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Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

For real z, the function ⌊z⌋  is the greatest integer less than or equal to z.

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

⌊z⌋ is a non-analytical function; it is a piecewise constant function which is defined in the whole complex z-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Sets of discontinuity

Series representations

Exponential Fourier series

Other series representations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Argument involving related functions

Addition formulas

Multiple arguments

Products, sums, and powers of the direct function

Sums of the direct function

Related transformations

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

In a distributional sense, for x∈ℝ .

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

For the direct function itself

In the following formulas a∈ℝ.

Integral transforms

Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Mellin transforms

Summation

Finite summation

Infinite summation

Representations through equivalent functions

With related functions

With Round

With Ceiling

With IntegerPart

With FractionalPart

With Mod

With Quotient

With elementary functions

Zeros

Theorems

The Arnold cat map

{x,y}⟹{x+y-⌊x+y⌋,x+2y-⌊x+2y⌋}.

History

– C.F. Gauss (1808)
– J. Liouville (1838)
– K.E. Iverson (1962) suggested the notation ⌊z⌋

The function ⌊z⌋ is encountered often in mathematics and the natural sciences.

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