Question 7.1.4: Using a Table of Conformal Mappings. Use Appendix III to fin......
Using a Table of Conformal Mappings
Use Appendix III to find a conformal mapping from the infinite horizontal strip 0 ≤ y ≤ 2, −∞ < x < ∞, onto the upper half-plane v ≥ 0. Under this mapping, what is the image of the negative x-axis?
Entry H-2 in Appendix III gives a mapping from an infinite horizontal strip onto the upper half-plane. Setting a = 2, we obtain the desired mapping w = e^{πz/2}. From H-2 we also see that the points labeled D and E = 0 on the negative x-axis in the z-plane are mapped onto the points D’ and E’ = 1 on the positive u-axis in the w-plane. Noting the relative positions of these points, we conclude that the negative x-axis is mapped onto the interval (0, 1] in the u-axis by w = e^{πz/2}. See Figure 7.5. This observation can also be verified using parametrizations.
