HarmonicNumber

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

(Subscript[H, z]) is an analytical function of z which is defined over the whole complex z-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function (Subscript[H, z]) has an infinite set of singular points:
a) z==-k/;k∈(SuperPlus[ℕ]), are the simple poles with residues -1;
b) z==(OverTilde[∞]) is the point of convergence of poles, which is similar to considering (OverTilde[∞]) as an essential singular point.

Branch points

The function (Subscript[H, z]) does not have branch points.

Branch cuts

The function (Subscript[H, z]) does not have branch cuts.

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==0

Expansions at z==(Subscript[z, 0])/; (Subscript[z, 0])≠-n

Expansions at z==-n

Asymptotic series expansions

Residue representations

Other series representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Limit representations

Generating functions

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Multiple arguments

Products, sums, and powers of the direct function

Sums of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

Complex characteristics

Real part

Imaginary part

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Summation

Finite summation

Infinite summation

G.Huvent (2006)

G.Huvent (2006)

G.Huvent (2006)

G.Huvent (2006)

G.Huvent (2006)

G.Huvent (2006)

G.Huvent (2006)

G.Huvent (2006)

G.Huvent (2006)

G.Huvent (2006)

G.Huvent (2006)

G.Huvent (2006)

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Theorems

The average number of comparisions done within the Quicksort algorithm for sorting

The average number of comparisons done within the Quicksort algorithm for sorting a list of n randomly ordered numbers is 2(n+1)((Subscript[H, n+1])-1).

The stacking of books (or coins) of equal lengths (diameters)

n books (or coins) of equal length (diameter) l can be stacked on top of each other without falling in such a way that the nth book hangs over by ((Subscript[H, n])-1)l/2.

The piecewise linear potential in the time independent Schrödinger equation

The piecewise linear potential V(x)==-(∑_(k=1)^∞)(π^2)(k^2) (θ(x-(Subscript[H, k-1])) θ((Subscript[H, k])-x)+θ(-x-(Subscript[H, k-1])) θ((Subscript[H, k])+x)) in the time–independent Schrödinger equation -ψ''(x)+V(x)ψ(x)==E ψ(x), -∞<x<∞ has a bound state at E=0 in the continuous spectrum. Its wave function is

ψ(x)==(∑_(k=1)^∞)((((-1))^k)/k)((θ(x-(Subscript[H, k-1])) θ((Subscript[H, k])-x)  sin(k π (x-(Subscript[H, k])))+θ(x+(Subscript[H, k])) θ(-(Subscript[H, k-1])-x)  sin(k π (x+(Subscript[H, k])))).

History

– Pythagoras of Samos (569-475 B.C) discovered the harmonic series
– Richard Suiseth (14th century)and  Nicole d’Oresme (1350) studied the harmonic series and found its divergence
– Pietro Mengoli (1647) proved divergence of the harmonic series and that the harmonic series with alternating signs converges to log (2)
– Nicolaus Mercator (1668) studied the series corresponding to the series of log(1+z)
– Jacob Bernoulli (1689) again proved divirgence of harmonic series
– G. W. Leibniz (1673)
– L. Euler (1740)
– Jacob Bernoulli (1744)

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