Homogeneous Functions
The polynomial function, where the degree of each term is n and
f(x,y) = a0xn + a1 xn-1 + a2xn-2y2 +…….+ anyn
is called a homogeneous function of degree n x and y. Again.
f(x,y) = a0xn + a1 xn-1 + a2xn-2y2 +…….+ anyn
= xn [a0 + a1 (y/x) + a2 (y/x)2 +…..+ + an(y/x)n]
= xnf(y/x)
Thus, a function f(x,y) is said to be homogeneous function of degree n in x and y, if
f(x,y) = xn f(y/x)
Definition:
A function z = f(x,y) is said to be homogeneous of degree n (n being a constant) if, for any
real number λ
F(λ× , λy ) = λn f(x,y)
Thus, if both x and y are multiplied by the same real number, then the resulting function value is a power of the number times the function value f(x,y).
Example : If f(x, y) = 2x2y+ xy2 – y3
F(λ× , λy ) = 2( λx)2 ( λx) + (λx) (λy)2 -(λy)3
=2λ³x2y+ λ³xy2 – λ³y3
= λ³f(x,y)
