Quotient

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

quotient(m,n) is the integer quotient of m and n.

Specific values

Specialized values

For fixed m

For fixed n

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

quotient(m,n) is a nonanalytical function; it is a piecewise continuous function which is defined over (ℂ^2).

Symmetries and periodicities

Parity

quotient(m,n) is an even function.

Periodicity

No periodicity

Sets of discontinuity

The function quotient(m,n) is a piecewise continuous function with jumps on the curves Re((m/n))=k∨Im((m/n))=l/;k,l∈ℤ. The functional property quotient(m,n)==quotient((m/n),1)==⌊(m/n)⌋ makes the behaviour of the quotient(m,n) similar to behavior of ⌊(m/n)⌋.

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

Related transformations

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

With respect to m

With respect to n

Fractional integro-differentiation

With respect to m

With respect to n

Integration

Indefinite integration

Involving only one direct function with respect to m

Involving one direct function and elementary functions with respect to m

Involving only one direct function with respect to n

Involving one direct function and elementary functions with respect to n

Definite integration

For the direct function with respect to m

For the direct function with respect to n

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 Round

With Ceiling

With IntegerPart

With FractionalPart

With Mod

With elementary functions

Zeros

History

–  J. Nemorarius (1237)
– the word "quotient" appears for the first time around 1250 in the writings of Meister Gernadus.

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