Conformal Mapping: Methods and Applications

By Roland Schinzinger and Patricio A. A. Laura

Beginning with a brief survey of some basic mathematical concepts, this graduate-level text proceeds to discussions of a selection of mapping functions, numerical methods and mathematical models, nonplanar fields and nonuniform media, static fields in electricity and magnetism, and transmission lines and waveguides. Other topics include vibrating membranes and acoustics, transverse vibrations and buckling of plates, stresses and strains in an elastic medium, steady state heat conduction in doubly connected regions, transient heat transfer in isotropic and anisotropic media, and fluid flow. Revision of 1991 ed. 247 figures. 38 tables. Appendices.

• Language: English
• Publisher: Dover Publications
• Release date: Apr 30, 2012
• ISBN: 9780486150741

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Chapter 1

INTRODUCTION AND OVERVIEW

This book presents examples of successful applications of conformal transformations by the authors and others in various fields of engineering and applied science. Our intention is to interest the community of engineers, physicists, and applied mathematicians in this field which we find so useful and fascinating.

We assume that many readers would like a brief review--or perhaps even a first introductory tour--to the classical methods of conformal mapping. Such basic material will also acquaint the reader with the notation adopted by the authors. (An abbreviated list of symbols is included in Appendix A-1.)

Accordingly, we begin the book and most of the chapters on applications with field problems which satisfy Laplace’s equation in two dimensions for a uniform medium. Thereafter we examine more complicated equations and media. Initially, then, we would consider a problem solved when we have identified equipotential lines and lines of force or flow in sufficient detail to interpret important field phenomena locally or globally.

The results will be sets of orthogonal lines u=constant and v=constant in, say, an x – y plane where u(x,y) and v(x,y) satisfy Laplace’s equation. The equipotential and flux lines can be interpreted as coordinates u and v of another plane to which the original field has been transformed in such a manner that the field lines eventually form a rectangular or polar grid depending on the choice of coordinate system.

An interpretation in terms of another coordinate system is easy once a solution has been found. But the trick is to find the solution in the first place. The task is facilitated by designating both the original x – y plane (in which the physical problem is presented) and the u – v plane (of the transformed model) as complex planes: z = x + i y and w = u + i v. The transformation or mapping function is then a complex function of a complex variable, w = f(z), as indicated in Fig. 1.1. Such functions, if fairly well behaved (i.e. analytic), will map a region in one complex plane onto another complex plane in a way which preserves the magnitude and direction of angles between intersecting arcs. Thus Laplace’s equation for potential fields is satisfied in the transformed plane as well. The partial differential equations describing wave propagation, diffusion, and vibration, along with boundary conditions, can also be transformed.

Figure 1.1 The physical plane z and the model plane w.

The z-plane is also known as the original or problem plane and the w-plane as the computational or standard plane (the latter when the transformed image is the unit disk or a square).

Let us return for a moment to Figure 1.1. There we have labeled the transformations as w = f(z) and z = f-1(w), implying forward and inverse directions. We will also use the notation w=w(z), z=z(w), or z → w and w → z. Whether we employ a mapping in one direction or the other will depend on the task at hand. If an existing physical configuration is to be analyzed, its transformation into the model plane will be carried out by w = w(z). On the other hand, an ideal model which satisfies certain performance criteria may be first fashioned in the w-plane. Thereafter it can be brought to realization in the z-plane by a transformation z = z(w). We may compare this inverse transformation w → z as an act of synthesis or engineering design in the idealized sense. Often engineers must resort to repeated analysis imbedded in a trial and error process to carry out their design tasks. In conformal mapping a similar situation frequently occurs, but in an opposite sense. While a forward mapping is usually desired, it may be easier to carry out trial mappings starting from standard model images (such as unit circles and squares) and to make adjustments to fit the given configurations in the problem plane. For this reason the transformations presented by us may appear as w(z), z(w), or (where necessary) in both forms.

