Quotient of Homogeneous Functions

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Theorem

Let $M \left({x, y}\right)$ and $N \left({x, y}\right)$ be homogeneous functions of the same degree.


Then:

$\dfrac {M \left({x, y}\right)} {N \left({x, y}\right)}$

is homogeneous of degree zero.


Proof

Let:

$Q \left({x, y}\right) = \dfrac {M \left({x, y}\right)} {N \left({x, y}\right)}$

where $M$ and $N$ are homogeneous functions of degree $n$.


Let $t \in \R$. Then:

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle Q \left({tx, ty}\right)\) \(=\) \(\displaystyle \) \(\displaystyle \frac {M \left({tx, ty}\right)} {N \left({tx, ty}\right)}\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \frac {t^n M \left({x, y}\right)} {t^n N \left({x, y}\right)}\) \(\displaystyle \) \(\displaystyle \)          as $M$ and $N$ are homogeneous of degree $n$          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle t^0 \frac {M \left({x, y}\right)} {N \left({x, y}\right)}\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle t^0 Q \left({x, y}\right)\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle Q \left({x, y}\right)\) \(\displaystyle \) \(\displaystyle \)                    

The result follows from the definition.

$\blacksquare$


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