Mathematical Methods for Physicists: A concise introduction - Site Map

APPENDIX 1 PRELIMINARIES
Problem A1.6
Using the rules of exponents, prove that ln …mn† ˆln m ‡ ln n:
Problem A1.7
Prove that:
p
…a† sin 2 x ˆ 1
2 …1 cos 2x†; cos2 x ˆ 1
2
…1 ‡ cos 2x†, and (b) A cos x ‡
B sin x ˆ
A 2 ‡ B 2 sin…x ‡ †, where tan ˆ A=B
Problem A1.8
Prove that: …a† cosh 2 x sinh 2 x ˆ 1, and (b) 2 x ‡ tanh 2 x ˆ 1.
Limits
We are sometimes required to ®nd the limit of a function f …x† as x approaches
some particular value :
lim f …x† ˆl:
x!
This means that if jx j is small enough, j f …x†lj can be made as small as we
please. A more precise analytic description of lim x! f …x† ˆl is the following:
For any ">0 (however small) we can always ®nd a number
(which, in general, depends upon ") such that jf …x†lj