1.1 STRUCTURE OF THE BOOK

The book is organized as follows. Chapters 2 and 3 are of an introductory nature. The reader who is familiar with analytic function theory and the classical approaches to conformal mapping can bypass these chapters and move right on to the modern work beginning in Chapter 4. Those acquainted with the latest numerical techniques will want to start with Chapter 5 (physical models) or Chapter 6. Chapter 2 describes the classical methods of conformal mapping applied to two-dimensional fields in uniform media. It also serves as a review of the basic properties of analytic functions--those functions of a complex variable which can transform orthogonal grids in one plane to orthogonal grids in another plane.

Figure 1.2 The bilinear function

It is widely used for mapping adjacent or included circles into pairs of concentric circles. Application example: heat flow through the thermal insulation separating two pipes, one inside of and parallel to the other.

Representative examples of basic but useful mapping functions as exemplified in Figures 1.2 and 1.3 are given in Chapter 3. We limit ourselves there to fields in singly connected regions as in Figure 1.3 and to easily handled doubly connected regions as in Figure 1.2.

Figure 1.3 A composite mapping function

This function can be derived by applying the Schwarz-Christoffel transformation. Shown is the fringing field in the endzone of an electric capacitor.

In Chapter 4 we present the first departure from the idealized cases under review in Chapter 3. The boundaries, in particular, will be difficult to describe mathematically, or they will be composed of portions which are readily transformed individually but not jointly. See Figure 1.4 for such a deceptively simple looking example. In these cases numerical approximation methods will be required. Accordingly that chapter will contain a review and discussion of recent developments in the numerical generation of mapping functions.

Chapter 5 presents an overview of the functional relationships which govern potential fields and related physical phenomena in the various areas of application to be covered in the following chapters. In many cases there are analogies among fields of different types. It is then not necessary to transform boundary configurations of a given sort again and again. The reader will be able to use the results of many transformations in, say, heat flow problems to solve a problem with like boundaries in electrostatics.

Figure 1.4 Numerical result for a doubly connected region

The square and circle are easily transformed separately but not jointly. A suitable transformation is found by numerical means. Example: heat flow in a graphite brick of a graphite-moderated nuclear reactor.

Figure 1.5 Vector components in more than two dimensions.

The curved waveguide supports an electromagnetic wave traveling in the direction of Poynting vector P. The electric field vector zhas x and y components, the magnetic field vector a ζ component. Conformal mapping is possible if the permeability in the w-plane is modified by the metric coefficient m as indicated.

In Chapter 6 we examine methods to overcome some of the problems presented by field components in a third dimension and by nonuniformity of the medium. An example of this is shown in Figure 1.5 where the curved boundaries of the waveguide cause the electric field to have components in two directions (in the plane of the paper). Combined with the magnetic field vector in the third direction (normal to the plane of the paper), this leads to a departure from propagation in a plane wave mode. Conformal mapping may nevertheless be carried out if the medium is modified by paying heed to the metric coefficients which usually disappear through cancellation in the two dimensional case.

Another example describes the reverse of the above process: Starting out with an anisotropic medium, we render it isotropic in the model plane by means of the metric coefficient. The result is a modification of the boundary. See Figure 1.6.

The main body of the book consists of the chapters on applications. These are arranged as follows:

7 Static fields in electricity and magnetism

8 Transmission lines and wave guides

9 Vibrating membranes and acoustics

10 Transverse vibrations and buckling of plates

11 Stresses and strains in an elastic medium

12 Steady state heat conduction

13 Transient heat conduction

14 Fluid flow

It should be stressed here that many examples depart from the simple functional relationships expressed by Laplace’s equation.

Chapter 15 concludes our presentation with a discussion of the use of conformal mapping in conjunction with other methods of solving boundary value problems.

There are several useful appendices which should be consulted while reading the book: A-1 gives a list of symbols which recur regularly, A-2 is an index of transformations, and A-3 is a selected bibliography and list of references. A-4 is the name/author index and A-5 is the subject index.

