Mathematical Methods for Physicists: A concise introduction - Site Map

TAYLOR'S EXPANSION
where f 0 …† means that f …x† has been di€erentiated and then we have put x ˆ ;
and by f 00 …† we mean that we have found f 00 …x† and then put x ˆ , and so on.
Substituting these into (A1.8) we obtain
f …x† ˆf …†‡f 0 …†…x †‡ 1 2! f 00 …†…x † 2 ‡‡ 1 n! f …n† …†…x † n ‡:
…A1:9†
This is the Taylor series for f …x† about x ˆ . The Maclaurin series for f …x† is the
Taylor series about the origin. Putting ˆ 0 in (A1.9), we obtain the Maclaurin
series for f …x†:
f …x† ˆf …0†‡f 0 …0†x ‡ 1 2! f 00 …0†x 2 ‡ 1 3! f F…0†x3 ‡‡ 1 n! f …n† …0†x n ‡:
…A1:10†
Example A1.15
Find the Maclaurin series expansion of the exponential function e x .
Solution: Here f …x† ˆe x . Di€erentiating, we have f …n† …0† ˆ1 for all
n; n ˆ 1; 2; 3 .... Then, by Eq. (A1.10), we have
e x ˆ 1 ‡ x ‡ 1 2! x2 ‡ 1 3! x3 ‡ ˆX1 x n
; 1< x < 1:
n!
The following series are frequently employed in practice:
…1† sin x ˆ x x3
3! ‡ x5
5! x7
x 2n1
7! ‡…1†n1 ‡;
…2n 1†!
1< x < 1:
…2† cos x ˆ 1 x2
2! ‡ x4
4! x6
x 2n2
6! ‡…1†n1 ‡;
…2n 2†!
1 < x < 1:
…3† e x ˆ 1 ‡ x ‡ x2
2! ‡ x3 xn1
‡‡ ‡; 1 < x < 1:
3! …n 1†!
…4† lnj…1 ‡ xj ˆx x2
2 ‡ x3
3 x4
x n
4 ‡…1†n1 ‡;
n
1 < x 1:
…5† 1 2 ln 1 ‡ x
1 x 3 ‡ x5
5 ‡ x7 x2n1
‡‡ ‡;
7 2n 1
1 < x < 1:
…6† tan 1 x ˆ x x3
3 ‡ x5
5 x7
x 2n1
7 ‡…1†n1 ‡; 1 x 1:
2n 1
…7† …1 ‡ x† p ˆ 1 ‡ px ‡
nˆ0
p…p 1†
x 2 ‡‡
2!
525
p…p 1†…p n ‡ 1†
x n ‡:
n!