May 2024
Intermediate to advanced
481 pages
17h 15m
English
Content preview from Complex Analysis
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“ca: ch_08” — 2012/3/27 — 21:13 — page 399 — #53
Bilinear and Conformal Transformations
|
399
If C
1
and C
2
are two smooth curves passing through the point z
0
and
1
and
2
are their image curves,
respectively, then from equation (8.45), we obtain
φ
2
− φ
1
= k
(
θ
2
− θ
1
)
Thus, the angle from
1
to
2
is k times the angle from C
1
to C
2
in magnitude, though the orientation
is preserved.
Note: The transformation is conformal at z
0
when k = 1 and z
0
is a critical point when k ≥ 2.
8.11.3 Scale Factors
The scale factor of the conformal transformation w = f
(
z
)
at a point z = z
0
is obtained by taking modulus
of f
(z
0
). By definition of derivatives, we have
f
(z
0
)
=
lim
z→z
0
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