APPENDIX D

Conformal Mappings

Elementary Mappings

E-1

 Two graphs of w equals z + z subscript 0 are shown. The first graph has a square graphed on an x y plane. The center of the square lies at the origin (0, 0). The second graph has a square graphed on a u v plane. The square lies in the first quadrant. The center of the square is marked with a dot and labeled z subscript 0.

E-2

 Two graphs of w equals e^(i times theta) times z are shown. The first graph has a square graphed on an x y plane. The bottom left vertex of the square lies at the origin (0, 0). The second graph has a square graphed on a u v plane. The diagonal of the square lies along the positive v axis from the origin. The angle made by the square at the origin to the positive u axis is labeled theta.

E-3

 Two graphs of w equals alpha times z, alpha greater than 0 are shown. The first graph has a square graphed on an x y plane. The center of the square lies at the origin (0, 0). The second graph has a square graphed on a u v plane. The center of the square lies at the origin (0, 0). The size of the square is smaller than the square in the first graph.

E-4

 Two graphs of w equals z^alpha, alpha greater than 0 are shown. The first graph has two lines graphed on an x y plane. The first line starts at the origin labeled B, goes up and to the right, and ends at the top right of the first quadrant. A point C is marked on this line. The second line is horizontal. It starts from the origin, goes to the right, and ends at the right on the positive x axis. A point A is marked on this line. The region of the plane between the two lines is shaded. The angle between the two lines at point B is labeled theta subscript 0. The second graph has two lines graphed on a u v plane. The first line starts at the origin labeled B prime, goes up and to the left, and ends at the top right of the second quadrant. A point C prime is marked on this line. The second line is horizontal. It starts from the origin, goes to the right, and ends at the right on the positive u axis. A point A rpime is marked on this line. The region of the plane between the two lines is shaded. The angle between the two lines at point B prime is labeled alpha times theta subscript 0.

E-5

 Two graphs of w equals e^z and z equals Ln w are shown. The first graph has two horizontal lines graphed on an x y plane. The first line lies on the x axis from the left end labeled C to the right end labeled D. The second line passes through y equals pi times i. It starts from the left end of the second quadrant labeled B to the right end labeled A. The region of the plane between the two lines is shaded. The second graph has a horizontal line graphed on a u v plane. It lies on the u axis and starts from the left end labeled A prime to the right end labeled D prime through B prime, origin, and C prime. The region of the plane above the line is shaded.

E-6

 Two graphs of w equals sin z and z equals sin^1 w are shown. The first graph has two vertical lines graphed on an x y plane. The first line passes through the point labeled A at x equals negative pi over 2. It starts from the bottom of the third quadrant labeled C to the top of the second quadrant labeled B. The second line passes through the point labeled D at x equals pi over 2. It starts from the bottom of the fourth quadrant labeled F to the top of the first quadrant labeled E. The region of the plane between the two lines is shaded. The second graph has two horizontal lines graphed on a u v plane. The first line lies on the negative u axis and starts from the left end labeled B prime and C prime and ends at the point labeled A prime at x equals negative 1. The second line lies on the positive u axis and starts from the point labeled D prime at x equals 1 and ends at the right end labeled E prime and F prime. The entire region of the plane is shaded.

E-7

E-8

E-9

Mappings to Half-Planes

H-1

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