Round

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

For real z , the function ⌊x⌉ is the integer closest to z (if z≠±(1/2), ±(3/2),…).

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

⌊z⌉ is a nonanalytical function; it is a piecewise constant function which is defined over the whole complex z-plane.

Symmetries and periodicities

Parity

⌊z⌉ is an odd function.

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

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

Representations through equivalent functions

With related functions

With Floor

With Ceiling

With IntegerPart

With FractionalPart

With Mod

With Quotient

With elementary functions

Zeros

History

– C. F. Gauss (1808)
– J. Liouville (1838)
– J. Hastad (1988) suggested the notation ⌊z⌉

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