Handbook of Conformal Mappings and Applications

Content: Cover --
Half Title --
Title Page --
Copyright Page --
Table of Contents --
Preface --
Notations, Definitions, and Acronyms --
Part 1: Theory and Conformal Maps --
1: Introduction --
1.1 Historical Background --
1.2 Modern Developments --
1.3 In Retrospect --
2: Conformal Mapping --
2.1 Definitions --
2.1.1 Analytic Functions --
2.1.2 Integration --
2.1.3 Fatou\'s Lemma --
2.2 Jordan Contour --
2.2.1 Hölder Condition --
2.3 Metric Spaces --
2.4 Basic Theorems --
2.4.1 Singularities --
2.4.2 Residues --
2.4.3 Boundary Values for Cauchy Integral --
2.4.4 Argument Principle --
2.4.5 Plemelj Formulas --
2.5 Harmonic Functions --
2.5.1 Harmonic Conjugate --
2.5.2 Capacity --
2.6 Univalent Functions --
2.6.1 Conformality and Uniqueness --
2.6.2 Conformal and Isogonal Mappings --
2.6.3 Conformal Mapping of an Area Element --
2.6.4 Analytic Continuation --
2.6.5 Chain Property --
Map 2.1. --
Map 2.2. --
Map 2.3. --
Map 2.4. --
Map 2.5. --
Map 2.6. --
Map 2.7. --
Map 2.8. --
Map 2.9. --
2.6.6 Schwarz Reflection Principle --
2.6.7 Conformal Equivalence --
2.6.8 Riemann Sphere --
2.6.9 Bieberbach Conjecture --
2.6.10 Mercator\'s Projection --
2.7 Taylor Series Approximations --
2.7.1 Interior of the Unit Circle --
Map 2.10. --
3: Linear and Bilinear Transformations --
3.1 Definitions of Certain Curves --
3.1.1 Line --
3.1.2 Circle --
3.1.3 Ellipse --
3.1.4 Hyperbola --
3.1.5 Rectangular Hyperbola --
3.1.6 Parabola --
3.1.7 Cassini\'s Ovals and Lemniscate --
3.1.8 Cardioid and Limaçons --
3.2 Bilinear Transformations --
3.2.1 Fixed Points --
3.2.2 Linear Transformation --
3.2.3 Composition of Bilinear Transformations --
Map 3.1. Involutory Transformation --
Map 3.2. Three Points onto Three Points --
Map 3.3. Sequence of Bilinear Transformations --
3.3 Cross-Ratio --
3.3.1 Symmetric Points --
3.3.2 Symmetry Principle --
3.3.3 Special Cases --
Map 3.4. Map 3.5. --
Map 3.6. --
Map 3.7. --
Map 3.8. --
Map 3.9. --
Map 3.10. --
Map 3.11. --
Map 3.12. --
Map 3.13. --
Map 3.14. --
Map 3.15. --
Map 3.16. --
Map 3.17. --
Map 3.18. --
Map 3.19. --
Map 3.20. --
Map 3.21. --
Map 3.22. --
Map 3.23. --
Map 3.24. --
Map 3.25. --
Map 3.26. --
Map 3.27. --
Map 3.28. --
Map 3.29. --
Map 3.30. --
Map 3.31. --
Map 3.32. --
Map 3.33. --
Map 3.34. --
Map 3.35. --
Map 3.36. --
Map 3.37. --
Map 3.38. --
Map 3.39. --
Map 3.40. --
Map 3.41. --
Map 3.42. --
Map 3.43. --
Map 3.44. --
Map 3.45. --
Map 3.46. --
Map 3.47. --
Map 3.48. --
Map 3.49. --
Map 3.50(a)-(d). Cassini\'s ovals --
Map 3.51. Cardioid and Limaçon --
Map 3.52. Cardioid and Generalized Cardioids --
3.4 Straight Lines and Circles --
Map 3.53. Lines parallel to the axes --
Map 3.54. Other lines and circles --
Map 3.55. Circle onto another circle --
Map 3.56(a)-(e). Three points onto three points --
Map 3.57. Straight line onto straight line --
Map 3.58. Angle onto itself, with arms interchanged --
Map 3.59. Straight line onto circle --
Map 3.60. Circle onto straight line --
Map 3.61. Circle and line in contact onto two parallel lines --
