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 415 — #69
Bilinear and Conformal Transformations
|
415
8. Prove that the transformation w =
1
π
sin
−1
z +
i
π
sin
−1
1
z
+
1 + i
2
transforms the upper half plane
Im z > 0 onto the right angle channel in the first quadrant which is bounded by the coordinate
axes and the rays x ≥ 1, y = 1 and y ≥ 1, x = 1.
ANSWERS
1. w =
a
π
sin
−1
z
3. w =
2c
π
sin
−1
z
5. w =
i
π
√
z
2
− 1 +
sin
−1
z
π
+
1
2
6. w = A
(
z − z
1
)
α
1
π
−1
(
z − z
2
)
α
2
π
−1
...
(
z − z
n
)
α
n
π
−1
dz+C, where A and C are constants
SUMMARY
K The transformation w = z +b is the translation of the axes but the shape and size remain the same
in this case.
K In case of transformation w = az, any image in
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