The harmonic number H_n,1 with (red line) with its asymptotic limit γ + ln[x] (blue line).

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: Image File history File links HarmonicNumbers. ... Image File history File links HarmonicNumbers. ... For other meanings of mathematics or math, see mathematics (disambiguation). ...

This also equals n times the inverse of the harmonic mean of these natural numbers. In mathematics, the harmonic mean is one of several methods of calculating an average. ...

Harmonic numbers were studied in antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function and appear in various expressions for various special functions. To meet Wikipedias quality standards, this article or section may require cleanup. ... See harmonic series (music) for the (related) musical concept. ... In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In mathematics, several functions are important enough to deserve their own name. ...

Introduction

The generalized harmonic number of order n of m is given by

Note that the limit as n tends to infinity exists if m > 1.

Other notations occasionally used include

The special case of m = 1 is simply called a harmonic number and is frequently written without the superscript, as

In the limit of , the generalized harmonic number converges to the Riemann zeta function In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...

The related sum occurs in the study of Bernoulli numbers; the harmonic numbers also appear in the study of Stirling numbers. In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... In mathematics, Stirling numbers arise in a variety of combinatorics problems. ...

For m = 1, the asymptotic expansion is given by

where γ is the Euler-Mascheroni constant The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is γ ≈ 0. ...

Applications

The harmonic numbers appear in several calculation formulas, such as the digamma function: In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. ...

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: Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

This representation can be easily shown to satisfy the recurrence relation by the formula

and then

inside the integral.

Generalizations

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 In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ... In mathematics, a difference operator maps a function, f(x), to another function, f(x + a) − f(x + b). ...

which is just the Newton's generalized binomial theorem. The interpolating function is in fact just the digamma function: In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. ...

where ψ(x) is the digamma, and γ is the Euler-Mascheroni constant. The integration process may be repeated to obtain

Generating functions

A generating function for the generalized harmonic numbers is In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

where Li_m(z) is the polylogarithm, and | z | < 1. As a special case, one has The polylogarithm (also known as Jonquiéres function) is a special function that is defined for all complex numbers s and z where |z| < 1 by: The special cases s = 2 and s = 3 are called the dilogarithm (also referred to as Spences function) and trilogarithm respectively. ...

where ln(z) is the logarithm. An exponential generating function is Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

where Ein(z) is the entire exponential integral. Note that In mathematics, the exponential integral Ei(x) is defined as Since 1/t diverges at t=0, the above integral has to be understood in terms of the Cauchy principal value. ...

where Γ(0,z) is the incomplete gamma function. In mathematics, the gamma function is defined by a definite integral. ...

References

• Arthur T. Benjamin, Gregory O. Preston, Jennifer J. Quinn, A Stirling Encounter with Harmonic Numbers, (2002) Mathematics Magazine, 75 (2) pp 95-103.
• Donald Knuth. The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89683-4. Section 1.2.7: Harmonic Numbers, pp.75–79.
• Ed Sandifer, How Euler Did It -- Estimating the Basel problem (2003)
• Weisstein, Eric W., Harmonic Number at MathWorld.
• Peter Paule and Carsten Schneider, Computer Proofs of a New Family of Harmonic Number Identities, (2003) Adv. in Appl. Math. 31(2), pp. 359-378.
• Wenchang CHU, A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apery Numbers, (2004) The Electronic Journal of Combinatorics, 11, #N15.

This article incorporates material from Harmonic number on PlanetMath, which is licensed under the GFDL. Donald Knuth at a reception for the Open Content Alliance. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...