Example 10.1.  Show that the mapping  [Graphics:Images/ConformalMappingMod_gr_54.gif]  is conformal at the points [Graphics:Images/ConformalMappingMod_gr_55.gif], [Graphics:Images/ConformalMappingMod_gr_56.gif], [Graphics:Images/ConformalMappingMod_gr_57.gif] and , [Graphics:Images/ConformalMappingMod_gr_58.gif] and determine the angle of rotation given by [Graphics:Images/ConformalMappingMod_gr_59.gif] at the given points.

          [Graphics:Images/ConformalMappingMod.0_gr_7.gif]          [Graphics:Images/ConformalMappingMod.0_gr_8.gif]

Explore Solution 10.1.

Enter the function f(z) = cos(z).

[Graphics:../Images/ConformalMappingMod_gr_66.gif]



[Graphics:../Images/ConformalMappingMod_gr_67.gif]

(a) Consider the mapping  w = cos(z)  near [Graphics:../Images/ConformalMappingMod_gr_68.gif].

[Graphics:../Images/ConformalMappingMod_gr_69.gif]




[Graphics:../Images/ConformalMappingMod_gr_70.gif]

 

 

 

The image of a small rectangle about  [Graphics:../Images/ConformalMappingMod_gr_71.gif]  is a curvilinear region about [Graphics:../Images/ConformalMappingMod_gr_72.gif].

[Graphics:../Images/ConformalMappingMod_gr_73.gif]




[Graphics:../Images/ConformalMappingMod_gr_75.gif]

[Graphics:../Images/ConformalMappingMod_gr_77.gif]

 

 

 

 

 

(b) Consider the mapping  w = cos(z)  near [Graphics:../Images/ConformalMappingMod_gr_78.gif].

[Graphics:../Images/ConformalMappingMod_gr_79.gif]




[Graphics:../Images/ConformalMappingMod_gr_80.gif]

 

 

 

The image of a small rectangle about  [Graphics:../Images/ConformalMappingMod_gr_81.gif] is a curvilinear region about [Graphics:../Images/ConformalMappingMod_gr_82.gif].  

[Graphics:../Images/ConformalMappingMod_gr_83.gif]




[Graphics:../Images/ConformalMappingMod_gr_85.gif]

[Graphics:../Images/ConformalMappingMod_gr_87.gif]

 

 

 

 

 

(c) Consider the mapping  w = cos(z)  near [Graphics:../Images/ConformalMappingMod_gr_88.gif].

[Graphics:../Images/ConformalMappingMod_gr_89.gif]




[Graphics:../Images/ConformalMappingMod_gr_90.gif]

 

 

 

The image of a small rectangle about  [Graphics:../Images/ConformalMappingMod_gr_91.gif]  is a curvilinear region about  [Graphics:../Images/ConformalMappingMod_gr_92.gif].  

[Graphics:../Images/ConformalMappingMod_gr_93.gif]




[Graphics:../Images/ConformalMappingMod_gr_95.gif]

[Graphics:../Images/ConformalMappingMod_gr_97.gif]

 

 

 

 

 

(d) Consider the mapping  w = cos(z)  near [Graphics:../Images/ConformalMappingMod_gr_98.gif].

[Graphics:../Images/ConformalMappingMod_gr_99.gif]




[Graphics:../Images/ConformalMappingMod_gr_100.gif]

 

 

 

The image of a small rectangle about  z = 1- i  is a curvilinear region about [Graphics:../Images/ConformalMappingMod_gr_101.gif].  

[Graphics:../Images/ConformalMappingMod_gr_102.gif]




[Graphics:../Images/ConformalMappingMod_gr_104.gif]

[Graphics:../Images/ConformalMappingMod_gr_106.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell