Handbook of Conformal Mappings and Applications 1st Edition by Prem Kythe 1138748471 9781138748477
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ISBN 10: 1138748471
ISBN 13: 9781138748477
Author: Prem K. Kythe
The subject of conformal mappings is a major part of geometric function theory that gained prominence after the publication of the Riemann mapping theorem — for every simply connected domain of the extended complex plane there is a univalent and meromorphic function that maps such a domain conformally onto the unit disk. The Handbook of Conformal Mappings and Applications is a compendium of at least all known conformal maps to date, with diagrams and description, and all possible applications in different scientific disciplines, such as: fluid flows, heat transfer, acoustics, electromagnetic fields as static fields in electricity and magnetism, various mathematical models and methods, including solutions of certain integral equations.
Table of contents:
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. w = −iz
Map 2.2. w = z2
Map 2.3. w = z2 − 1
Map 2.4. w = z
Map 2.5. w = iz
Map 2.6. w = 1−z1+z
Map 2.7. w = z − 1
Map 2.8. w = 2z
Map 2.9. w = z+3i2
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. w = z−z0z−1/z¯0
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. w = az + b
Map 3.5. w = 2z − 2i
Map 3.6. w = 1/z
Map 3.7. w = 1/z
Map 3.8. w = 1/z
Map 3.9. w = 1/z
Map 3.10. w = k/z
Map 3.11. w = a/z
Map 3.12. w = z + 1/z
Map 3.13. w = z + 1/z
Map 3.14. w = z + 1/z
Map 3.15. w = z−αz+α
Map 3.16. w = z+1z−1
Map 3.17. w = 1+z1−z
Map 3.18. w = 1+z1−z
Map 3.19. w = z−1z+1
Map 3.20. w = 1−z1+z
Map 3.21. w = 1−z1+z
Map 3.22. w = z−iz+i
Map 3.23. w = z−iz+i
Map 3.24. w = i+zi−z
Map 3.25. w = i1−z1+z
Map 3.26. w = −iz+iz−i
Map 3.27. w = z2z−8
Map 3.28. w = zz−1
Map 3.29. w = zz−(1+i)
Map 3.30. w = ziz+2
Map 3.31. w−w0=ℜ{ eiτz−z0−rαα¯z−αz0−r }
Map 3.32. w = eiτz−αα¯z−1
Map 3.33. w = eiθz−z01−z¯0z
Map 3.34. w = ρeiαz−z1z−z2
Map 3.35. w = 11−z
Map 3.36. w = 1z−a
Map 3.37. w = 2irz−a
Map 3.38. w = r2r2−r1(2r1z+i)
Map 3.39. w = ir2r1+r2(1−2r1z)
Map 3.40. w = (1+z1−z)2
Map 3.41. w = (z + 1)α + (z − 1)α
Map 3.42. w = (1+zm1−zm)2
Map 3.43. w = iz2+1iz2−1
Map 3.44. w = e2miarccot(p)(z+1z−1)m
Map 3.45. w = z−aaz−1
Map 3.46. w = z−aaz−1
Map 3.47. w = eiλz−z0z−z¯0
Map 3.48. w = eiλz−z01−z¯0z
Map 3.49. w = eiβz−αα¯z−1
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. w = Kσz(iσ−z1)z−iσ
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. w = R1z−iσz+iσ
Map 3.69. w = tR1r1z−sz−t
3.5 Ellipses and Hyperbolas
Map 3.70. Ellipses
Map 3.71. z = 12[ (a−b)w+a+bw ]
Map 3.72. Hyperbolas
4 Algebraic Functions
4.1 Polynomials
Map 4.1. w = (z − a)n, n > 1
Map 4.2. w = zn
Map 4.3. w = zm, m≥12
Map 4.4. w = (z−a1)(z−a2)⋯(z−a)n )(1−a¯1z)(1−a¯2z)⋯(1−a¯nz)
Map 4.5. w = z2
Map 4.6. w = z2
Map 4.7. w = z2
Map 4.8. w = z2
Map 4.9. w = z2
Map 4.10. w = z2
Map 4.11. w = z2
Map 4.12. w = z2
