Is there such a thing as a homogenous function of degree n < 0?(adsbygoogle = window.adsbygoogle || []).push({});

Considering functions of two variables, the expression:

[tex]f(x,y) = \frac{y}{x}[/tex]

is homogeneous in degree 0, since:

[tex]f(tx,ty) = \frac{ty}{tx} = \frac{y}{x} = f(x,y) = t^0 \cdot f(x,y)[/tex]

and the expression:

[tex]f(x,y) = x[/tex]

is homogenous in degree 1 since:

[tex]f(tx,ty) = tx = t^1 \cdot f(x,y)[/tex]

and the expression:

[tex]f(x,y) = x^3y^2[/tex]

is homogeneous in degree 5, since:

[tex]f(tx,ty) = t^5 \cdot f(x,y)[/tex]

I suppose that in the same way I could construct a function something like:

[tex]f(x,y) = \frac{y}{x^2}[/tex]

So that:

[tex]f(tx,ty) = t^{-1} \cdot f(x,y)[/tex]

or in other words, "homogenous" in degree n = -1.

Does this ever really come up much?

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# Homogenous Functions

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