The harmonic number of order of is defined as
Note that may be equal to , provided .
If , we may just write as (this is a common notation).
If and then the sum is the Riemann zeta function.
If , then we get what is known simply as“the harmonic number”, and it has many important properties. For example, it has asymptotic expansion where is Euler’s constant.
It is possible11See “The Art of computer programming” vol. 2 by D. Knuth to define harmonic numbers for non-integral . This is done by means of the series .
|Date of creation||2013-03-22 13:01:28|
|Last modified on||2013-03-22 13:01:28|
|Last modified by||mathcam (2727)|
|Defines||harmonic number of order|