Is the given function Homogenous?

Asked Nov 09 '23 at 11:23

Active Nov 09 '23 at 18:09

Viewed 48 times

$g(x, y)=x^2/y^3$

Now a function $f(x,y)$ is said to be a homogenous function of degree $n$ if,

$f(tx,ty)=(t^n) f(x,y)$ where $t$ is some scalar

For example $f(x,y)= x^2+y^2$

Now for some scalar $t$

$f(tx,ty)=(t^2)(x^2+y^2)=(t^2)f(x,y)$

Thus $f$ is a homogenous function of degree 2 Now for the example in which I have a difficulty ($g(x,y)=x^2/y^3$)

$$g(tx,ty)=(t^{-1})g(x,y)$$

So, does that mean that $g$ is homogenous function of degree $-1$?

And can the degree of a homogenous function be negative like in this case it comes out to be -1?

edited Nov 09 '23 at 18:09

Gonçalo

asked Nov 09 '23 at 11:23

Amoeba_37

Yeah, that's what it means, and your working is fine. – Sarvesh Ravichandran Iyer Nov 09 '23 at 11:26
@SarveshRavichandranIyer The degree of a homogenous function can be negative? – Sebastiano Nov 09 '23 at 12:36
1

@Sebastiano According to that definition, I can't see why that can't be the case. For example, $f(x) = x^{-1}$ is homogenous of degree $-1$ on its domain $\mathbb R \setminus {0}$. – Sarvesh Ravichandran Iyer Nov 09 '23 at 12:53
@SarveshRavichandranIyer the function that you have defined in Italy we call an algebraic fraction or fractional monomial. It is evident that the degree is negative. Thank you. – Sebastiano Nov 09 '23 at 12:55
1

@Sebastiano You're welcome! After some google searches, it's evident that negative degree homogenous functions are not studied much : however, from the example I gave it's clear that some obvious functions fall in that category. – Sarvesh Ravichandran Iyer Nov 09 '23 at 12:59

0 Answers0