Map 4.13. w = z2
Map 4.14. w = z2
Map 4.15. w = z2
Map 4.16. w = z2
Map 4.17. w = z2
Map 4.18. w = az2 + bz + c
Map 4.19. w = a2z2 + b, a ≠ 0
Map 4.20. w = azα + bzβ, β < 0 < α, ab ≠ 0
Map 4.21. w = zα + bzβ, β > α > 0, ab ≠ 0
Map 4.22. w = azα + bz−α, α > 0, ab ≠ 0
Map 4.23. w = zn
Map 4.24. w = zα
Map 4.25. w = z1/2
Map 4.26. w = z
Map 4.27. w = z2/3
Map 4.28. w = z−α, α > 0
Map 4.29. w = z−α, α > 1
Map 4.30. w = zπ/α
Map 4.31. w = zα − zβ, β > α > 0
Map 4.32. w = zia
Map 4.33. w = az + b/z, or w+2kw−2k=(z+k/az−k/a)2, k=ab≠0
Map 4.34. w = zeiα−a2z eiα
Map 4.35. w = z−a2z
Map 4.36. w = w = 12z+βz, β≠0
Map 4.37. w = −za+az
Map 4.38. w = za+az
Map 4.39. w = −z2a2−a2z2
Map 4.40. w = z + i/z
Map 4.41. w = i(√z − √p)
Map 4.42. w = 2pz−1
Map 4.43. w = z−12p−i12p
Map 4.44. w = −(za)π/β−(az)π/β
Map 4.45a. w = az2+bz+cz+d, a≠0
Map 4.45b. w = az2+bz+cz2+dz+f, a ≠ 0
Map 4.45c. w = i{ b(1+z)2+2(1−z) }2(1+z)
Map 4.46. w = iz2+2az−a2z2−2az−a2
Map 4.47. w = z2+2cz−c2z2−2cz−c2
Map 4.48. w = (1+zπ/β)2−i(1−zπ/β)2(1+zπ/β)2+i(1−zπ/β)2
Map 4.49. wb = 1 − za, a > 0, b > 0, a ≠ 1, b ≠ 1
Map 4.50. w = (z+cz−c)2
Map 4.51a. w = −(1+1+z2z)2
Map 4.51b. w = z1+z2
Map 4.51c. w = (z/c+1z/c−1)2
Map 4.51d. w2 = z−1z+1
Map 4.52. w = 1−z2
Map 4.53. w = ((z/c)π/α+1(z/c)π/α−1)2
Map 4.54. w = (1+z3)2−i(1−z3)2(1+z3)2+i(1−z3)2
4.1.1 Regions Bounded by Two Circular Arcs
Map 4.55. w = k(z−z0z−ζ0)π/α, k ≠ 0
Map 4.56. w = ξπ/α
Map 4.57. w = c(zz¯1iσ−z)π/α, c > 0, σ > 0
Map 4.58. w = ce−iβπ/α(c+zc−z)π/α, c > 0, β > 0, 0 < α < α + β < π
5 Exponential Family of Functions
5.1 Exponentials
5.1.1 de Moivre’s Theorem
Map 5.1. w = ez
Map 5.2. z = log w
5.2 Specific Cases of Mappings
Map 5.3. w = ez
Map 5.4. w = r eiθ
Map 5.5. w = ez
Map 5.6. w = ez
Map 5.7. w = ez
Map 5.8. w = log z, w = ez, w = cosh z
Map 5.9. w = ez, w = eiz
Map 5.10. w = eπz/a
Map 5.11. Composites of w = eaz
Map 5.12. w = eπz/a
Map 5.13. w = eaz − cebz, a > b > 0, c > 0
Map 5.14. w = eαz − e(α−1)z, 0 < α < 1
Map 5.15. w = aefz + begz
Map 5.16. w = z + ez
Map 5.17. w = αeaz + βebz
Map 5.18. w = eiπz/p
Map 5.19. w = 1+ez
Map 5.20a. w = log(az + b)
Map 5.20b. w = log z
Map 5.20c. w = log za = a log z, a ≥ 1
Map 5.21. w = sec2 z
Map 5.22. w = log (zα − 1), α > 1
Map 5.23. w = log (zα + a)
Map 5.24. w = logzα+z−α2
Map 5.25. w = logz2+12z
Map 5.26. w = logz+1z−12logz
5.3 Other Related Functions
Map 5.27. w = sin z
Map 5.28. w = sin z
Map 5.29. w = sin z
Map 5.30. w = cos z
Map 5.31. w = sin (az)
Map 5.32. w=eπz/a−eπz0/aeπz/a+eπz¯0/a
Map 5.33. w = ln z
Map 5.34. w = ln z
Map 5.35a. w = logz−1z+1
Map 5.35b. w = logz−1z+1
Map 5.36. w = logz+1z−1
Map 5.37. w = log[ i1−z1+z ]
Map 5.38. w = logaez+bcez+d
Map 5.39. w = logz−mz+m+logmz−1mz+1
Map 5.40. w = log {ez + (e2z − 1)1/2}
Map 5.41. w = log sin z = log cosh(iz − iπ/2)
Map 5.42. w = log tan z
Map 5.43. w = sinπza
Map 5.44. w = sinπz2a
Map 5.45. w = cosπza
Map 5.46. w = cosπza
Map 5.47. w = cos (a log z)
Map 5.48. w = 1a−cosz
