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

 Two graphs of w equals 1 over z are shown. The first graph has a circle graphed on an x y plane. The center of the circle lies on the positive x axis. The y axis is tangential to the circle at its left end. Three points A, B, and C are labeled on the circumference of the circle. The point A lies on the positive x axis, point B in the first quadrant, and point C in the fourth quadrant. The region of the plane inside the circle is shaded. The second graph has a vertical line graphed on a u v plane. It starts from the bottom of the fourth quadrant labeled B prime and goes up to the top labeled C prime through A prime which lies on the positive u axis. The region of the plane to the right of the line is shaded.

E-8

 Two graphs of w equals log subscript e mod(z) + i Arg z, a greater than 1 are shown. The first graph has two curves graphed on an x y plane. The curves are concentric semicircular in form and their centers lie at the origin. The first curve starts on the negative x axis at E, reaches the positive y axis at F, and ends on the positive x axis at a. The second curve starts on the negative x axis at D, reaches the positive y axis at C, and ends on the positive x axis at b. The region of the plane between the two curves is shaded. The second graph has a rectangle graphed on a u v plane. It is vertical and the bottom lies on the positive u axis. The corners are labeled, from bottom left in counter clockwise direction, as ln a, ln b, D prime, and E prime. The center of the side ln b and D prime is labeled C prime. The center of the side ln a and E prime is labeled F prime. The height of the rectangle is pi times i. The region of the plane inside the rectangle is shaded.

E-9

 Two graphs of w equals cos h z are shown. The first graph has three lines graphed on an x y plane. The first line is horizontal and starts from a point on the right end of the positive x axis labeled A and ends at the origin labeled B. The second line is vertical and starts from the origin and ends at a point labeled C on the positive y axis at y equals pi times i. The third line is horizontal and starts from the point C and ends at the right of the first quadrant labeled D. The region of the plane between the three lines is shaded. The second graph has a horizontal line graphed on a u v plane. The line lies on the u axis and starts from the left end labeled D prime, goes through C prime at x equals negative 1 and B prime at x equals 1, and ends at the right end labeled A prime. The region of the plane above the line is shaded.

Mappings to Half-Planes

H-1

 Two graphs of w equals i times ((1 minus z) over (1 + z)) are shown. The first graph has a circle A B C D graphed on an x y plane. The center of the circle lies at the origin. The radius of the circle is 1. The region of the plane inside the circle is shaded. The second graph has a horizontal line graphed on a u v plane. The line lies on the u axis and starts from the left end, goes through D prime at x equals negative 1, A prime, and B prime at x equals 1, and ends at the right end. The region of the plane above the line is shaded.

H-2

 Two graphs of w equals e^(pi times z over a) are shown. The first graph has two lines graphed on an x y plane. The first line is horizontal and lies on the x axis. It starts from the left end of the x axis labeled D, goes through the origin labeled E, and ends at the right end labeled F. The second line is horizontal and lies above the first line. It starts from the left end of the second quadrant labeled C, goes through the point (0, a i) labeled B, and ends at the right end of the first quadrant labeled A. The region of the plane between the lines is shaded and labeled width equals a. The second graph has a horizontal line graphed on a u v plane. The line lies on the u axis and starts from the left end labeled A prime, goes through B prime at x equals negative 1, C prime, D prime, and E prime at x equals 1, and ends at the right end labeled F prime. The region of the plane above the line is shaded.

H-3

 Two graphs of w equals (a over 2) (z + (1 over z)) are shown. The first graph has a function of three parts graphed on an x y plane. The first part is a horizontal line and lies on the negative x axis. It starts from the left end labeled A, and ends at x equals negative 1. The second part is a semicircle of radius 1 that starts at (negative 1, 0), goes up and to the right to (0, 1) labeled B, goes down and to the right, and ends at (1, 0). The third part is a horizontal line and lies on the positive x axis. It starts from (1, 0), and ends at the right end labeled C. The region of the plane above the lines and semicircle is shaded. The second graph has a horizontal line graphed on a u v plane. The line lies on the u axis and starts from the left end labeled A prime, goes through x equals negative a, the origin labeled B prime, x equals a, and ends at the right end labeled C prime. The region of the plane above the line is shaded.

