SSM-Ch17 - Chap. 17 Conformal Mapping Overview. This...

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Chap . 17 Conformal Mapping Overview . This chapter consists of five sections. In Sec. 17.1 you see that the mapping given by an analytic function f u z U is conformal (angle-preserving in both magnitude and sense of angles between curves; Theorem 1). Exceptions are points at which the derivative is zero. Example: f u z U u z 2 , f U u z U u 2 z u 0 at z u 0, where angles are doubled. Sections 17.2 and 17.3 deal with the mapping by an important class of functions w u u az ± b U / u cz ± d U with constant a , b , c , d . Section 17.4 shows mapping by complex trigonometric and hyperbolic functions. The optional Sec. 17.5 concerns an ingenious idea of making multivalued complex relations single-valued on suitable domains of definition, called Riemann surfaces . To fully understand specific mappings, make graphs or sketches. Sec . 17 . 1 Geometry of Analytic Functions : Conformal Mapping Problem Set 17 . 1 . Page 733 3 . Mapping by w u f ( z ) u _ z . For instance, the segment from u 0, 0 U to any u x , y U u u 0, 0 U makes the angle u u arctan u y / x U with the x -axis. Its image is the segment from u 0, 0 U to u x , U y U , which makes the angle u ² u arctan u U y / x U u U arctan u y / x U u U u . Note that f u z U is not analytic. Make a sketch. 5 . Rotation . The image of z u re i u ( r ³ 0) under the mapping w u iz is w u iz u e i U /2 re i u u re i u ² where u ² u u ± U /2, so that this mapping is indeed a rotation about 0 through an angle U /2 in the positive sense (a counter-clockwise rotation) . In the answer on p. A42 the points on a line x u c are z u x ± iy u c ± iy , so that w u iz u i u c ± iy U u U y ± ic . For instance, z u x u c on the real axis is mapped onto w u ic on the imaginary axis. Similarly, for y u k you have z u x ± ik , hence iz u U k ± ix . 7 . Angular regions , sectors . The double inequality U U /4 ´ Arg z ´ U /4 determines the angular region R between the bisecting lines of the first and fourth quadrants. Since w u z 3 triples angles at the origin, the image of R is the complex plane without the angular region between the bisecting lines of the second and third quadrants; this image of R is an angular region of angle 3 U /2, in which the image of the given region lies, restricted to | w | ´ 1/8 since | z | ´ 1/2. z
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This note was uploaded on 04/08/2012 for the course MT 423 taught by Professor M.lee during the Spring '12 term at National Taiwan University.

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SSM-Ch17 - Chap. 17 Conformal Mapping Overview. This...

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