Our style of writing may appear to be inconsistent: in one section we skim the surface, in another we appear almost bogged down with minutiae. The reason is our desire to give the reader a feeling now and then of how a particular methodology came about, while at other times we wish to stress the breadth of present or possible applications.

Figure 1.6 A nonuniform medium.

A parallel plate capacitor with an anisotropic dielectric is transformed into an isotropic capacitor of different thickness.

We now turn to such an overview. It is an illustration of the nonclassical uses of conformal mapping by means of examples of the type presented in Chapters 7 through 14. In several chapters the modern approaches are preceded by discussion of some classical problems to highlight the basic physical concepts. The chapters may also include examples of how new technologies benefit from conformal mapping, be it of the classical or nonclassical type.

1.2 MODERN APPLICATIONS OF CONFORMAL MAPPING

Classical applications of conformal mapping to many stationary problems of mathematical physics go back over a century and continue to the present. As was indicated, these applications usually deal with solutions of Laplace’s equation which remains invariant if the original plane is subjected to a conformal transformation. Consequently, complicated configurations can be transformed into more convenient ones without modification of the governing partial differential equation.

Applications of conformal mapping techniques in certain areas, such as the mathematical theory of elasticity, are considerably more complex. In other cases the physics may not be the problem, but the shape of the boundary is. In both respects new ground has been broken. A brief sampling of such nonclassical approaches follows. Specific examples, detailed solutions, and additional references are provided in Chapters 7 through 14.

Electromagnetics

As integrated circuits are pushed to new limits of performance, more exacting information is required on the characteristics of microcircuit components. Promising numerical approaches have been applied by Trefethen [1981, 1984] and others to planar semiconductor segments such as resistors and Hall effect transducers, while the more classical approaches continue to be of importance in the design of microstrip lines and antennas. See Chapter 7.

In waveguide analysis the seminal contributions of Meinke [1949-1,2; 1963], which involve trading boundary shapes (or the effects of a third dimension) for uniformity of the medium, are being followed by investigators interested in propagation along lossy transmission lines on the one hand [Schinzinger & Ametani, 1974, 1978] and in originally anisotropic media on the other [Kurase & Terakado, 1979]. See Chapter 8.

Other new applications include surface wave devices, ion optics, and magnetohydrodynamics. Examples of practical significance in the use of conformal transformations occur in designing the electric deflecting fields for ink jet printers and the magnetic heads for tape recorders. The effects of quasistatic electromagnetic fields at conductor edges constitute a recurring problem, particular in large scale integrated circuits [Yung, 1987; Chaudhry, 1989].

Vibrating Membranes and Acoustics

It is interesting to note that the method used to extend planar field mapping to a third dimension as applied to electromagnetic waveguides was first proposed by Routh in 1860 when he analyzed the motion of a heterogeneous membrane. Another link to electromagnetics is found in the equations which govern acoustic wave propagation.

In both areas--vibrating plates and acoustic waveguides--interesting results have been achieved, including those reported over the years by Laura and coworkers. Interesting examples include star shaped boundaries [Laura & Shahady, 1966] and eigenvalue optimization [Laura & Cortinez, 1986]. A method for solving three-dimensional axially symmetric problems related to the diffraction and radiation from a general class of bodies of revolution was developed by Pond [1970]. The method depends on the conformal transformation of the region outside the meridian profile of the body onto the region outside a circle. The required boundary value problem is formulated in spherical coordinates in the transformed space. In this form, Galerkin’s method can be applied to obtain a functional approximation for the solution of the boundary value problem.

Berger [1975] obtained a numerical solution for the transient vibration of an arbitrary shell of revolution, surrounded by an acoustic medium. The region external to the shell’s generating curve is mapped conformally onto the region external to the unit circle. Large amplitude oscillations were examined by Banerjee and Datta [1979].