H-4

 Two graphs of w equals cos (pi times z over a) are shown. The first graph has three lines graphed on an x y plane. The first line is vertical and lies on the positive y axis. It starts from the top of the positive y axis labeled A and ends at the origin labeled B. The second line is horizontal and lies on the positive x axis. It starts from the origin and ends at a point labeled C on the positive x axis at x equals a. The third line is vertical and starts from the point C and ends at the top of the first quadrant labeled D. The region of the plane covered inside the three lines is shaded and labeled width equals a. The second graph has a horizontal line graphed on a u v plane. The line lies on the u axis and starts from the left end labeled D prime, goes through C prime at x equals negative 1, B prime at x equals 1, and ends at the right end labeled A prime. The region of the plane above the line is shaded.

H-5

 Two graphs of w equals ((1 + z) over (1 minus z))^2 are shown. The first graph has a function of two parts graphed on an x y plane. The first part is a semicircle of radius 1 that starts at (1, 0) labeled A, goes up and to the left to (0, 1) labeled B, goes down and to the left, and ends at (negative 1, 0) labeled C. The second part is a horizontal line and lies on the x axis. It starts at C, goes through the origin labeled D, and ends at A. The region of the plane between the line and semicircle is shaded. The second graph has a horizontal line graphed on a u v plane. The line lies on the u axis and starts from the left end labeled A prime, goes through the point labeled B prime at x equals negative 1, the origin labeled C prime, the point labeled D prime at x equals 1, and ends at the right end of the positive x axis. The region of the plane above the line is shaded.

H-6

 Two graphs of w equals (e-superscript(pi over z) + e-superscript(negative pi over z)) over (e-superscript(pi over z) minus e-superscript(negative pi over z)) are shown. The first graph has a circle and a line graphed on an x y plane. The circle B C D has it center on the positive y axis. The x axis is tangential to the circle at its bottom. The second part is a horizontal line and lies on the x axis. It starts at the left end labeled A, goes through the origin, and ends at the right end labeled E. The region of the plane above the line excluding the circle is shaded. The second graph has a horizontal line graphed on a u v plane. The line lies on the u axis and starts from the left end labeled A prime, goes through the point labeled B prime at x equals negative 1, the origin labeled C prime, the point labeled D prime at x equals 1, and ends at the right end of the positive x axis labeled E prime. The region of the plane above the line is shaded.

Mappings to Circular Regions

C-1

 Two graphs of w equals (z minus a) over (a z minus 1) are shown. The first graph has two circles graphed on an x y plane. The first circle labeled A has it center at the origin. The radius of the circle is 1. The second circle labeled B has it center on the positive x axis. The diameter labeled b c lies on the positive x axis. The entire region of the plane excluding the circles is shaded. The graph is captioned a equals (b c + 1 + sqrt((b^2 minus 1)(c^2 minus 1))) over (b + c). The second graph has two concentric circles graphed on a u v plane. The circles are centered at the origin. The radius of the first circle labeled B prime is r subscript 0. The radius of the second circle labeled A prime is 1. The region of the plane between the circles is shaded. The graph is captioned r subscript 0 equals (b c minus 1 minus sqrt((b^2 minus 1)(c^2 minus 1))) over (c minus b).

C-2

 Two graphs of w equals (z minus a) over (a z minus 1) are shown. The first graph has two concentric circles graphed on an x y plane. The circles are centered at the origin. The radius of the first circle labeled A is 1. The diameter of the second circle labeled B is b c that lies on the x axis. The region of the plane between the circles is shaded. The graph is captioned a equals (1 + b c + sqrt((1 minus b^2)(1 minus c^2))) over (c + b). The second graph has two concentric circles graphed on a u v plane. The radius of the first circle labeled B prime is r subscript 0. The radius of the second circle labeled A prime is 1. The region of the plane between the circles is shaded. The graph is captioned r subscript 0 equals (1 minus b c + sqrt((1 minus b^2)(1 minus c^2))) over (c minus b).