Map 5.49. w = tan(arccos z24)
Map 5.50. w = tan2(z/2)
Map 5.51. w = tan2(a2z)
Map 5.52. w = sec z
Map 5.53. w = cosh z
Map 5.54. w = cosh(πz/a)
Map 5.55. w = icosh(π12z/p−14)
Map 5.56(a)-(f). w = tanh z
Map 5.57. w = tanh zz
Map 5.58. w = tanh2π4zp
Map 5.59. w = coth(z/2)
Map 5.60. w = coth(π/z)
Map 5.61. w = z + c coth z
Map 5.62. w = coth zz
Map 5.63. w = −2 arccot(z/c)
Map 5.64. w = iπ + z − log z
Map 5.65. w = −iπ + 2 log z − z2
Map 5.66. w = z(z + 1)1/2 + log(z+1)1/2−1(z+1)1/2+1
Map 5.67. w = iklog1+ikt1−ikt+log1+t1−t
Map 5.68. w = z−1z+2c log z
Map 5.69. w = log cothz2
Map 5.70. w = ±sinh z
Map 5.71. w = log sinh (z+β)sinh z=log(a+coth z)−log(a2−1)
Map 5.72. w = k logk1−k + log 2(1 − k) + iπ − k log(z + 1)−(1−k)log(z − 1)
Map 5.73. w = tan(1narccos zn/2)
Map 5.74. w = tanh−1 z − b arctan(z/b)
Map 5.75. w = 2π{ tanh−1pz−p arctanz }
Map 5.76. w = hπ[ (z2−1)1/2+cosh−1z ]
Map 5.77. w = cosh−1(2z−k−1k−1)−1kcosh−1((k+1)z−2k(k−1)z)
Map 5.78. w = (1+z1−z)a/(iπ)
5.4 Complex Exponential Function
6 Joukowski Airfoils
6.1 Joukowski Maps
Map 6.1. w = 12(z+1/z)
Map 6.2. w = 12(z+1z)
Map 6.3. w = 12(z+1z)
Map 6.4(a)-(e). z = w + w2−1
Map 6.5. w = a2(z+1z)
Map 6.6. w = 12(a+baz+a(a−b)z)
Map 6.7. w = 12(ze−α+z−1eα)
Map 6.8. w = z+a2z
Map 6.9. w = aπ(1+w+w2)
Map 6.10. w = 12(z+1z)
Map 6.11. w = z+c2z
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
Map 7.53. Curvilinear triangle with two right angles onto the upper half-plane
Map 7.54. Interior of a triangle onto the upper half-plane
Map 7.55. Interior of a rectangle onto the upper half-plane
Map 7.56. Upper half-plane with a path onto the upper half-plane
Map 7.57. Upper half-plane onto upper half-plane minus an isosceles triangle
Map 7.58. Interior of an ellipse onto the unit disk
Map 7.59. Triangle onto the upper half-plane
Map 7.60. Upper half-plane onto a square
Map 7.61. Shaded region onto the upper half-plane
Map 7.62. Special case of Map 7.61
Map 7.63. Rectangle onto upper-half plane
Map 7.64. w = A∫z0z(t−z2)3/4(t−z4)1/4(t−z1)(t−z3)dt+B
Map 7.65. πw = cosh−1 z = arcsin(1/z) + π/2
Map 7.66. w = i1−z2
Map 7.67. w = 1−b2log(z−1)+1+b2log(z+1)
7.7 Inverse Schwarz-Christoffel Mapping
Part 2: Numerical Conformal Mapping
8 Schwarz-Christoffel Integrals
8.1 Parameter Problem
8.1.1 Kantorovich’s Method
8.2 Newton’s Method
8.3 Numerical Computations
Map 8.1. Schwarz-Christoffel integral for the unit disk
8.3.1 Removal of Singularities
8.3.2 Trefethen’s Method
8.4 Minimum Area Problem
8.4.1 Bergman Kernel
8.5 Numerical Methods for Minimum Area Problem
8.5.1 Ritz Method
8.5.2 Bergman Kernel Method
Map 8.2. w = z−z0z−1/z¯0
8.6 Minimum Boundary Problem
8.6.1 Ritz Method for Problem II
8.7 Numerical Parameter Problem
8.7.1 Parameter Method
8.7.2 van Dyke’s Method
8.7.3 Trefethen’s Method
8.7.4 Foster-Anderson’s Method
8.7.5 Unit Disk onto Polygon
8.7.6 Polygon onto Unit Disk
8.8 Generalized Schwarz-Christoffel Parameter Problem
8.9 Proofs
9 Nearly Circular Regions
9.1 Small Parameter Expansions
Map 9.1. w = f(z, λ)
9.2 Method of Infinite Systems
Map 9.2. Coefficients of w = f(z, λ)
Map 9.3. w = f(z, λ)
Map 9.4. w = f(z, λ) for an ellipse
9.3 Special Cases
Map 9.5. Unit disk onto the boundary Γ
Map 9.6. Unit disk onto the interior of an ellipse
Map 9.7. Family of squares zz¯+k(z2−z¯24)2=1 onto the unit disk
9.4 Exterior Regions
Map 9.8. Nearly circular boundary onto the exterior of a circle
Map 9.9. Nearly circular boundary onto the interior of a circle
Map 9.10. Exterior of an ellipse onto exterior of a circle
Map 9.11. Exterior of the square −1≤ x, y ≤ 1 onto |w| < R
Map 9.12. Nearly circular ellipse onto the unit disk
Map 9.13. Unit disk onto a nearly circular region
Map 9.14. Unit disk onto a nearly circular region
10 Integral Equation Methods
10.1 Neumann Kernel
10.2 Interior Regions
10.2.1 Lichtenstein’s Integral Equation
10.2.2 Gershgorin’s Integral Equation
10.2.3 Carrier’s Integral Equation
10.3 Exterior Regions
10.3.1 Banin’s Integral Equation
10.3.2 Warschawki-Stiefel’s Integral Equation
10.3.3 Interior and Exterior Maps
10.4 Iterative Method
10.5 Degenerate Kernel
10.6 Szegö Kernel
10.6.1 Generalized Szegö and Bergman Kernels
11 Theodorsen’s Integral Equation
11.1 Classical Iterative Method
11.2 Convergence
11.3 Theodorsen’s Method
11.4 Integral Representation
11.5 Starlike Regions
11.6 Exterior Regions
11.7 Trigonometric Interpolation
11.8 Wegmann’s Method
11.9 Newton’s Method
11.10 Kantorovich’s Method
11.11 Fornberg’s Method of Successive Approximation
11.11.1 Taylor’s Series Method
11.12 Lichtenstein-Gershgorin Method
11.13 Theodorsen-Garrick Method
11.14 Variational Method
11.14.1 Minimization Process
11.14.2 Orthogonalization
11.14.3 Minimization of the Perimeter
12 Symm’s Integral Equation
12.1 Symm’s Integral Equation
12.1.1 Interior Regions
12.1.2 Exterior Regions
12.2 Orthonormal Polynomial Method
12.3 Modified OPN Method
12.4 Lagrange Interpolation
12.5 Spline Approximations
13 Airfoils and Singularities
13.1 Nearly Circular Approximations
13.2 James’s Method
13.2.1 Single-Element Airfoils
13.2.2 von Karmen-Trefftz Transformations
13.2.3 Two-Element Airfoils
13.2.4 Multi-Element Airfoils
Map 13.1. w = lz2+2mz+npz2+2qz+r
Map 13.2. w = z + a2/z
Map 13.3. w = z + a2/z
Map 13.4. w = (zn+1zn−1)2
13.3 Arbenz’s Integral Equation
Map 13.5. Unit square onto the unit disk
13.4 Boundary Corner
13.5 Singularity Behavior
13.6 Pole-Type Singularities
Map 13.6. z = γ(ζ)
Map 13.7. w = | α |αf1(z)−α1−α¯f1(z), α=f1(ae2)
Map 13.8. Cubic Arc
Map 13.9. Circular Arc
13.7 Exterior Regions
Map 13.10. w = βζ−αζ−α¯, ζ=sn(z,k), ℑ{α} > 0
Map 13.11. Rectangle
Map 13.12. Bean-shaped Region
14 Doubly Connected Regions
14.1 Annular Regions
Map 14.1. Annulus onto a parallel strip, w = log z
Map 14.2. w = 2cz−1+1−4c2(1−1−4c2)z−2c
Map 14.3. w = ℘(logzr)
Map 14.4. w = sn(2Klog R/rlogzRr), R>r
Map 14.5. w = ℘(logz+cz−c+12loga+ca−c)