Chapter 9 on Vibrating Membranes and Acoustics offers many other applications, including a determination of the scattering of acoustic signals from rough ocean surfaces.

Transverse Vibrations and Buckling of Plates

Aggarwala examined how complex distributions of loading affect simply supported plates of rectilinear shape [1954]. Later he [1966, 1967] and others extended the investigations to clamped and more arbitrarily shaped plates. Rounded corners were examined by Ercoli [1985].

As pointed out by Laura and coworkers referred to in Chapter 10, solution of the eigenvalue problem governing the stability of a thin elastic plate subjected to hydrostatic in-plane loading is easily accomplished when the boundary configuration is natural to one of the common coordinates systems or when we can conformally transform the given domain onto a simpler one, i.e., the unit circle.

They also showed that the determination of an upper bound of the critical in-plane loading of simply supported plates of rectilinear sides is quite straightforward if a theorem by Szegö [1952] is used. The same approach is valid when determining natural frequencies of vibrations of plates of complicated boundary shape.

The conformal mapping technique has been applied for some time to determine the fundamental frequencies of vibrating plates of complicated boundary shape carrying concentrated masses, in-plane forces, stepped variation of thickness, etc. Early contributors were Routh [1860] and Munakata [1954]. Recent applications include nuclear reactors and printed circuit boards [Szilard, 1974; Blevins, 1979].

The grain of a solid propellant rocket motor is usually formed into a circular cylinder bonded to a thin case. This grain quite commonly has a star-shaped internal perforation. The mathematical solution of boundary or eigenvalue problems becomes quite complicated for such exotic geometric configurations. This difficulty can be alleviated to a large extent by conformally transforming the grain cross section into a simpler region such as a circle or annulus. Several studies which were performed on axial shear vibrations of solid and hollow bars were based on the use of conformal mapping and variational or bounding techniques [Laura, Gutierrez, et al., 1976-1980].

Elasticity

The use of conformal mapping in the theory of elasticity is based on the early work of M. Kolossoff [1908, 1915] and N. I. Muskhelishvili [1915, 1946]. Good overviews are found in the books by Green & Zerna [1968] and Goodier & Hodge [1958].

Stresses in beams having holes of arbitrary shape and subjected to pure bending are of particular interest since Wilson [1963] and Richardson [1965] produced promising numerical results with complicated configurations. Problems related to cracks and their propagation are also being solved using conformal mapping in conjunction with other techniques. For examples see Bowie and Isida [in Sih, 1973] and Aifantis and Gerberich [1978].

Thermoelastic problems arising out of heat flow around holes of various shapes were successfully solved in the early 1960s [Florence & Goodier, 1960; Deresiewicz, 1961]; more recently, important work was reported by Takeuti and colleagues [1970, 1980]. Thermal stresses in the presence of ribbon like inclusions were investigated by Sumi [1981].

Elasticity problems are discussed in Chapter 11, including applications to anisotropic media.

Heat Transfer

Chapter 12 addresses heat conduction in the steady state and Chapter 13 examines it in the transient case.

The analysis of flow and heat transfer in ducts of arbitrary shape has been the subject of investigation for many years. A common application of such conduits is in space vehicles. A very general approach has been developed by Sparrow and Haji-Sheikh [1966] and several practical cases were covered in their excellent paper. A different approach was followed by Casarella et al [1966, 1971]. While their technique is not as general as the one mentioned above, it is more convenient for their specific problem.

Laura and his coworkers have made good use of conformal mapping techniques as reported in Chapters 12 and 13. They applied it to transient heat conduction through bars of arbitrary cross-section, such as hollow bars with corrugated surfaces [Laura, Bergmann, & Cortinez, 1988], and in orthotropic two-dimensional media. Applications include composite media taking into account heat generation effects. Nuclear fuel elements for reactors is just one example [Sánchez Sarmiento & Laura, 1981].