C-3

 Two graphs of w equals e^z are shown. The first graph has three lines graphed on an x y plane. The first line is horizontal and starts from a point on the left end of the second quadrant labeled A and ends at a point (0, pi times i) labeled B. The second line is vertical and starts from B, goes through E, and ends at the origin labeled D. The third line is horizontal and starts from D and ends at the left end labeled C on the negative x axis. The region of the plane between the three lines is shaded. The second graph has a function of two parts graphed on a u v plane. The first part is a horizontal line and lies on the u axis. It starts at a point on the negative u axis labeled B prime, goes through the origin labeled A prime and C prime, and ends at D prime on the positive u axis. The second part is a semicircle of radius 1 that starts at B prime, goes up and to the right to (0, 1) labeled E prime, goes down and to the right, and ends at D prime. The region of the plane inside the semicircle is shaded.

C-4

 Two graphs of w equals (i minus z) over (i + z) are shown. The first graph has a horizontal line graphed on an x y plane. The line lies on the x axis and starts from the left end labeled A, goes through the point labeled B at x equals negative 1, the point labeled C at x equals 1, and ends at the right end of the positive x axis labeled D. The region of the plane above the line is shaded. The second graph has a circle A prime B prime C prime D prime graphed on a u v plane. The center of the circle lies at the origin. The radius of the circle is 1. The region of the plane inside the circle is shaded.

C-5

 Two graphs of w equals i times (z^2 + 2iz + 1) over (z^2 minus 2iz + 1) are shown. The first graph has a function of two parts graphed on an x y plane. The first part is a semicircle of radius 1 that starts at B on the negative x axis, goes up and to the right to (0, 1) labeled A, goes down and to the right, and ends at D on the positive x axis. The second part is a horizontal line and lies on the u axis. It starts at B, goes through the origin C, and ends at D. The region of the plane between the semicircle and line is shaded. The second graph has a circle A prime B prime C prime D prime graphed on a u v plane. The center of the circle lies at the origin. The radius of the circle is 1. The region of the plane inside the circle is shaded.

Miscellaneous Mappings

M-1

 Two graphs of w equals z + e^z + 1 are shown. The first graph has two lines graphed on an x y plane. The first line is horizontal and lies above the x axis. It starts from the left end labeled A of the second quadrant, goes through the point labeled B on the positive y axis, and ends at the right end labeled C of the first quadrant. The line is labeled y equals pi. The second line is horizontal and lies below the x axis. It starts from the left end labeled D of the third quadrant, goes through the point labeled E on the negative y axis, and ends at the right end labeled F of the fourth quadrant. The line is labeled y equals negative pi. The region of the plane between the lines is shaded. The second graph has two lines graphed on a u v plane. The first line is horizontal and lies above the u axis. It starts from the left end labeled A prime and C prime of the second quadrant, and ends at the point labeled B prime on the positive v axis at v equals pi times i. The second line is horizontal and lies below the u axis. It starts from the left end labeled F prime and D prime of the third quadrant, and ends at the point labeled E prime on the negative v axis at v equals negative pi times i. The entire region of the plane is shaded.

M-2

 Two graphs of w equals (a over pi)[((z^2 minus 1)^(1 over 2)) + (cos h^negative 1 z)] are shown. The first graph has a horizontal line graphed on an x y plane. The line lies on the x axis and starts from the left end labeled A, goes through the point labeled B at x equals negative 1, the point labeled C at x equals 1, and ends at the right end of the positive x axis labeled D. The region of the plane above the line is shaded. The second graph has two lines graphed on a u v plane. The first line is horizontal and lies above the u axis. It starts from the left end labeled A prime of the second quadrant, and ends at the point labeled B prime on the positive v axis at v equals a times i. The second line is horizontal and lies on the u axis. It starts from the left end of the negative u axis, goes through the origin labeled C prime, and ends at the point labeled D prime on the right end of the positive u axis. The region of the plane above the line A prime B prime and above the segment C prime D prime is shaded.