Map 3.62. Two circles in contact onto two parallel lines (inner contact) --
Map 3.63. Two circles in outer contact onto two parallel lines (outer contact) --
Map 3.64. Two \'circles\' intersecting at two points onto two intersecting lines --
Map 3.65. --
Map 3.66. Disjoint circles and straight line onto two concentric circles --
Map 3.67. Two circles, without common point, onto two concentric circles --
Map 3.68. --
Map 3.69. --
3.5 Ellipses and Hyperbolas --
Map 3.70. Ellipses --
Map 3.71. --
Map 3.72. Hyperbolas --
4: Algebraic Functions --
4.1 Polynomials --
Map 4.1. --
Map 4.2. --
Map 4.3. --
Map 4.4. --
Map 4.5. --
Map 4.6. --
Map 4.7. --
Map 4.8. --
Map 4.9. --
Map 4.10. --
Map 4.11. --
Map 4.12. --
Map 4.13. --
Map 4.14. Map 4.15. --
Map 4.16. --
Map 4.17. --
Map 4.18. --
Map 4.19. --
Map 4.20. --
Map 4.21. --
Map 4.22. --
Map 4.23. --
Map 4.24. --
Map 4.25. --
Map 4.26. --
Map 4.27. --
Map 4.28. --
Map 4.29. --
Map 4.30. --
Map 4.31. --
Map 4.32. --
Map 4.33. --
Map 4.34. --
Map 4.35. --
Map 4.36. --
Map 4.37. --
Map 4.38. --
Map 4.39. --
Map 4.40. --
Map 4.41. --
Map 4.42. --
Map 4.43. --
Map 4.44. --
Map 4.45a. --
Map 4.45b. --
Map 4.45c. --
Map 4.46. --
Map 4.47. --
Map 4.48. --
Map 4.49. --
Map 4.50. --
Map 4.51a. --
Map 4.51b. --
Map 4.51c. --
Map 4.51d. --
Map 4.52. --
Map 4.53. --
Map 4.54. --
4.1.1 Regions Bounded by Two Circular Arcs --
Map 4.55. --
Map 4.56. --
Map 4.57. --
Map 4.58. --
5: Exponential Family of Functions --
5.1 Exponentials --
5.1.1 de Moivre\'s Theorem --
Map 5.1. --
Map 5.2. --
5.2 Specific Cases of Mappings --
Map 5.3. --
Map 5.4. --
Map 5.5. --
Map 5.6. --
Map 5.7. --
Map 5.8. --
Map 5.9. --
Map 5.10. --
Map 5.11. --
Map 5.12. --
Map 5.13. --
Map 5.14. --
Map 5.15. --
Map 5.16. --
Map 5.17. --
Map 5.18. --
Map 5.19. --
Map 5.20a. --
Map 5.20b. --
Map 5.20c. --
Map 5.21. --
Map 5.22. --
Map 5.23. --
Map 5.24. --
Map 5.25. --
Map 5.26. --
5.3 Other Related Functions --
Map 5.27. --
Map 5.28. --
Map 5.29. --
Map 5.30. --
Map 5.31. --
Map 5.32. --
Map 5.33. --
Map 5.34. --
Map 5.35a. --
Map 5.35b. --
Map 5.36. --
Map 5.37. --
Map 5.38. --
Map 5.39. --
Map 5.40. --
Map 5.41. --
Map 5.42. --
Map 5.43. --
Map 5.44. --
Map 5.45. --
Map 5.46. --
Map 5.47. --
Map 5.48. --
Map 5.49. --
Map 5.50. --
Map 5.51. --
Map 5.52. --
Map 5.53. --
Map 5.54. --
Map 5.55. --
Map 5.56(a)-(f) --
Map 5.57. --
Map 5.58. --
Map 5.59. --
Map 5.60. --
Map 5.61. --
Map 5.62. --
Map 5.63. --
Map 5.64. --
Map 5.65. --
Map 5.66. --
Map 5.67. --
Map 5.68. --
Map 5.69. --
Map 5.70. --
Map 5.71. --
Map 5.72. --
Map 5.73. --
Map 5.74. --
Map 5.75. --
Map 5.76. --
Map 5.77. Map 5.78. --
5.4 Complex Exponential Function --
6: Joukowski Airfoils --
6.1 Joukowski Maps --
Map 6.1. --
Map 6.2. --
Map 6.3. --
Map 6.4(a)-(e). --
Map 6.5. --
Map 6.6. --
Map 6.7. --
Map 6.8. --
Map 6.9. --
Map 6.10. --
Map 6.11. --
7: Schwarz-Christoffel Transformation --
7.1 Schwarz-Christoffel Transformations --
Map 7.1. Upper half-plane or unit circle onto a polygon --
7.1.1 Triangles --
Map 7.2. Triangles onto the upper half-plane --
7.1.2 Size of the Angles --
Map 7.3. Polygon onto the upper half-plane --
