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**Unformatted text preview: **Figure 7.16. The Effect of ζ = z 2 on Various Domains. while the horizontal strip S − π,π = braceleftbig − π < Im z < π bracerightbig of width 2 π is mapped onto the domain Ω ∗ = Ω − π,π = braceleftbig − π < ph ζ < π bracerightbig = C \ { Im z = 0 , Re z ≤ } obtained by cutting the complex plane along the negative real axis. On the other hand, vertical lines Re z = a are mapped to circles | ζ | = e a . Thus, a vertical strip a < Re z < b is mapped to an annulus e a < | ζ | < e b , albeit many-to- one, since the strip is effectively wrapped around and around the annulus. The rectangle R = braceleftbig a < x < b, − π < y < π bracerightbig of height 2 π is mapped in a one-to-one fashion on an annulus that has been cut along the negative real axis. See Figure 7.15. Finally, we note that no domain is mapped to the unit disk D = {| ζ | < 1 } (or, indeed, any other domain that contains 0) because the exponential function is never zero: ζ = e z negationslash = 0. Example 7.23. The squaring map ζ = g ( z ) = z 2 , or ξ = x 2 − y 2 , η = 2 xy, (7 . 56) is analytic on all of C , but is not one-to-one. Its inverse is the square root function z = √ ζ , which, as we noted in Section 7.1, is doubly-valued, except at the origin z = 0. Furthermore, its derivative g ′ ( z ) = 2 z vanishes at z = 0, violating the invertibility condition (7.49). However, once we restrict g ( z ) to a simply connected subdomain Ω that does not contain 0, the function g ( z ) = z 2 does define a one-to-one mapping, whose inverse z = g − 1 ( ζ ) = √ ζ is a well-defined, analytic and single-valued branch of the square root...

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