M-3

 Two graphs of w equals (2a over pi)[((z^2 minus 1)^(1 over 2)) + (sin^negative 1 (1 over z))] are shown. The first graph has a horizontal line graphed on an x y plane. The line lies on the x axis and starts from the left end labeled A, goes through the point labeled B at x equals negative 1, the point labeled C at x equals 1, and ends at the right end of the positive x axis labeled D. The region of the plane above the line is shaded. The second graph has four lines graphed on a u v plane. The first line is horizontal and lies on the negative u axis. It starts from the left end labeled A prime, and ends at the point labeled B prime at u equals negative a. The second line is horizontal and lies on the positive u axis. It starts from the point labeled C prime at u equals a, and ends at the point labeled D prime on the right end of the positive u axis. The third line is vertical and starts from B prime, goes down, and ends at the bottom of the third quadrant. The fourth line is vertical and starts from C prime, goes down, and ends at the bottom of the fourth quadrant. The region of the plane above the u axis and between the two vertical lines is shaded.

M-4

 Two graphs of w equals a(z^2 minus 1)^(1 over 2) are shown. The first graph has a horizontal line graphed on an x y plane. The line lies on the x axis and starts from the left end labeled A, goes through the point labeled B at x equals negative 1, the origin labeled C, the point labeled D at x equals 1, and ends at the right end of the positive x axis labeled E. The region of the plane above the line is shaded. The second graph has two lines graphed on a u v plane. The first line is horizontal and lies on the u axis. It starts from the left end labeled A prime, goes through the origin labeled B prime and D prime, and ends at the point labeled E prime on the right end of the positive u axis. The second line is vertical and starts from the origin, goes up, and ends at the point labeled C prime on the positive v axis at v equals a times i. The region of the plane above the line A prime B prime D prime E prime is shaded.

M-5

 Two graphs of w equals 2 zeta + Ln ((zeta minus 1) over (zeta + 1)) and zeta equals (z + 1)^(1 over 2) are shown. The first graph has a horizontal line graphed on an x y plane. The line lies on the x axis and starts from the left end labeled A, goes through the point labeled B at x equals negative 1, the origin labeled C and D, and ends at the point labeled E on the right end of the positive x axis. The region of the plane above the line is shaded. The second graph has three lines graphed on a u v plane. The first line is vertical and starts at the point labeled A prime on the positive v axis, goes down, and ends at the point labeled B prime at v equals pi times i. The second line is horizontal and lies above the u axis. It starts from B prime, goes to the left, and ends at the left end labeled C prime of the second quadrant. The third line is horizontal and lies on the u axis. It starts from the left end of the negative u axis labeled D prime, goes through the origin, and ends at the point labeled E prime on the right end of the positive u axis. The region of the plane that is above the line D prime E prime, below the line C prime B prime, and to the right of the line A prime B prime is shaded.

M-6

 Two graphs of w equals i Ln ((1 + i zeta) over (1 minus i zeta)) + Ln ((1 + zeta) over (1 minus zeta)) and zeta equals ((z minus 1) over (z + 1))^(1 over 2) are shown. The first graph has a horizontal line graphed on an x y plane. The line lies on the x axis and starts from the left end labeled A, goes through the point labeled B at x equals negative 1, the origin labeled C and D, the point labeled E at x equals 1, and ends at the point labeled F on the right end of the positive x axis. The region of the plane above the line is shaded. The second graph has four lines graphed on a u v plane. The first line is horizontal and starts at the point labeled A prime on the right end of the first quadrant, goes to the left through the positive v axis at v equals pi times i, and ends at the point (negative pi, pi times i) labeled B prime in the second quadrant. The second line is vertical and starts from B prime, goes down through (negative pi, 0), and ends at the point labeled C prime at the bottom of the third quadrant. The third line is vertical and lies on the negative v axis. It starts from the bottom of the negative v axis labeled D prime, goes up, and ends at the origin labeled E prime. The fourth line is horizontal and lies on the positive u axis. It starts from E prime, and ends at the point labeled F prime on the right end of the positive u axis. The region of the plane covered inside the four lines is shaded.

