Conformal mapping

A special operation in mathematics in which a set of points in one coordinate system is mapped or transformed into a corresponding set in another coordinate system, preserving the angle of intersection between pairs of curves.

A mapping or transformation of a set E of points in the xy plane onto a set F in the uv plane is a correspondence, Eqs. (1), that is defined for each

 

 

 (1) 

point (x,y) in E and sends it to a point (u,v) in F, so that each point in F is the image of some point in E. A mapping is one to one if distinct points in E are transformed to distinct points in F. A mapping is conformal if it is one to one and it preserves the magnitudes and orientations of the angles between curves (Fig. 1). Conformal mappings preserve the shape but not the size of small figures.

 

 

Fig. 1  Angle-preserving property of a conformal mapping.

 

 

 

 

 

 

Fig. 2  Illustration for proof that analytic functions preserve angles.

 

 

 

 

 

 

Relation to analytic functions

 

If the points (x,y) and (u,v) are viewed as the complex numbers z = x + iy and w = u + iv, the mapping becomes a function of a complex variable: w = f(z). It is an important fact that a one-to-one mapping is conformal if and only if the function f is analytic and its derivative f  ′(z) is never equal to zero. This can be seen by considering a function f that is analytic near z0 = x0 + iy0, with f  ′(z0) ≠ 0, and mapping a curve C passing through z0 to a curve Γ passing through w0 = f(z0). Then, as a point z tends to z0 along C, the angle arg{z – z0} tends to the angle α between C and the line y = y0 (Fig. 2). But by assumption, expression (2) is valid. In

 

 

 (2) 

particular, arg{ f(z) – w0}– arg{z – z0} tends to a limit γ = arg{ f  ′(z0)}. In other words, arg{w – w0} → β as w → w0 along Γ, where β = α + γ. But any other curve through z0 is rotated by the same angle γ, so that the mapping preserves the angle between the two curves. In a similar manner, it can be shown that a conformal mapping is necessarily given by an analytic function. See also: Complex numbers and complex variables

 

Examples

 

Some examples of functions that provide conformal mappings (and one that is not conformal) will now be given.

The function w = (z – 1)/(z + 1) maps the right half-plane, defined by Re{z} > 0, conformally onto the unit disk, defined by |w| < 1.

The function w = log z maps the right half-plane conformally onto the horizontal strip defined by −π/2 < Im{w} < π/2.

The function w = z2 maps the upper semidisk, defined by Im{z} > 0 and |z| < 1, conformally onto the unit disk |w| < 1 with the segment 0 ≤ u < 1 removed. It doubles angles at the origin, but this is a boundary point which does not lie in the semidisk.

The function w = z of complex conjugation preserves the magnitudes but not the orientations of angles between curves. It is nowhere conformal.

The function w = z + 1/z maps the unit disk |z| < 1 conformally onto the extended complex plane (including the point at infinity) with the line segment −2 ≤ u ≤ 2 removed.

The Koebe function k(z) = z(1 – z)-2 maps the unit disk conformally onto the complex plane with the half-line – ∞ < u ≤ – 1/4 removed.

 

 

Linear fractional transformations

 

A linear fractional transformation is a function of the form given by Eq. (3), where a, b, c, and d are complex

 

 

 (3) 

constants. It is also known as a Möbius function. One example is the function w = (z – 1)/(z + 1), discussed above. Simpler examples are magnifications, given by Eq. (4), rotations, given by Eq. (5), and inversion, given by Eq. (6). Every

 

 

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 (5) 

 

 

 

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linear fractional transformation is a composition of linear fractional transformations of these three special types. Thus each linear fractional transformation provides a conformal mapping of the extended complex plane onto itself, and in fact the linear fractional transformations are the only such mappings. There is a unique linear fractional transformation which carries three prescribed (distinct) points z1, z2, z3 to prescribed images w1, w2, w3. The most general conformal mapping of the unit disk onto itself is a linear fractional transformation of the form given by Eq. (7).

 

 

 (7) 

 

Each linear fractional transformation carries circles to circles and symmetric points to symmetric points. Here a circle means a circle or a line. Two points are said to be symmetric with respect to a circle if they lie on the same ray from the center and the product of their distances from the center is equal to the square of the radius. Two points are symmetric with respect to a line if the line is the perpendicular bisector of the segment joining the two points. As an instance of this general property of linear fractional transformations, the mapping w = (z − 1)/(z + 1) sends the family of circles of Apollonius with symmetric points 1 and −1 (defined by requiring that on each circle the quotient of the distances from 1 and −1 be constant) onto the family of all circles centered at the origin. It carries the orthogonal family of curves, consisting of all circles through the points 1 and −1, onto the family of all lines through the origin (Fig. 3).

 

 

Fig. 3  Circles of Apollonius and members of the orthogonal family of curves.