Map 14.6. w = sn(Kλlogz+cz−c+ρ)
Map 14.7. w = −ζ(ilogzRr)+iηπlogzRr
Map 14.8. w = ζ(χ) + ηχπ
Map 14.9. Non-concentric annulus onto symmetric annulus
14.2 Area Theorem
Map 14.10. t = L sn(2iKπlogζρ+K, k), k=L2
Map 14.11. Dirichlet problem for the annulus
Map 14.12. Neumann problem for the annulus
14.3 Source Density
14.3.1 Numerical Computation of µ(ζ)
Map 14.13. A pair of limaçons
14.4 Dipole Distribution
Map 14.14. Symmetric doubly connected region onto the unit disk or an annulus
Map 14.15. Nonsymmetric horizontal lines
14.5 Gaier’s Variational Method
Map 14.16. Circle in a square
Map 14.17. Square in a square
Map 14.18. A cross in a square
14.6 Mapping onto Annulus
14.6.1 Numerical Methods
15 Multiply Connected Regions
15.1 Some Useful Results
15.2 Dirichlet Problem
15.3 Mikhlin’s Integral Equation
15.4 Mayo’s Method
15.5 Fast Poisson Solver
Part 3: Applications
16 Grid Generation
16.1 Computational Region
16.1.1 Coordinate Transformations
16.1.2 Orthogonal Method
16.2 Inlet Configurations
Map 16.1. One-step method
Map 16.2. Two-step method
16.3 Cascade Configurations
Map 16.3. Ordinary type of grids
Map 16.4. Flow of parallel streamlines
Map 16.5. Flow streamlines starting from infinity
Map 16.6. Transonic flow: Design of an airfoil
17 Field Theories
17.1 Mathematical Models
17.2 Initial and Boundary Conditions
17.3 Classification of Second-Order Equations
17.4 Superposition Principle
17.5 Parabolic Equations
17.6 Hyperbolic Equations
17.6.1 Second-Order Hyperbolic Equations in One Space Variable
17.6.2 General Case
17.7 Elliptic Equations
17.7.1 Boundary Value Problems for Laplace’s Equation
17.7.2 Cylindrical Polar Coordinates
17.7.3 Transformation of Laplace’s Equation
17.8 Poisson’s Equation
17.8.1 Particular Solutions of Poisson’s Equation
17.8.2 Poisson’s Equation in Polar Coordinate System
17.8.3 Helmholtz Equation
17.8.4 Fourth-Order Stationary Equations
17.9 Two-Dimensional Flows
17.9.1 Laplace’s Equation under a Conformal Map
17.9.2 Sources and Sinks
18 Fluid Flows
18.1 Viscous Laminar Flows
18.2 External Flows
18.2.1 Creeping Flow around a Sphere
18.2.2 Boundary Layer Flow over a Flat Plate
18.2.3 Ocean Flow
18.3 Ideal Fluid Flows
18.3.1 Complex Potential Flow
18.4 Potential Flows
18.4.1 Potential Flow of Ideal Fluids
18.4.2 Flow over a Plate
18.4.3 Potential Flow at Separation
18.5 Boundary Layer Flows
18.5.1 Presence of Circulation
18.6 Kirchhoff’s Flow Problem
18.6.1 Original Kirchhoff’s Flow Problem
18.7 Streamlines in Fluid Flows
18.7.1 Stagnation Point
18.7.2 Velocity Potential in Fluid Flows
18.8 Conformal Mapping of Flow Patterns
18.9 Joukowski Maps
18.10 Airfoils
18.10.1 Blasius Theorem
18.10.2 Boundary Layer Flow
18.10.3 Circulation
18.10.4 Lift
19 Heat Transfer
19.1 Heat Flow
19.1.1 Method of Separation of Variables
19.2 Heat Transfer
19.2.1 Ill-Posed Problems
19.3 Conformal Transformations
19.4 Poisson’s Integral Formulas
19.4.1 First Poisson’s Integral Formula
