Definition:Absolutely Symmetric Function/Definition 2

Definition

Let $f: \R^n \to \R$ be a real-valued function.

Let $f$ be such that, for all $\mathbf x := \tuple {x_1, x_2, \ldots, x_n} \in \R^n$:

$\map f {\mathbf x} = \map f {\mathbf y}$

where $\mathbf y$ is a permutation of $\tuple {x_1, x_2, \ldots, x_n}$.

Then $f$ is an absolutely symmetric function.

Examples

Arbitrary Example $1$

Let $f: \R^3 \to \R$ be the real-valued function defined as:

$\forall \tuple {x, y, z} \in \R^3: \map f {x, y, z} = x^2 + y^2 + 2 x y z$

Then $f$ is an absolutely symmetric function.

Arbitrary Example $2$

Let $f: \R^2 \to \R$ be the real-valued function defined as:

$\forall \tuple {x, y} \in \R^2: \map f {x, y} = x^2 + 2 x y + y^2$

Then $f$ is an absolutely symmetric function.

Also known as

An absolutely symmetric function is also known as a totally symmetric function.

Some sources use the term symmetric function,

Also see

• Equivalence of Definitions of Absolutely Symmetric Function

• Results about absolutely symmetric functions can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): absolutely symmetric
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): symmetric function: 2.