In a related field, Siegel and coworkers [1970-1,2; Goldstein, 1969] developed a conformal mapping method for analyzing two-dimensional transient and steady-state solidification problems. The method was applied to the solidification which takes place on a cold plate of finite width immersed in a flowing liquid and to the solidification inside a cooled rectangular channel containing a warm flowing liquid. The investigations dealt with the transient and steady-state shapes of the frozen regions.

It is interesting to point out that the transient shapes of the frozen region were found by mapping the region into a potential plane and then determining the time varying conformal transformation between the potential and physical planes.

Siegel and Snyder [1981] also applied conformal mapping to the analysis of heat transfer in porous regions with curved boundaries.

Fluid Flow

Conformal mapping has long played an important role in hydrodynamics and aerodynamics. For the contributions to aerodynamics by Kutta and Joukowski, see Tani [1979]. Initially the problems were restricted to fluids which are neither viscous nor compressible. The Joukowski profiles were--and continue to be--useful models for the analysis of airfoils. The Schwarz-Christoffel transformation proved helpful in the tracing of free streamlines in hydrodynamics problems. A velocity plane, the hodograph, was conceived by Helmholtz and Kirchhoff for free streamline flow without boundaries as in efflux and deflected jets. Hodographic transformations still are useful tools today.

Airfoil design by conformal mapping continues to be of importance in the aircraft industry [Ives, 1976]. Current airfoil applications are found in the design of transonic wings [Chen et al, 1985] and blade design for helicopters, windmills, hydroplanes, turbines, and spoilers. Many modern applications deal with foils in cascade. It is interesting to note that the important numerical conformal mapping method of Theodorsen [1931, 1933] arose in work on wing sections.

While flow of water through porous media [Sokolnikoff, 1941; Fox & McNamee, 1948] remains a classical textbook example, its occurrence in practice remains no less current and physically risky. Intrusion of salt water [Hunt, 1985] and erosion under weirs [Kumar et al, 1982] are recent examples. Other interesting applications mentioned in Chapter 14 include ships’ hulls, ocean wave movement and aerosol flow, magnetohydrodynamics, wakes, and transient phenomena.

Other Areas

The application chapters do not cover all possible areas in which conformal mapping is used today. Therefore the following brief remarks will have to suffice to let the reader catch a glimpse of some interesting applications which are not covered in Chapters 7 – 14.

Much of the work cited below is not directly related to topological considerations of field patterns. Instead, conformal transformations may have been applied primarily as a tool to solve equations. The Smith chart, which is used in transmission line analysis to find reflection factors based on line and load impedances, is an example of such an application if one views the transformation from an unbounded, cartesian impedance chart to a bounded, polar chart as a conformal mapping (of the bilinear type). Similar examples arise in uses of the complex frequency plane to determine the stability of automatic control systems. A design method for regulators that uses conformal mapping for pole assignment was recently described by Kim and Furuta [1988].

In particle physics the method of pole extrapolations is used to obtain the coupling constants of elementary particles. As described by Ciulli and Fisher [1961], conformal transformations can be used effectively to map energy or scattering angle; then the extrapolation to the desired pole can be performed by expanding the amplitudes or differential cross-sections in the mapped variables. See Locher and Mizutani [1978] and Langbein [1977] for further references and details on obtaining the best expansions. Other reports on scattering processes in various contexts include the papers by Fuda et al [1980], who address parameterization of unknown discontinuities in N/D equations, by Klarsfeld [1981], who evaluates relativistic scattering amplitudes in the physical region, and by Parida and coworkers [1978-1980], who address diffraction scattering. The work of Parida et al is critiqued in a paper by Hokler and Stefanescu [1982], followed by a rebuttal [Parida, 1982]. The scattering of electromagnetic and acoustic waves on a less microscopic level is discussed by J. J. Bowman et al [1969, p 37] and the references they cite.