19.4.2 Poisson’s Formula for the Unit Disk
19.4.3 Second Poisson’s Integral Formula
19.5 Diffusion Equation
19.5.1 Finite Difference Method
19.5.2 Steady-State Temperature
19.6 High Temperature Effects
19.6.1 Hypersonic Flow and High Temperature Effects
19.6.2 Shock Layer
19.6.3 Flow through a Constriction
19.7 Transient Problems
20 Vibrations and Acoustics
20.1 Wave Propagation and Dispersion
20.1.1 Wave Propagation
20.1.2 Wave Dispersion
20.1.3 Damped Waves
20.1.4 Thermal Waves
20.2 Vibrations of Strings
20.3 Vibrations of Membranes
20.3.1 Force at the Boundary of a Membrane
20.3.2 Fundamental Frequency of a Membrane
20.3.3 Free Vibrations of Membranes
Map 20.1. One-step method
20.4 Acoustics
20.4.1 Harmonic Analysis
20.4.2 Huygens Principle
20.4.3 Audio Signals
20.4.4 Acoustic Waveguides
21 Electromagnetic Field
21.1 Electromagnetic Field
21.2 Electrostatic Field
21.2.1 Electric Potential
21.3 Electric Field
21.3.1 Hall Effect
21.4 Electromagnetic Waves
21.5 Electric Capacitors
Map 21.1a. z = aπ(1+w+ew)
Map 21.1b. z = aπ[ 1−ew−w ]+ia
Map 21.1c. w = 2 log ζ
21.5.1 Capacitance between Two Conductors
21.5.2 Problem of Two Cylinders
21.6 AC Circuits
21.6.1 Two-Dimensional Phase Diagrams
21.7 Laplace’s Equation
21.7.1 Electric Potential and Field between Two Infinite Sheets
21.7.2 Poisson’s Equation
22 Transmission Lines and Waveguides
22.1 Maxwell’s Equations
22.2 Wave Propagation
22.3 Transmission Lines
22.3.1 Lossless Transmission Lines
22.3.2 Infinitely Long Transmission Line
22.3.3 VSWR
22.3.4 Parallel Wire Transmission Lines
22.3.5 Transmission with Lossy Conductors
22.3.6 Pipe-Enclosed Cable
22.3.7 Homogeneous Strip Lines
22.4 Conformal Mapping and Electric Transmission
22.5 Conformal Mapping and Waveguides
22.6 Helmholtz Equation and Rib-Shaped Waveguide
22.6.1 Homotopy Perturbation Method
22.7 Coplanar Waveguides
22.7.1 Conductors
22.7.2 CPW Directional Couplers
22.7.3 Cell Simulation
22.8 Nonuniform Waveguides
22.8.1 Conformal Mapping of the Transverse Section of the Waveguide
23 Elastic Medium
23.1 Stress and Strain
23.2 Stress Function
23.3 Infinite Plate and Conformal Mapping
Map 23.1. z = M (w + µw1−n)
24 Finite Element Method
24.1 Weak Variational Form
24.2 Galerkin Method
24.3 Rayleigh-Ritz Method
24.4 Linear Three-Node Triangular Elements
24.5 Single Dependent Variable Problems
24.5.1 Local Weak Formulation
24.5.2 Finite Element Equation
24.5.3 Evaluation of Stiffness Matrix and Load Vector
24.5.4 Evaluation of Boundary Integrals
24.5.5 Assembly of Element Matrices
24.6 Fluid Flows
24.7 Free Axial Vibrations of an Elastic Rod
24.8 Electric Potential
24.8.1 Axisymmetric Exterior-Field Problems
Map 24.1. z = c tanh(w/2)
Map 24.2. z = aw+ib
24.9 Waveguide
24.10 Motor Design
24.11 Other Applications
25 Computer Programs and Resources
25.1 Numerical Methods
25.2 Software
25.2.1 Finite Difference Approximations
25.3 Computer Codes
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