The best-known properties and formulas of the number theory functions

Simple values at zero

The rounding and congruence functions , , , , and have zero values at zero:

Specific values for specialized variables

The values of five rounding and congruence functions , , , , and at some fixed points or for specialized variables and infinities are shown in the following table:

The values of mod function , and at some fixed points or for specialized variables are shown here:

Analyticity

All seven rounding and congruence functions (floor function , round function , ceiling function , integer part , fractional part , mod function , and the quotient function ) are not analytical functions. They are defined for all complex values of their arguments and . The functions , , , and are piecewise constant functions and the functions , , and are piecewise continuous functions.

Periodicity

The rounding and congruence functions , , , , , and are not periodic functions.

is a periodic function with respect to with period :

Parity and symmetry

Four rounding and congruence functions (round function , integer part , fractional part , and mod function ) are odd functions. The quotient function is an even function:

The rounding and congruence functions , , , , and have the following mirror symmetry:

Sets of discontinuity

The floor and ceiling functions and are piecewise constant functions with unit jumps on the lines . The functions (and ) are continuous from the right (from the left) on the intervals and from above (from below) on the intervals .

The function is a piecewise constant function with unit jumps on the lines . The function is continuous from the right on the intervals , and from the left on the intervals . The function is continuous from above on the intervals , and from below on the intervals .

The function (and ) is a piecewise constant (continuous) function with unit jumps on the lines . The functions and are continuous from the right on the intervals , and from the left on the intervals . The functions and are continuous from above on the intervals , and from below on the intervals .

The functions and are piecewise continuous functions with jumps on the curves . The functional properties and make the behavior of that functions similar to the behavior of floor function .

The previous described properties can be described in more detail by the formulas from the following table:

Series representations

The rounding and congruence functions , , , , , , and have the following series representations:

Transformations and argument simplifications (arguments involving basic arithmetic operations)

The values of rounding and congruence functions , , , , and at the points , , can also be represented by the following formulas:

The values of the functions and at the points , , , , , and have the following representations:

Transformations and argument simplifications (arguments involving related functions)

Compositions of rounding and congruence functions , , , , , , and with the rounding and congruence functions in many cases lead to very simple zero results:

Addition formulas

The rounding and congruence functions , , , , and satisfy the following addition formulas:

Multiple arguments

The rounding and congruence functions , , , , , and have the following relations for multiple arguments:

Sums of the direct function

Sums of the floor and ceiling functions and satisfy the following relations:

Identities

All rounding and congruence functions satisfy numerous identities, for example:

Complex characteristics

Complex characteristics (real and imaginary parts and , absolute value , argument , complex conjugate , and signum ) of the rounding and congruence functions can be represented in the forms shown in the following tables:

Differentiation

Derivatives of the rounding and congruence functions , , , , , , and can be evaluated in the classical and distributional sense. In the last case, all variables should be real and results include the Dirac delta function. All rounding and congruence functions also have fractional derivatives. All these derivatives can be represented as shown in the following tables:

Indefinite integration

Simple indefinite integrals of the rounding and congruence functions , , , , , , and have the following representations:

Definite integration

Some definite integrals of the rounding and congruence functions , , , , , , and can be evaluated and are shown in the following table:

Integral transforms

All Fourier transforms of the rounding and congruence functions , , , , , , and can be evaluated in a distributional sense and include the Dirac delta function:

Laplace and Mellin integral transforms of the rounding and congruence functions , , , , , , and can be evaluated in the classical sense:

Summation

Sometimes finite and infinite sums including rounding and congruence functions have rather simple representations, for example:

Zeros

Zeros of rounding and congruence functions are given as follows: