Mathematical Methods for Physicists: A concise introduction - Site Map

APPENDIX 1 PRELIMINARIES
Problem A1.4
Show that
1
2 ‡ 1 4 ‡ 1 8 ‡‡ 1 1 for all positive integers n > 1:
n1
2
If x 1 ; x 2 ; ...; x n are n positive numbers, their arithmetic mean is de®ned by
A ˆ 1
n
X n
kˆ1
and their geometric mean by
s
Y
G ˆ n
n
ˆ
x k ˆ x1 ‡ x 2 ‡‡x n
n
x k
kˆ1
np x 1 x 2 x n ;
where P and Q are the summation and product signs. The harmonic mean H is
sometimes useful and it is de®ned by
1
H ˆ 1 X n 1
ˆ 1
1
‡ 1 ‡‡ 1
:
n x
kˆ1 k n x 1 x 2 x n
There is a basic inequality among the three means: A G H, the equality sign
occurring when x 1 ˆ x 2 ˆˆx n .
Problem A1.5
If x 1 and x 2 are two positive numbers, show that A G H.
Functions
We assume that the reader is familiar with the concept of functions and the
process of graphing functions.
A polynomial of degree n is a function of the form
f …x† ˆp n …x† ˆa 0 x n ‡ a 1 x n1 ‡ a 2 x n2 ‡‡a n …a j ˆ constant; a 0 6ˆ 0†:
A polynomial can be di€erentiated and integrated. Although we have written
a j ˆ constant, they might still be functions of some other variable independent
of x. For example,
t 3 x 3 ‡ sin tx 2 p
‡
t x ‡ t
is a polynomial function of x (of degree 3) and each of the as is a function of a
certain variable t: a 0 ˆ t 3 ; a 1 ˆ sin t; a 2 ˆ t 1=2 ; a 3 ˆ t.
The polynomial equation f …x† ˆ0 has exactly n roots provided we count repetitions.
For example, x 3 3x 2 ‡ 3x 1 ˆ 0 can be written …x 1† 3 ˆ 0 so that
the three roots are 1, 1, 1. Note that here we have used the binomial theorem
…a ‡ x† n ˆ a n ‡ na n1 n…n 1†
x ‡ a n2 x 2 ‡‡x n :
2!
508