“Hermite's identity”的意思、由来-开放百科全书

词条

Hermite's identity

释义

Proof
References

{{dablink|This article is not about Hermite's cotangent identity.}}

In mathematics, Hermite's identity, named after Charles Hermite, gives the value of a summation involving the floor function. It states that for every real number x and for every positive integer n the following identity holds:^[1]^[2]

Proof

Split into its integer part and fractional part, . There is exactly one with

By subtracting the same integer from inside the floor operations on the left and right sides of this inequality, it may be rewritten as

Therefore,

and multiplying both sides by gives

Now if the summation from Hermite's identity is split into two parts at index , it becomes

References

1. ^{{citation|title=Mathematical Miniatures|volume=43|series=New Mathematical Library|first1=Svetoslav|last1=Savchev|first2=Titu|last2=Andreescu|publisher=Mathematical Association of America|year=2003|isbn=9780883856451|chapter=12 Hermite's Identity|pages=41–44}}.
2. ^{{citation | last = Matsuoka | first = Yoshio | doi = 10.2307/2311413 | issue = 10 | journal = The American Mathematical Monthly | mr = 1533020 | page = 1115 | title = Classroom Notes: On a Proof of Hermite's Identity | volume = 71 | year = 1964}}.

2 : Mathematical identities|Articles containing proofs

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。