M-7

 Two graphs of w equals z + Ln z + 1 are shown. The first graph has a horizontal line graphed on an x y plane. The line lies on the x axis and starts from the left end labeled A, goes through the point labeled B at x equals negative 1, the origin labeled C and D, the point labeled E at x equals 1, and ends at the point labeled F on the right end of the positive x axis. The region of the plane above the line is shaded. The second graph has two lines graphed on a u v plane. The first line is horizontal and lies above the u axis. It starts from the left end labeled A prime of the second quadrant, and ends at the point labeled B prime on the positive v axis at v equals pi times i. The second line is horizontal and lies on the u axis. It starts from the left end of the negative u axis labeled C prime and D prime, goes through the origin, the point E prime, and ends at the point labeled F prime on the right end of the positive u axis. The region of the plane above the line C prime D prime E prime F prime is shaded.

M-8

 Two graphs of w equals (e^z + 1) over (e^z minus 1) are shown. The first graph has three lines graphed on an x y plane. The first line is horizontal and lies above the positive x axis. It starts from the right end labeled A of the first quadrant, and ends at the point labeled B and C on the positive y axis. The line is labeled y equals pi. The second line is vertical and lies on the y axis. It starts from B, goes down through the origin labeled D and E, and ends at the point labeled F and G on the negative y axis. The third line is horizontal and lies below the positive x axis. It starts from G, and ends at the right end labeled H of the fourth quadrant. The line is labeled y equals negative pi. The region of the plane between the lines is shaded. The second graph has two lines graphed on an x y plane. The first line is horizontal and lies on the positive u axis. It starts from the point labeled A prime and H prime on the positive u axis at u equals 1, and ends at the origin labeled B prime and G prime. The second line is vertical and lies on the v axis. It starts from the point labeled D prime on the negative v axis, goes up through the origin labeled C prime and F prime, and ends at the point labeled E prime on the positive v axis. The region of the plane to the right of the line D prime C prime F prime E prime is shaded.

M-9

 Two graphs of w equals (pi times i) minus (1 over 2)[Ln(z + 1) + Ln(z minus 1)] are shown. The first graph has a horizontal line graphed on an x y plane. The line lies on the x axis and starts from the left end labeled A, goes through the point labeled B, the point (negative 1, 0), the point C, the origin labeled D, the point labeled E, the point (1, 0), the point F, and ends at the point labeled G on the right end of the positive x axis. The region of the plane above the line is shaded. The second graph has three lines graphed on a u v plane. The first line is horizontal and lies on the u axis. It starts from the left end labeled A prime, goes through the origin, and ends at the right end labeled B prime on the positive u axis. The second line is horizontal and lies above the positive u axis. It starts from the right end labeled C prime and E prime of the first quadrant, and ends at the point labeled D prime on the positive v axis at v equals pi times i over 2. The third line is horizontal and lies above the u axis. It starts from the right end labeled F prime of the first quadrant, goes through the positive v axis at v equals pi times i, and ends at the left end labeled G prime of the second quadrant. The region of the plane between the lines A prime B prime and F prime G prime is shaded.

M-10

 Two graphs of w equals (1 minus i) (z minus i) over (z minus 1) are shown. The first graph has two circles graphed on an x y plane. The first circle A B C has its center at the origin. The radius of the circle is 1. The second circle D E F has its center on the positive x axis at (a, 0). The two circles intersect at (1, 0) on the positive x axis. The graph is captioned 0 less than a less than 1. The region of the plane between the two circles is shaded. The second graph has two lines graphed on a u v plane. The first line is horizontal and lies on the u axis. It starts from the left end, goes through the point labeled A prime, the origin, the point labeled B prime, and ends at the right end labeled C prime on the positive u axis. The second line is horizontal and lies above the positive u axis. It starts from the right end, goes through the point labeled D prime, the point (0, a over (1 minus a)), the point E prime, and ends at the point labeled F prime at the right end of the first quadrant. The region of the plane between the two lines is shaded.