💠Intro to Complex Analysis Unit 7 – Conformal Mappings in Complex Analysis

Key Concepts and Definitions

• Conformal mapping preserves angles and orientation between curves in the complex plane
• Biholomorphic function is another term for a conformal mapping that is also bijective (one-to-one and onto)
• Analytic functions have the property that they are infinitely differentiable and equal to their own Taylor series
  • Entire functions are analytic functions defined on the whole complex plane
• Harmonic functions satisfy Laplace's equation $\nabla^2 f = 0$ and are the real and imaginary parts of analytic functions
• Isothermal coordinates are a parametrization of a surface that preserves angles and shapes locally
• Riemann mapping theorem states that any simply connected domain in the complex plane, other than the entire plane itself, can be conformally mapped onto the open unit disk

Historical Context and Applications

• Conformal mappings have roots in the work of Lagrange, Gauss, and Riemann in the 18th and 19th centuries
• Riemann introduced the concept of Riemann surfaces and the Riemann mapping theorem in his doctoral thesis (1851)
• Conformal mappings find applications in various fields such as fluid dynamics, electromagnetism, and heat transfer
  • They can simplify boundary conditions and geometry in these problems
• In cartography, conformal projections (Mercator projection) preserve angles and shapes of small regions on a map
• Conformal mappings are used in computer graphics and image processing for texture mapping and image warping
• In aerodynamics, Joukowsky transform is used to study the flow around airfoils by mapping the airfoil to a circle

Conformal Mapping Fundamentals

• Conformal mappings are angle-preserving transformations in the complex plane
• They are realized by analytic functions with non-zero derivative
• The mapping $w = f(z)$ is conformal at a point $z_0$ if $f'(z_0) \neq 0$
  • The angle of intersection between two curves is preserved under the mapping
• Conformality is a local property that holds in a neighborhood of each point where the derivative is non-zero
• The composition of two conformal mappings is also a conformal mapping
• The inverse of a conformal mapping, if it exists, is also conformal

Types of Conformal Transformations

• Translation $w = z + c$ shifts the complex plane by a constant $c$
• Rotation $w = e^{i\theta}z$ rotates the plane by an angle $\theta$
• Scaling $w = kz$ enlarges or shrinks the plane by a factor $k$
• Inversion $w = 1/z$ reflects the plane across the unit circle and exchanges the interior and exterior of the circle
• Möbius transformations are rational functions of the form $w = \frac{az+b}{cz+d}$ with $ad-bc \neq 0$
  • They map circles and lines to circles and lines
• Exponential function $w = e^z$ maps horizontal lines to circles centered at the origin and vertical lines to radial lines
• Logarithm function $w = \log z$ is the inverse of the exponential function and maps the right half-plane to a vertical strip

Analytic Functions and Conformality

• Analytic functions satisfy the Cauchy-Riemann equations $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
  • $u$ and $v$ are the real and imaginary parts of the function $f(z) = u(x,y) + iv(x,y)$
• If $f(z)$ is analytic and $f'(z) \neq 0$, then $f(z)$ is conformal
• Conformality implies that the function locally preserves angles, shapes, and orientation
• The Jacobian matrix of a conformal mapping is a scalar multiple of a rotation matrix at each point
• Harmonic conjugates are the real and imaginary parts of an analytic function and satisfy the Cauchy-Riemann equations

Preservation of Angles and Shapes

• Conformal mappings preserve angles between intersecting curves
  • The angle of intersection is measured by the argument of the derivative of the mapping
• Orthogonal curves (curves intersecting at right angles) are mapped to orthogonal curves under a conformal mapping
• Infinitesimal circles are mapped to infinitesimal circles, preserving the local shape
• The mapping is locally isotropic, meaning it stretches or shrinks the plane equally in all directions at each point
• Conformal mappings do not necessarily preserve global shapes or sizes, only local angles and shapes
• The conformal modulus, defined as the ratio of the radii of concentric circles, is invariant under conformal mappings

Mapping Techniques and Examples

• Schwarz-Christoffel transformation maps the upper half-plane to a polygon by integrating a certain differential equation
  • It is used to solve problems in fluid dynamics and electrostatics involving polygonal boundaries
• Joukowsky transform $w = z + 1/z$ maps the exterior of the unit circle to the exterior of a Joukowsky airfoil
• Szegő kernel is a conformal mapping of the unit disk onto a simply connected domain with smooth boundary
• Koebe 1/4 theorem states that the image of the unit disk under any univalent function contains a disk of radius 1/4
• Conformal mappings can be constructed using the Riemann mapping theorem and the Schwarz lemma
• Elliptic functions, such as the Weierstrass $\wp$-function, provide conformal mappings of rectangular regions to the complex plane

Challenges and Common Mistakes

• Ensuring the analyticity and non-vanishing derivative of the mapping function
• Dealing with singularities and branch cuts when constructing conformal mappings
• Choosing appropriate branch cuts and domains for multi-valued functions (logarithm, square root)
• Correctly identifying and interpreting the images of curves and regions under the mapping
• Applying the chain rule correctly when composing conformal mappings
• Remembering that conformality is a local property and may not hold globally
• Distinguishing between conformal and isometric mappings (preserve both angles and distances)