Mathematical Methods for Physicists: A concise introduction - Site Map

HOMOGENEOUS FUNCTIONS
which becomes, with the help of Eqs. (A1.12) and (A1.16),
dI…†
d
ˆ
Z b
a
@f …x;†
dx ‡ f …b;† db da
f …a;†
@
d d ;
…A1:17†
which is known as Leibnitz's rule for di€erentiating a de®nite integral. If a and b,
the limits of integration, do not depend on , then Eq. (A1.17) reduces to
dI…†
d
ˆ d
d
Z b
a
f …x;†dx ˆ
Z b
a
@f …x;†
dx:
@
Problem A1.26
Z
2
If I…† ˆ sin…x†=xdx, ®nd dI=d.
0
Homogeneous functions
A homogeneous function f …x 1 ; x 2 ; ...; x n † of the kth degree is de®ned by the
relation
f …x 1 ;x 2 ; ...;x n †ˆ k f …x 1 ; x 2 ; ...; x n †:
For example, x 3 ‡ 3x 2 y y 3 is homogeneous of the third degree in the variables x
and y.
If f …x 1 ; x 2 ; ...; x n ) is homogeneous of degree k then it is straightforward to
show that
X n
jˆ1
x j
@f
@x j
ˆ kf :
This is known as Euler's theorem on homogeneous functions.
Problem A1.27
Show that Euler's theorem on homogeneous functions is true.
Taylor series for functions of two independent variables
The ideas involved in Taylor series for functions of one variable can be generalized.
For example, consider a function of two variables (x; y). If all the nth partial
derivatives of f …x; y† are continuous in a closed region and if the …n ‡ 1)st partial
535