A method to map the obliquely intersecting equipotential and current flow lines in Hall effect sensors has found use in analyzing the oblique reflection from boundaries in two-dimensional Brownian motion [Trefethen & Williams, 1986]. This procedure can also be extended to certain queueing problems in traffic analysis.

In a 1977 paper, Yang defined a conformal mapping of gage fields. He found that the mapping preserves sourcelessness in four dimensions for any signatures of the metric. Orthogonality is also preserved, as expected.

An interesting problem in geophysics was solved by Roy et al [1982] by means of the Schwarz-Christoffel method of conformal mapping. Telluric currents or earth currents flow through the earth as large sheet currents. They may be considered as being fed by two infinitely long rotating line electrodes located at infinite distance serving as current sources. The problem consists of computing telluric fields and their gradients over a step fault.

A two-dimensional model of a magnetosphere with a magnetic dipole direction parallel to the flow of solar wind plasma may adequately describe the interactions which occur in the solar system. Biernat et al [1981] applied conformal mapping to analyze this model.

The problem of modeling a gravitational or magnetic mass is addressed by Strakhov [1978]. He employed series expansions to find suitable mapping functions.

Toroidal plasma containment systems generally have noncircular cross-sections because of the influence of high-ß and the shape of the conducting walls. Numerically constructed conformal mappings greatly facilitate the solution of equilibrium and stability problems by exploiting the analytical knowledge from simpler configurations, according to Goedbloed [1981].

In optics a bag model can successfully describe the effective potential between opposite, localized color changes when they are widely separated. Steinhardt [1980] applied this model to the case of two quark-antiquark pairs in a line and showed how the bag surface, the energy of the configuration, and the color field distribution can be found through a series of conformal mappings.

et al, 1986]. Wechsler and Zimmermann [1988] described a pattern recognition scheme which combines conformal mapping with an associative memory.

1.3 GROWTH IN SCOPE OF APPLICATIONS

The conformal transformation method has been successfully applied in a wide variety of fields as one can deduce from this brief survey: from acoustic waveguide studies to plate buckling theory; from the analysis of vibrating printed circuit boards to the determination of the temperature field in the nuclear fuel arrangement of fast breeder reactors; from studies on air flow past multielement airfoils to the solution of geophysical and oceanographic problems; from microstrip design to analysis of the earth-ionosphere waveguide. This represents significant progress since Ptolemy conformally transformed the celestial sphere into a plane.

The pace of activity is quickening and the papers being published are growing steadily in scope and number. Publications listed in Mathematical Reviews are currently appearing at the rate of about 150 a year, over 2500 since 1973, and publications listed in COMPENDEX or INSPEC number about 100 a year, over 1700 since 1973. There are ample opportunities for the practitioners in science and engineering to exploit the many recent accomplishments of their colleagues in mathematics, while mathematicians can find much stimulation in the application problems posed by engineers and physicists.

Chapter 2

BASIC MATHEMATICAL CONCEPTS

In this chapter the properties of functions of complex variables are reviewed to the extent necessary for an understanding of the capabilities and limitations of conformal mapping methods.

2.1 TRANSFORMATION OF COORDINATES

The simplest transformations involve the selection of new coordinate systems when the ones in which a problem has been specified are not convenient for analysis. Such transformations need not involve complex numbers. Consider for example the ellipse shown in various orientations and sizes in Figures 2.1(a) through 2.1(d). The particular coordinate axes in (a) and (b) fail to exploit symmetry; in (c) the ellipse may be too small; in (d) the representation suffers from a mixture of the foregoing shortcomings. Instead of laboring with the original x,y coordinate systems shown, one could select in each case a new set of axes u,v which appear as broken lines in Figures 2.1 (a), (b), and (d) to obtain the effect of Figure 2.1(e).

