The_Cambridge_Handbook_of_Physics_Formulas

28 Mathematics
Power series
Binomial
series a
(1+x) n =1+nx+ n(n−1)
2!
( )
Binomial
n n n!
C
coefficient b r ≡ ≡
r r!(n−r)!
Binomial
theorem
(a+b) n =
n∑
k=0
x 2 + n(n−1)(n−2) x 3 +··· (2.120)
3!
(2.121)
( n
k)
a n−k b k (2.122)
Taylor series
(about a) c
Taylor series
(3-D)
Maclaurin
series
f(a+x)=f(a)+xf (1) (a)+ x2
2! f(2) (a)+···+ xn−1
(n−1)! f(n−1) (a)+··· (2.123)
f(a+x)=f(a)+(x·∇)f| a + (x·∇)2
2!
f| a + (x·∇)3 f| a +··· (2.124)
3!
f(x)=f(0)+xf (1) (0)+ x2
2! f(2) (0)+···+ xn−1
(n−1)! f(n−1) (0)+··· (2.125)
a If n is a positive integer the series terminates and is valid for all x. Otherwise the (infinite) series is convergent for
|x| < 1.
b The coefficient of x r in the binomial series.
c xf (n) (a) isx times the nth derivative of the function f(x) with respect to x evaluated at a, taken as well behaved
around a. (x·∇) n f| a is its extension to three dimensions.
Limits
n c x n → 0 as n →∞ if |x| < 1 (for any fixed c) (2.126)
x n /n! → 0 as n →∞ (for any fixed x) (2.127)
(1+x/n) n → e x as n →∞ (2.128)
xlnx → 0 as x → 0 (2.129)
sinx
x
→ 1 as x → 0 (2.130)
f(x)
If f(a)=g(a)=0 or ∞ then lim
x→a g(x) = f(1) (a)
g (1) (a)
(l’Hôpital’s rule) (2.131)