Mathematical Methods for Physicists: A concise introduction - Site Map

COMPLEX INTEGRATION
Figure 6.12.
f 0 f …z
…z 0 †ˆlim 0 ‡ h†f …z 0 †
h!0 h
I
1
ˆ lim
h!0 2ih
C
1
z …z 0 ‡ h† 1
z z 0
f …z†dz:
Now
1 1
h z …z 0 ‡ h† 1
1
ˆ
z z 0 …z z 0 † 2 ‡ h
…z z 0 h†…z z 0 † 2 :
Thus,
f 0 …z 0 †ˆ 1 I
2i C
f …z†
…z z 0 † 2 dz ‡ 1
2i lim h h!0
IC
f …z†
…z z 0 h†…z z 0 † 2 dz:
The proof follows if the limit on the right hand side approaches zero as h ! 0. To
show this, let us draw a small circle of radius centered at z 0 (Fig. 6.12), then
1
2i lim
h!0 h IC
f …z†
…z z 0 h†…z z 0 † 2 dz ˆ 1
2i lim h h!0
I
f …z†
…z z 0 h†…z z 0 † 2 dz:
Now choose h so small (in absolute value) that z 0 ‡ h lies in and jhj =2 ˆ =2. Next, as f …z† is analytic in R, we can ®nd a positive
number M such that j f …z†j M. And the length of is 2. Thus,
I
h
2i
f …z†dz
…z z 0 h†…z z 0 † 2
jj h M…2†
2 …=2†… 2 † ˆ 2jhjM
2 ! 0 as h ! 0;
proving the formula for f 0 …z 0 †.
263