Figure 2.1(a) reveals that a translation of the ellipse (or, equivalently, a selection of new axes u,v parallel to the original x,y, system) would transform the configuration of Figure 2.1 (a) into that of Figure 2.1(e). The transformation would be governed by

(2.1)

Similarly one could transform Figure 2.1(b) into (e) by rotating it through an angle of δ; Figure 2.1(c) into (e) by scaling; and Figure 2.1(d) into (e) by a combination of the foregoing taken in unison or sequentially. This latter, more general transformation might take the form

(2.2)

Figure 2.1 On Transformation of Coordinates

The desired final image (e) is obtained from (a) by translation, from (b) by rotation, from (c) by magnification, and from (d) by applying all of the foregoing.

When an analysis has already been carried out on the original configuration, the answers can give a clue as to what transformation one may wish to use the next time. For instance, Maxwell found the solution of the potential field problem associated with the fringing flux of a parallel plate capacitor as in Figure 2.2 to be of the form

(2.3)

This is the same transformation as given earlier in Figure 1.3, except that we have elected to separate the real and imaginary parts of z and w to facilitate the following discussion. Each line v=constant corresponds to a potential line in Figure 2.2 and each line u=constant corresponds to a flux or flow line. (Later, in Chapter 5, we will introduce the less ambiguous term flux tube to describe the flow bounded by two adjacent flux lines when shown in a plane.) For each point generated by the intersection of lines of u=constant and v=constant in the u,v-plane there is a corresponding point in the x,y-plane given by Equations (2.3). Since lines of constant u and v form a simple grid in the u,v-plane as shown in Figure 1.3(b), an analysis of capacitance, electric field strength, potential, etc. is carried out with ease in the u,v-plane. The results can then be transformed into the physical x,y-plane. We need not repeat Maxwell’s original derivation every time a similar problem is to be solved, even if another kind of potential field is involved (such as the one encountered in viscous fluid flow) as long as the boundary configuration is of the same general form.

Figure 2.2 Lines of Flux (u) and Lines of Equipotential (v) in the Top Half of Figure 1.3.

Since we deal with orthogonal field lines (potential and flux) in our field problems, and since coordinate systems in general consist of orthogonal grids as well, it is not far fetched to think of a solution as defining a coordinate transformation. It is in this spirit that tables of transformations of coordinates have been especially constructed for analyses dealing with potential fields. Often configurations in three dimensions with axial symmetry are included. [Moon and Spencer, 1971; Korn and Korn, 1961]

Returning to our two-dimensional examples, consider the desired u,v coordinate system which is shown superimposed on the original x,y-plane in Figure 2.3. Let that illustration be a magnification of an infinitesimal segment from Figure 2.2. We note that small deviations in x,y must be related to the corresponding deviations in u,v by similar triangles, yielding the relationships expressed by Equations (2.4) and (2.5):

(2.4)

also

(2.5)

Figure 2.3 Small Changes in x,y and Corresponding Changes in u,v.

After differentiating Equations (2.4) and eliminating the mixed partial derivative terms--and performing similar operations on Equations (2.5)--we find that the transformation functions u(x,y) and v(x,y), as well as x(u,y) and y(u,v), satisfy Laplace’s equation in two dimensions: and

(2.6)

We have noted a few of the conditions which a transformation must satisfy if orthogonality between the potential and flux lines is to be maintained during the process of transformation, but we have not drawn on complex analysis so far. Why bother then with complex variables and their functions? The answer has to do with our main question: How do we find suitable transformations? Rarely can we expect to find a ready-made transformation such as the one based on Maxwell’s solution for the semi-infinite parallel plates. The task is made considerably easier if we express x,y and u,v as components of complex variables z and w:

(2.7)

or, in polar notation

(2.8)

The subscripts z and w will be omitted when polar coordinates are used with only one of the planes and the context is unambiguous.

