Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
If y ˆ 0, the required limit is lim x!0 x=x ˆ 1. On the other hand, if
x ˆ 0, the required limit is 1. Then since the limit depends on the manner
in which z ! 0, the derivative does not exist and so f …z† ˆz* is non-analytic
everywhere.
Example 6.8
Given f …z† ˆ2z 2 1, ®nd f 0 …z† at z 0 ˆ 1 i.
Solution:
f 0 …z 0 †ˆf 0 …2z 2 1†‰2…1 i† 2 1Š
…1 i† ˆ lim
z!1i z …1 i†
2‰z …1 i†Š‰z ‡…1 i†Š
ˆ lim
z!1i z …1 i†
ˆ lim 2‰z ‡…1 i†Š ˆ 4…1 i†:
z!1i
The rules for di€erentiating sums, products, and quotients are, in general, the
same for complex functions as for real-valued functions. That is, if f 0 …z 0 † and
g 0 …z 0 † exist, then:
(1) … f ‡ g† 0 …z 0 †ˆf 0 …z 0 †‡g 0 …z 0 †;
(2) … fg† 0 …z 0 †ˆf 0 …z 0 †g…z 0 †‡f …z 0 †g 0 …z 0 †;
f 0
(3) …z †ˆg…z 0† f 0 …z 0 †f …z 0 †g 0 …z 0 †
g 0
g…z 0 † 2 ; if g 0 …z 0 †6ˆ0:
The Cauchy±Riemann conditions
We call f …z† analytic at z 0 ,iff 0 …z† exists for all z in some neighborhood of z 0 ;
and f …z† is analytic in a region R if it is analytic at every point of R. Cauchy and
Riemann provided us with a simple but extremely important test for the analyticity
of f …z†. To deduce the Cauchy±Riemann conditions for the analyticity of
f …z†, let us return to Eq. (6.10):
f 0 f …z
…z 0 †ˆ lim 0 ‡ z†f …z 0 †
:
z!0 z
If we write f …z† ˆu…x; y†‡iv…x; y†, this becomes
f 0 …z† ˆ
lim
x;y!0
u…x ‡ x; y ‡ y†u…x; y†‡i…same for v†
:
x ‡ iy
There are of course an in®nite number of ways to approach a point z on a twodimensional
surface. Let us consider two possible approaches ± along x and along
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