Page 346 - Determinants and Their Applications in Mathematical Physics

A.9 The Euler and Modified Euler Theorems on Homogeneous Functions 331

The first is due to Euler. The second is similar in nature to Euler’s and can
be obtained from it by means of a change of variable.
The function f is said to be homogeneous of degree s in its variables if
f(λx 0 ,λx 1 ,λx 2 ,...,λx n )= λ f. (A.9.2)
s
Theorem A.5 (Euler). If the variables are independent and f is differ-
entiable with respect to each of its variables and is also homogeneous of
degree s in its variables, then
∂f
n

= sf.
x r
r=0 ∂x r
The proof is well known.
The function f is said to be homogeneous of degree s in the suffixes of
its variables if
2
f(x 0 ,λx 1 ,λ x 2 ,...,λ x n )= λ f. (A.9.3)
n
s
Theorem A.6 (Modified Euler). If the variables are independent and f
is differentiable with respect to each of its variables and is also homogeneous
of degree s in the suffixes of its variables, then
n
∂f
= sf.
rx r
r=1 ∂x r
Proof. Put
u r = λ x r , 0 ≤ r ≤ n [in (A.9.3)].
r
Then,
f(u 0 ,u 1 ,u 2 ,...,u n ) ≡ λ f.
s
Differentiating both sides with respect to λ,
n
∂f du r s−1
= sλ f,
∂u r dλ
r=0
n
∂f r−1 s−1
rλ x r = sλ f.
r=0 ∂u r
Put λ = 1. Then, u r = x r and the theorem appears.
A proof can also be obtained from Theorem A.5 with the aid of the
change of variable

v r = x .
r
r
Both these theorems are applied in Section 4.8.7 on double-sum relations
for Hankelians.