Integration with variables Notes_Part_10

Integration with variables Notes_Part_10 - Figure 7.16. The...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 02/10/2012 for the course MATH 5587 taught by Professor Olver during the Fall '10 term at University of Central Florida.

Page1 / 3

Integration with variables Notes_Part_10 - Figure 7.16. The...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online