $e^{x} = \sum_{k \in ℤ_{\geq 0}} \frac{x^{k}}{k!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots .$

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

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

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

The logarithm is $\log (1 + x) = \sum_{k \in ℤ_{> 0}} {(- 1)}^{k - 1} \frac{x^{k}}{k} .$

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

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

Notes and References

The binomial theorem and 'Pascals triangle' are useful computational tools for multiplying out algebraic expressions. The exponential function "is the most important function in mathematics" [Ru, Prologue]. The theorems showing that the exponential function is a homomorphism and that the formal inverse to the exponential function is log are found in [Bou, Alg. Ch. IV § 4 no. 10].

References

[Bou] N. Bourbaki, Algèbre, Chapitre ?: ??????????? MR?????.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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