Unformatted text preview: e upper half plane to the lower half plane
(c) Maps the second quadrant onto the righthalf plane
(d) What happens to the strip {x+iy  0 ≤ y ≤ 1, x ≥ 0} under the map f(z) = iez?
2. A Möbius transformation is a complex function of the form
az + b
f(z) =
.
cz + d
(a) Find a Möbius transformation f with the property that f(1) = 1, f(0) = i, and f(1) =
1.
(b) Prove that your function is the only possible Möbius transformation with this
property. (It is suggested you do some research in the Section 12.9 of the textbook.)
4. Contour Integrals & the CauchyGoursat Theorem
(§18.1–18.4 in the text)
A curve C in the complex plane C is a pair of piecewise continuous functions x = x(t), y
I
= y(t) for a ≤ t ≤ b. (This is just a piecewise continuous curve in 2dimensional space).
Given a curve C in a domain D ¯ C and a function f: DÆC , we can define the
I
I
corresponding contour integral,
Û
Ù f(z) dz
ı
C
as the limit of a Riemann sum of the form
£f(zi*)∆zi
associated with a partition a = t0 < t1 ≤ ... ≤ tn = b, where the limit is taken as max
{∆zi} Æ 0. If we write f(z) as u(x, y) + iv(x, y) and d...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.
 Fall '03
 StefanWaner
 Math, Algebra, Geometry, Complex Numbers

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