Mathematical Methods for Physicists: A concise introduction - Site Map

INFINITE SERIES
A ®nite discontinuity may occur at x ˆ . This will arise when
lim x!0 f …x† ˆl 1 , lim x!0 f …x† ˆl 2 , and l 1 6ˆ l 2 .
It is obvious that a continuous function will be bounded in any ®nite interval.
This means that we can ®nd numbers m and M independent of x and such that
m f …x†M for a x b. Furthermore, we expect to ®nd x 0 ; x 1 such that
f …x 0 †ˆm and f …x 1 †ˆM.
The order of magnitude of a function is indicated in terms of its variable. Thus,
if x is very small, and if f …x† ˆa 1 x ‡ a 2 x 2 ‡ a 3 x 3 ‡ (a k constant), its magnitude
is governed by the term in x and we write f …x† ˆO…x†. When a 1 ˆ 0, we
write f …x† ˆO…x 2 †, etc. When f …x† ˆO…x n †, then lim x!0 f f …x†=x n g is ®nite and/
or lim x!0 f f …x†=x n1 gˆ0.
A function f …x† is said to be di€erentiable or to possess a derivative at the point
x if lim h!0 ‰ f …x ‡ h†f …x†Š=h exists. We write this limit in various forms
df =dx; f 0 or Df , where D ˆ d… †=dx. Most of the functions in physics can be
successively di€erentiated a number of times. These successive derivatives are
written as f 0 …x†; f 00 …x†; ...; f n …x†; ...; or Df ; D 2 f ; ...; D n f ; ...:
Problem A1.9
If f …x† ˆx 2 , prove that: (a) lim x!2 f …x† ˆ4, and …b† f …x† is continuous at x ˆ 2.
In®nite series
p
In®nite series involve the notion of sequence in a simple way. For example, 2 is
irrational and can only be expressed as a non-recurring decimal 1:414 ...: We can
approximate to its value by a sequence of rationals, 1, 1.4, 1.41, 1.414, ...say fa
p n g
which is a countable set limit of a n whose values approach inde®nitely close to
p
2 .
Because of this we say
p
the limit of a n as n tends to in®nity exists and equals 2 ,
and write lim n!1 a n ˆ
2 .
In general, a sequence u 1 ; u 2 ; ...; fu n g is a function de®ned on the set of natural
numbers. The sequence is said to have the limit l or to converge to l, if given any
">0 there exists a number N > 0 such that ju n lj