1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

416 CHAPTER 10 • CONFORMAL MAPPING

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10.3.2 The Riemann Surface for w = (z^2 - 1)2

Using the other branch of the square root, we find that w = fz ( z) = -fi ( z) is a

one-to-one conformal mapping from the domain set D2, consisting of the z plane
slit along the segment -1 ::; x ::; 1, y = 0, onto the range set H2, consisting of

thew plane slit along the segment u = 0, -1::; v::; 1. The sets D 1 and H 1 for

fi (z) and Dz and Hz for fz (z) are shown in Figure 10.14.

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We obtain the Riemann surface for w = ( z^2 - 1)^2 by gluing the edges of D 1

and Dz together and the edges of Hi and Hz together. In the domain set, we

glue edges A to a, B to b, C to c, and D to d. In the image set, we glue edges A'

to a', B' to ti, C' to c', and D' to d'. The result is a Riemann domain surface

and Riemann image surface for the mapping, as illustrated in Figures 10.15(a)

and 10.15(b), respectively.

y v

u

y v

u

Figure 10.14 The mappings w = /1 (z) and w = fz (z ).