Map 7.4. Unit disk onto a polygon --
7.1.3 Transition between Two Parallel Lines in the z-plane --
7.2 Specific Transformations --
Map 7.5. Upper half-plane onto a triangle --
Map 7.6. Trigonal region --
Map 7.7. Upper half-plane onto a rectangle --
Map 7.8. Trapezoid onto the upper half-plane --
Map 7.9. Upper half-plane onto a square --
Map 7.10. Upper half-plane onto a quadrilateral --
Map 7.11. Horizontal parallel strip with a horizontal cut onto upper half-plane --
Map 7.12. Upper half-plane onto two parallel lines --
Map 7.13. Upper half-plane onto the semi-infite strip --
Map 7.14. Semi-infinite Strip onto upper half-plane --
Map 7.15. Trapezoid onto the upper half-plane --
Map 7.16. Upper half-plane onto the fourth quadrant plus the strip 0 <
 v <
 1 --
Map 7.17. Exterior of a thin straight slit onto the exterior of the unit circle --
7.3 Regions Exterior to and between Rectangles --
Map 7.18. Exterior of a rectangle onto the upper half-plane --
Map 7.19. Region between two rectangles onto the upper half-plane --
Maps 7.20a. Two concentric squares onto a parallel strip --
Maps 7.20b. Inner square diamond onto upper half strip --
7.4 Polygons with Round Corners --
Map 7.21. Polygon with round corner onto the upper half-plane --
Map 7.22. Curved boundary line onto the upper half-plane. Map 7.23. Two curved boundaries onto the upper half-plane --
Map 7.24. Four curved boundaries onto the upper half-plane --
7.5 Weierstrass Integral Equation --
Map 7.25. Square onto the upper half-plane or onto a circle --
Map 7.26. Interior of a rectangle onto the upper half-plane --
7.6 Rectangles and Other Regions --
Map 7.27. Square onto the lower half-plane --
Map 7.28. Rectangle onto a half-plane or onto a quarter plane --
Map 7.29. Square onto a quarter plane --
Map 7.30. Four sub-rectangles onto the right half of a Cassini\'s oval --
Map 7.31. Rectangle in Map 7.26 --
Map 7.32. Ellipse onto the unit circle --
Map 7.33. Exterior of a rectangle onto a quarter plane --
Map 7.34. Region exterior to two semi-infinite strip onto the lower half-plane --
Map 7.35. Region interior to a semi-infinite strip onto the upper half-plane --
Map 7.36. Mapping similar to Map 7.31 --
Map 7.37. Diamond-shaped square on the unit disk --
Map 7.38. Triangle onto the half-plane --
Map 7.39. Polygon (n ≥ 3) onto the unit circle --
Map 7.40. Equilateral triangle onto the half-plane --
Map 7.41. Two triangles onto the half-planes --
Map 7.42. Triangle with slits --
Map 7.43. Triangle with slits --
Map 7.44. Interior of a rectangle onto exterior of another rectangle --
Map 7.45. Interior of a rectangle onto exterior of two semi-infinite strips --
Map 7.46. Interior of a rectangle onto exterior of two semi-infinite strips --
Map 7.47. Rectangle onto a cut plane --
Map 7.48. Curvilinear triangle onto two intersecting circles --
Map 7.49. Two intersecting unit circles onto the whole plane --
Map 7.50. Region over upper-half of the unit circle onto the whole plane --
Map 7.51. Equilateral equiangular circular triangle onto a circle --
Map 7.52. Interior of a circular triangle onto the half-plane.