Complex Analysis - Maths KU

410 Chapter 7 Conformal Mappings
y
(a) A bounded polygonal region
y
(b) An unbounded polygonal region
Figure 7.19 Polygonal regions
(b) If T (1) = 1, then what, if anything, can be said about the coefficients a, b,
c, and d?
(c) If T (∞) =∞, then what, if anything, can be said about the coefficients a,
b, c, and d?
30. Use Problem 29 to show that if T is a linear fractional transformation and
T (0) = 0, T (1) = 1, and T (∞) =∞, then T must be the identity function.
That is, T (z) =z.
31. Use Theorem 7.4 to derive the mapping in entry H-1 in Appendix III.
32. Use Theorem 7.4 to derive the mapping in entry H-3 in Appendix III.
7.3 Schwarz-Christoffel Transformations
7.3
One problem that arises frequentlyin the studyof fluid flow is that of constructing the
flow of an ideal fluid that remains inside a polygonal domain D ′ . We will see in Section 7.5
that this problem can be solved byfinding a one-to-one complex mapping of the half-plane
y ≥ 0 onto the polygonal region that is a conformal mapping in the domain y>0. The
existence of such a mapping is guaranteed bythe Riemann mapping theorem discussed in the
Remarks at the end of Section 7.1. However, even though the Riemann mapping theorem
does assert the existence of a mapping, it gives no practical means of finding a formula for
the mapping. In this section we present the Schwarz-Christoffel formula, which provides an
explicit formula for the derivative of a conformal mapping from the upper half-plane onto
a polygonal region.
x
x
Polygonal Regions A polygonal region in the complex plane is
a region that is bounded bya simple, connected, piecewise smooth curve
consisting of a finite number of line segments. The boundarycurve of a
polygonal region is called a polygon and the endpoints of the line segments in
the polygon are called vertices of the polygon. If a polygon is a closed curve,
then the region enclosed bythe polygon is called a bounded polygonal
region, and a polygonal region that is not bounded is called an unbounded
polygonal region. See Figure 7.19. In the case of an unbounded polygonal
region, the ideal point ∞ is also called a vertex of the polygon.
Simple examples of polygonal regions include the region bounded by the
triangle with vertices 0, 1, and i, which is an example of a bounded polygonal
region, and the region defined by0 ≤ x ≤ 1, 0 ≤ y