Mathematical Methods for Physicists: A concise introduction - Site Map

APPENDIX 1 PRELIMINARIES
derivatives exist in the open region, then we can expand the function f …x; y† about
x ˆ x 0 ; y ˆ y 0 in the form
f …x 0 ‡ h; y 0 ‡ k† ˆf …x 0 ; y 0 †‡ h @
@x ‡ k @
f …x
@y 0 ; y 0 †
‡ 1
h @
2! @x ‡ k @ 2
f …x
@y 0 ; y 0 †
‡‡ 1
h @
n! @x ‡ k @ n
f …x
@y 0 ; y 0 †‡R n ;
where h ˆ x ˆ x x 0 ; k ˆ y ˆ y y 0 ; R n , the remainder after n terms, is given
by
1
R n ˆ h @
…n ‡ 1†! @x ‡ k @ n‡1
f …x
@y 0 ‡ h; y 0 ‡ k†; 0