 

 

 

 

 

 

 

Applications

 

Conformal mappings are important in two-dimensional problems of fluid flow, heat conduction, and potential theory. They provide suitable changes of coordinates for the analysis of difficult problems. For example, the problem of finding the steady-state distribution of temperature in a conducting plate requires the calculation of a harmonic function with prescribed boundary values. If the region can be mapped conformally onto the unit disk, the transformed problem is readily solved by the Poisson integral formula, and the required solution is the composition of the resulting harmonic function with the conformal mapping. The method works because a harmonic function of an analytic function is always harmonic. See also: Conduction (heat); Fluid-flow principles; Laplace's differential equation; Potentials

The term conformal applies in a more general context to the mapping of any surface onto another. A problem of great importance for navigation is to produce conformal mappings of a portion of the Earth's surface onto a portion of the plane. The Mercator and stereographic projections are conformal in this sense. See also: Map projections

 

Riemann mapping theorem

 

In 1851, G. F. B. Riemann enunciated the theorem that every open simply connected region in the complex plane except for the whole plane can be mapped conformally onto the unit disk. Riemann's proof was incomplete; the first valid proof was given by W. F. Osgood in 1900. Most proofs exhibit the required mapping as the solution of an extremal problem over an appropriate family of analytic univalent functions. (A univalent function is simply a one-to-one mapping.) For instance, the Riemann mapping maximizes | f ′(z0)| among all analytic univalent functions which map the given region into the unit disk: | f (z)| < 1. Here z0 is chosen arbitrarily in the region, and the Riemann mapping has f(z0) = 0.

 

Multiply connected regions

 

There is no exact analog of the Riemann mapping theorem for multiply connected regions. For instance, two annuli, r1 < |z| < r2 and R1 < |w| < R2, are conformally equivalent if and only if r2/r1 = R2/R1. Any doubly connected region can be mapped conformally onto a (possibly degenerate) annulus. Any finitely connected region (other than a punctured plane) can be mapped conformally onto the unit disk minus a system of concentric circular arcs, or onto the whole plane minus a system of parallel segments, or radial segments, or concentric circular arcs. Other canonical regions are bounded by arcs of lemniscates or logarithmic spirals, or by full circles.

 

Distortion theorems

 

Conformal mappings are often studied by considering the class S of functions f(z) which are analytic and univalent in the unit disk and have the normalizing properties f(0) = 0 and f ′(0) = 1. Alternatively, the class S may be defined as the class of all univalent power series of the form f(z) = z + a2z2 + a3z3 + · · · that are convergent for |z| < 1. The Koebe distortion theorem gives the sharp bounds, Eqs. (8),

 

 

 

 

 (8) 

for all functions f in S and all points z in the disk. The Koebe ¼-theorem asserts that each function f in S includes the full disk |w| < ¼ in its range. Suitable rotations of the Koebe function (discussed above) show that each of these bounds is the best possible. All of these statements can be deduced from a theorem of L. Bieberbach (1916) that |a2| ≤ 2 for all functions f in S, with equality occurring only for the Koebe function, given by Eq. (9),

 

 

 (9) 

or one of its rotations, given by e-i θk (e i θz for some value of θ. Bieberbach conjectured that in general |an| ≤ n for all n. For many years, the Bieberbach conjecture stood as a challenge and inspired the development of powerful methods in geometric function theory. After a long series of advances by many mathematicians, the final step in the proof of the Bieberbach conjecture was taken by L. de Branges in 1984.

 

Boundary correspondence

 

The open region inside a simple closed curve is called a Jordan region. Riemann's theorem ensures the existence of a conformal mapping of one Jordan region onto another. C. Carathéodory proved in 1913 that such a mapping can always be extended to the boundary and the extended mapping is a homeomorphism, or a bicontinuous one-to-one mapping, between the closures of the two regions. In fact, Carathéodory proved a much more general theorem which admits inaccessible boundary points and establishes a homeomorphic correspondence between “clusters” of boundary points, known as prime ends. For a conformal mapping of the unit disk onto a Jordan region with a rectifiable boundary, the Carathéodory extension preserves sets of measure zero (or zero “length”) on the boundary. It is also angle-preserving at almost every boundary point, that is, except for a set of points of measure zero. If the boundary of the Jordan region has a smoothly turning tangent direction, the derivative of the mapping function can also be extended continuously to the boundary. See also: Measure theory

 

Quasiconformal mappings

 

A theory of generalized conformal mappings, known as quasiconformal mappings, has evolved. Roughly speaking, a univalent function w = f (z) is said to be K-quasiconformal in a certain region if it maps infinitesimal circles to infinitesimal ellipses in which the ratios of major to minor axes are bounded above by a constant K ≥ 1. Equivalently, a mapping is quasiconformal if it distorts angles by no more than a fixed ratio. The simplest examples are linear mappings of the form u = ax + by, v = cx + dy, ad − bc ≠ 0, where a, b, c, and d are real constants. The 1-quasiconformal mappings are simply the conformal mappings. The notion of a quasiconformal mapping is readily extended to higher dimensions.

Peter L. Duren

 

Bibliography

 

 

  • L. Bieberbach, Conformal Mapping, 1952, reprint 2000
  • R. V. Churchill and J. W. Brown, Complex Variables and Applications, 6th ed., 1995
  • P. L. Duren, Univalent Functions, 1983
  • R. Schinziger and P. A. Laura, Conformal Mapping: Methods and Applications, 1991
  •  Alifazeli=egeology.blogfa.com