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# 3.5 Maximum Function

3.88   Definition (.) Let be an ordered field, and let . We define

Then

3.89   Definition ( .) Let be an ordered field, let and let be a function. Define by the rules

Hence, e.g., if ,

We write

where is a dummy index, and we think of as the largest of the numbers . By definition

and

3.90   Notation ( ) Let with . Then

3.91   Theorem. Let be an ordered field, let and let be a function. Then for all ,
 (3.92)

Proof: Let be the proposition form on such that is the proposition (3.92). Then says

i.e.,

Hence is true.

Now for all ,

We also have

so

By induction, is true for all .

3.93   Note. The notation for positive integer powers of was introduced by Descartes in 1637[15, vol 1,p 346]. Both Maple and Mathematica denote by a^n.

The notation for the factorial of was introduced by Christian Kramp in 1808[15, vol 2, p 66].

The use of the Greek letter to denote sums was introduced by Euler in 1755[15, vol 2,p 61]. Euler writes

The use of limits on sums was introduced by Augustin Cauchy(1789-1857). Cauchy used the notation to denote what we would write as [15, vol 2, p 61].

In Maple, the value of is denoted by sum(f(i),i=1..n). In Mathematica it is denoted by Sum[f[i],{i,1,n}].

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