Complex Analysis - Maths KU

7.3 Schwarz-Christoffel Transformations 419
Focus on Concepts
11. Use the Schwarz-Christoffel formula (6) to construct a conformal mapping from
the upper half-plane onto the polygonal region shown in gray in Figure 7.31.
Require that f(−1) = πi and f(1) = 0.
12. Use Schwarz-Christoffel formula (6) to construct a conformal mapping from
the upper half-plane onto the polygonal region shown in gray in Figure 7.32.
Require that f(−1) = −ai and f(1) = ai.
πi
v
Figure 7.31 Figure for Problem 11
u
ai
–ai
v
Figure 7.32 Figure for Problem 12
13. Use the Schwarz-Christoffel formula (6) to verify the conformal mapping in
entry M-3 of Appendix III by first constructing the derivative of a mapping of
the upper half-plane onto the polygonal region shown in gray in Figure 7.33.
Require that f(−1) = −a f(0) = v1i, and f(1) = a, and then let v1 →−∞
along the v-axis.
14. Use the Schwarz-Christoffel formula (6) to verify the conformal mapping in
entry M-4 of Appendix III by first constructing the derivative of a mapping of
the upper half-plane onto the polygonal region shown in gray in Figure 7.34.
Require that f(−1) = −u1, f(0) = ai, and f(1) = u1, and then let u1 → 0
along the u-axis.
v
–a a
v 1 i
Figure 7.33 Figure for Problem 13
Computer Lab Assignments
u
–u 1
v
ai
u 1
Figure 7.34 Figure for Problem 14
In Problems 15–18, use a CAS to approximate the images of the points z1 = i and
z2 =1+i under the given function.
15. w = f(z) is the mapping from Problem 3.
16. w = f(z) is the mapping from Problem 6.
17. w = f(z) is the mapping from Problem 8.
18. w = f(z) is the mapping from Problem 9.
u
u