Mathematical Methods for Physicists: A concise introduction - Site Map

APPENDIX 1 PRELIMINARIES
Di€erentiation of integrals
(1) Inde®nite integrals: We ®rst consider di€erentiation of inde®nite integrals. If
f …x;† is an integrable function of x and is a variable parameter, and if
Z
f …x;†dx ˆ G…x;†;
…A1:11†
then we have
@G…x;†=@x ˆ f …x;†:
…A1:12†
Furthermore, if f …x;† is such that
then we obtain
@
@x
@ 2 G…x;†
@x@
@G…x;†
@
ˆ @
@
ˆ @2 G…x;†
@@x ;
@G…x;†
@x
ˆ
@f …x;†
@
and integrating gives
Z @f …x;†
@
dx ˆ @G…x;† ; …A1:13†
@
which is valid provided @f …x;†=@ is continuous in x as well as .
(2) De®nite integrals: We now extend the above procedure to de®nite integrals:
I…† ˆ
Z b
a
f …x;†dx;
…A1:14†
where f …x;† is an integrable function of x in the interval a x b, anda and b
are in general continuous and di€erentiable (at least once) functions of . We now
have a relation similar to Eq. (A1.11):
I…† ˆ
Z b
a
f …x;†dx ˆ G…b;†G…a;†
…A1:15†
and, from Eq. (A1.13),
Z b
a
@f …x;†
@
Di€erentiating (A1.15) totally
dI…†
d
ˆ @G…b;†
@b
dx ˆ @G…b;†
@
db
d ‡ @G…b;†
@
534
@G…a;† : …A1:16†
@
@G…a;†
@a
da
d @G…a;† :
@