Mathematical Methods for Physicists: A concise introduction - Site Map

DIFFERENTIAL CALCULUS
A function f …z† is said to be continuous in a region R of the z plane if it is
continuous at all points of R.
Points in the z plane where f …z† fails to be continuous are called discontinuities
of f …z†, and f …z† is said to be discontinuous at these points. If lim z!z0 f …z† exists
but is not equal to f …z 0 †, we call the point z 0 a removable discontinuity, since by
rede®ning f …z 0 † to be the same as lim z!z0 f …z† the function becomes continuous.
To examine the continuity of f …z† at z ˆ1, we let z ˆ 1=w and examine the
continuity of f …1=w† at w ˆ 0.
Derivatives and analytic functions
Given a continuous, single-valued function of a complex variable f …z† in some
region R of the z plane, the derivative f 0 …z†… df=dz† at some ®xed point z 0 in R is
de®ned as
f 0 f …z
…z 0 †ˆ lim 0 ‡ z†f …z 0 †
; …6:10†
z!0 z
provided the limit exists independently of the manner in which z ! 0. Here
z ˆ z z 0 ,andz is any point of some neighborhood of z 0 .Iff 0 …z† exists at z 0
and every point z in some neighborhood of z 0 , then f …z† is said to be analytic at z 0 .
And f …z† is analytic in a region R of the complex z plane if it is analytic at every
point in R.
In order to be analytic, f …z† must be single-valued and continuous. It is
straightforward to see this. In view of Eq. (6.10), whenever f 0 …z 0 † exists, then
that is,
lim f …z f …z
‰ 0 ‡ z†f …z 0 † Š ˆ lim 0 ‡ z†f …z 0 †
lim
z!0 z!0 z
z ˆ 0
z!0
lim f …z† ˆf …z 0†:
z!0
Thus f is necessarily continuous at any point z 0 where its derivative exists. But the
converse is not necessarily true, as the following example shows.
Example 6.7
The function f …z† ˆz* is continuous at z 0 , but dz*=dz does not exist anywhere. By
de®nition,
dz*
dz ˆ lim …z ‡ z†* z* …x ‡ iy ‡ x ‡ iy†* …x ‡ iy†*
ˆ lim
z!0 z
x;y!0
x ‡ iy
ˆ
x iy‡x iy …x iy† x iy
lim
ˆ lim
x;y!0 x ‡ iy
x;y!0 x ‡ iy :
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