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Homogeneous Functions

zkee

New member
Feb 13, 2014
2
Hey people!

I'm confused as to why the ln(Y/X) part of the numerator is not considered in the calculation of the degree of numerator.

q3.jpg

Any help or websites to browse through for the answer would be appreciated!
 

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
The definition of an homogeneous function means that if $t \in \mathbb{R}$ then $f(tx,ty) = t^n f(x,y)$, where $n$ is the degree of homogeneity. When you do this to $g(x,y) = \ln (y/x)$ what you get is

$$g(tx,ty) = \ln \left( \frac{ty}{tx} \right) = \ln \left( \frac{y}{x} \right) = g(x,y).$$

The $t$'s cancel, therefore it makes no contribution. This is a degree zero homogeneous function. :)

Best wishes,

Fantini.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,869
Hey people!

I'm confused as to why the ln(Y/X) part of the numerator is not considered in the calculation of the degree of numerator.

Any help or websites to browse through for the answer would be appreciated!
Welcome to MHB, zkee! :)

The degree of y/x is 0.
Or in other words, $\ln(y/x)$ behaves like a constant.