Mathematical Methods for Physicists: A concise introduction - Site Map

DIFFERENTIAL CALCULUS
and
But
f …z 0 ‡ z†f …z 0 †
z
ˆ @u…x 0; y 0 †
‡ i @v…x 0; y 0 †
@x @x
‡ ‰ H…x y†‡iG…x y† Š
q
…x† 2 ‡…y† 2
x ‡ iy
ˆ 1:
q
…x† 2 ‡…y† 2
:
x ‡ iy
Thus, as z ! 0, we have …x; y† !…0; 0† and
f …z
lim 0 ‡ z†f …z 0 †
z!0 z
ˆ @u…x 0; y 0 †
@x
‡ i @v…x 0; y 0 †
;
@x
which shows that the limit and so f 0 …z 0 † exist. Since f …z† is di€erentiable at all
points in region R, f …z† is analytic at z 0 which is any point in R.
The Cauchy±Riemann equations turn out to be both necessary and sucient
conditions that f …z† ˆu…x; y†‡iv…x; y† be analytic. Analytic functions are also
called regular or holomorphic functions. If f …z† is analytic everywhere in the ®nite
z complex plane, it is called an entire function. A function f …z† is said to be
singular at z ˆ z 0 , if it is not di€erentiable there; the point z 0 is called a singular
point of f …z†.
Harmonic functions
If f …z† ˆu…x; y†‡iv…x; y† is analytic in some region of the z plane, then at every
point of the region the Cauchy±Riemann conditions are satis®ed:
and therefore
@u
@x ˆ @v
@y ; and @u
@y ˆ@v @x ;
@ 2 u
@x 2 ˆ @2 v
@x@y ; and @ 2 u
@y 2 ˆ @2 v
@y@x ;
provided these second derivatives exist. In fact, one can show that if f …z† is
analytic in some region R, all its derivatives exist and are continuous in R.
Equating the two cross terms, we obtain
throughout the region R.
@ 2 u
@x 2 ‡ @2 u
@y 2 ˆ 0
247
…6:12a†