Harmonic number, Harmonic number - Applications, Harmonic number - Generalizations, Harmonic number - Generating functions, Harmonic number - Introduction
ARTICLES RELATED TO Harmonic number - Applications
The harmonic numbers appear in several calculation formulas, such as the digamma function:
where γ is the Euler-Mascheroni constant 0.5772156649... This relation is also frequently used to define the extension of the harmonic numbers to non-integer n. The harmonic numbers are also frequently used to define γ, in that
although
converges more quickly.
An integral representation is given by Euler:
This representation can be easily shown to satisfy the recurrence relation by the formula:
...
The generalized harmonic number of order n of m is given by
.
Note that n may be equal to , provided m > 1.
And If , while , the harmonic series does not converge and hence the harmonic number does not exist.
Other notations occasionally used include
The special case of m = 1 is simply called a harmonic number and is frequently written w ...
Euler's integral formula for the harmonic numbers follows from the integral identity
which holds for general complex-valued s, for the suitably extended binomial coefficients. By choosing a=0, this formula gives both an integral and a series representation for a function that interpolates the harmonic numbers and extends a definition to the complex plane. This integral relation is easily derived by manipulating the Newton series
which is just the Newton's generalized binomial theorem. The interpolating function is in f ...