Let w = f(z) denote the function which transforms or maps a configuration from the z-plane onto the w-plane and let z = f – 1(w) perform the inverse operation. Then, for a starter, we can illustrate how much simpler the elementary transformations discussed at the beginning of this section can be written. Replace the y-coordinate in Figures 2.1(a) through (d) by the imaginary term i y so that we can write

for translation:

(2.9)

for rotation:

(2.10)

for scaling:

(2.11)

all combined:

(2.12)

One of the oldest transformations is the representation of the world’s surface on a flat sheet. Ptolemy used a stereographic projection (around 150 A.D.); the Mercator projection came 1400 years later. Euler interested himself in such projections at St. Petersburg Academy in the 1700s when the Academy was commissioned to produce more accurate maps of Russia. Euler employed a number of concepts, such as the complex plane, which would later become part of conformal mapping. Gauss also approached conformal mapping by way of geodesy (1820-1830).

2.2 TRANSFORMATION BY MEANS OF COMPLEX FUNCTIONS

Our task is now one of finding a complex function w = f(z) of the complex variable z which will take a configuration consisting of points, lines, angles, and regions (domains) in the complex z-plane and map it into a simpler and more readily analyzable configuration in the complex w-plane. Unless otherwise specified we will be concerned with planar fields. The particular form of the function f(z) will depend on the specific boundaries and the application at hand.

For our purposes it is important that the mapping be conformal, i.e., that the relative proportions of neighboring small line segments and the angles of intersecting line segments be preserved at the microscopic level during transformation. Any orthogonal set of field lines in the original or physical z-plane would therefore appear as another set of orthogonal lines in the w-plane which serves as the mathematical model.

The configurations in the z- and w-planes are images of each other. A mapping will be one-to-one if f(z) is linear or bilinear (a ratio of linear functions). If not, ambiguities may arise, but these are frequently resolved by knowledge of the physical realities underlying the mathematical construction or by introducing overlaid image planes (Riemann surfaces).

Finding a suitable transformation is not necessarily easy. One approach is to select a desired end product, such as a rectangular or circular configuration in the w-plane, and pick or construct any number of functions z = g(w) to see what image results in the z-plane. If we are lucky we can extract the forward transformation w = f(z) from what was essentially the inverse, z = g(w) = f – 1(w). This is not always easy and sometimes not even possible. The parallel plate capacitor presents us with such a problem. The equations for z = f – 1(w) or for its components x = x(u,v) and y = y(u,v) are given by equations which appeared earlier and which are repeated here,

(2.13)

(2.14)

but there is no convenient way of expressing w explicitly in terms of z.

However, it is not always necessary to have an explicit formulation for the forward mapping from z to w. As long as the boundaries and other key elements of the original field can be identified in the w-plane, the necessary field analyses can be carried out there and the results transferred to the original plane by means of the available f – 1(w).

Each of the elementary transformations described in Section 2.1 has an inverse which is easily found. The same is true of the widely applicable bilinear transformation

(2.15)

also known as the Moebius or linear fractional transformation, a particular example of which was given in Figure 1.2.

2.3 ANALYTIC FUNCTIONS

The properties we need in a mapping function are to be found in analytic functions. A complex valued function of a complex variable is said to be analytic in a region of interest if it is single-valued and differentiable in that region. (It may also be multiple valued as long as it is single valued on a Riemann surface.) The terms regular and holomorphic are also used for such a function. They refer to differentiability at a point and in the entire region respectively.

Single-valuedness is essential for unambiguous, one-to-one mappings. Differentiability is desired so that angles will not change under transformation (isogonality) and that all sides of a small triangle describing an angle will stretch or contract in the same ratio |dw/dzl and rotate by the same amount arg(dw/dz). Differentiability also implies continuity. Thus a set of orthogonal, continuous curves, each of which satisfies Laplace’s equation in the original plane, will satisfy Laplace’s equation when mapped onto the model plane.

Another way of defining analyticity is to say that f(z) is analytic at point p if in its neighborhood it can be expanded in a Taylor series,

(2.16)

Such a condition amounts to a shorthand version of requiring single valuedness, continuity, and differentiability.

Let us briefly review what we mean by