Question

Cum se demostreaza identitatea lui Hermite?

Answer 1

Avem asa
[x]+[x+1/2]=[2x] pentru orice x real
Notez ch n partea intreaga a lui x
[x]=n de aici rezulta n≤x<n+1
Si vom avea 2 cazuri:

1) Consideram ca x se afla in prima jumatate a intervalului;

n≤x<(2n+1)/2
n+1/2≤x+1/2<n+1

[x+1/2]=n
=>. [x]+[x+1/2]=2n

n≤x<(2n+1)/2 /•2
2n≤2x<2n+1
[2x]=2n

=>[x]+[x+1/2]=[2x]

2) x se afla in a cealalta jumatate a intervalului

(2n+1)/2≤x<n+1
n+1≤x+1/2<n+3/2
[x+1/2]=n+1
[x]+[x+1/2]=2n+1

(2n+1)/2≤x<n+1/•2
2n+1≤2x<2n+2
[2x]=2n+1

din cele doua cazuri => [x]+[x+1/2]=[2x]