CHAPTER 17

Conformal Mapping

Conformal mappings are invaluable to the engineer and physicist as an aid in solving problems in potential theory. They are a standard method for solving boundary value problems in two-dimensional potential theory and yield rich applications in electrostatics, heat flow, and fluid flow, as we shall see in Chapter 18.

The main feature of conformal mappings is that they are angle-preserving (except at some critical points) and allow a geometric approach to complex analysis. More details are as follows. Consider a complex w = f(z) function defined in a domain D of the z–plane; then to each point in D there corresponds a point in the w-plane. In this way we obtain a mapping of D onto the range of values of f (z) in the w- plane. In Sec. 17.1 we show that if f(z) is an analytic function, then the mapping given by w = f(z) is a conformal mapping, that is, it preserves angles, except at points where the derivative f′(z) is zero. (Such points are called critical points.)

Conformality appeared early in the history of construction of maps of the globe. Such maps can be either “conformal,” that is, give directions correctly, or “equiareal,” that is, give areas correctly except for a scale factor. However, the maps will always be distorted because they cannot have both properties, as can be proven, see [GenRef8] in App. 1. The designer of accurate maps then has to select which distortion to take into account.

Our study of conformality is similar to the approach used in calculus where we study properties of real functions y = f(x) and graph them. Here we study the properties of conformal mappings (Secs. 17.1–17.4) to get a deeper understanding of the properties of functions, most notably the ones discussed in Chap. 13. Chapter 17 ends with an introduction to Riemann surfaces, an ingenious geometric way of dealing with multivalued complex functions such as w = sqrt (z) and w = ln z.

So far we have covered two main approaches to solving problems in complex analysis. The first one was solving complex integrals by Cauchy's integral formula and was broadly covered by material in Chaps. 13 and 14. The second approach was to use Laurent series and solve complex integrals by residue integration in Chaps. 15 and 16. Now, in Chaps. 17 and 18, we develop a third approach, that is, the geometric approach of conformal mapping to solve boundary value problems in complex analysis.

Prerequisite: Chap. 13.

Sections that may be omitted in a shorter course:17.3 and 17.5.

References and Answers to Problems: App. 1 Part D, App. 2.

17.1 Geometry of Analytic Functions: Conformal Mapping

We shall see that conformal mappings are those mappings that preserve angles, except at critical points, and that these mappings are defined by analytic functions. A critical point occurs wherever the derivative of such a function is zero. To arrive at these results, we have to define terms more precisely.

A complex function

of a complex variable z gives a mapping of its domain of definition D in the complex z-plane into the complex w-plane or onto its range of values in that plane.¹ For any point z₀ in D the point w₀ = f(z₀) is called the image of z₀ with respect to f. More generally, for the points of a curve C in D the image points form the image of C; similarly for other point sets in D. Also, instead of the mapping by a function w = f(z) we shall say more briefly the mapping. w = f(z).

Fig. 378. Mapping w = z². Lines |z| = const, arg z = const and their images in the w-plane

Fig. 379. Images of x = const, y = const under w = z²

Conformal Mapping

A mapping w = f(z) is called conformal if it preserves angles between oriented curves in magnitude as well as in sense. Figure 380 shows what this means. The angle α (0 α π) between two intersecting curves C₁ and C₂ is defined to be the angle between their oriented tangents at the intersection point z₀. And conformality means that the images and of C₁ and C₂ make the same angle as the curves themselves in both magnitude and direction.

Fig. 381. Secant and tangent of the curve C

In the remainder of this section and in the next ones we shall consider various conformal mappings that are of practical interest, for instance, in modeling potential problems.

Fig. 382. Mapping by w = zⁿ

Fig. 384. Joukowski airfoil

Fig. 385. Mapping by w = e^z

Fig. 386. Mapping by w = e^z

Magnification Ratio. By the definition of the derivative we have

Therefore, the mapping w = f(z) magnifies (or shortens) the lengths of short lines by approximately the factor |f′(z₀)|. The image of a small figure conforms to the original figure in the sense that it has approximately the same shape. However, since f′(z) varies from point to point, a large figure may have an image whose shape is quite different from that of the original figure.

More on the Condition f′(z) ≠ 0. From (4) in Sec. 13.4 and the Cauchy–Riemann equations we obtain

that is,

This determinant is the so-called Jacobian (Sec. 10.3) of the transformation w = f(z) written in real form u = u(x, y), v = v(x, y). Hence f′(z₀) ≠ 0 implies that the Jacobian is not 0 at z₀. This condition is sufficient that the mapping w = f(z) in a sufficiently small neighborhood of z₀ is one-to-one or injective (different points have different images). See Ref. [GenRef4] in App. 1.

17.2 Linear Fractional Transformations (Möbius Transformations)

Conformal mappings can help in modeling and solving boundary value problems by first mapping regions conformally onto another. We shall explain this for standard regions (disks, half-planes, strips) in the next section. For this it is useful to know properties of special basic mappings. Accordingly, let us begin with the following very important class.

The next two sections discuss linear fractional transformations. The reason for our thorough study is that such transformations are useful in modeling and solving boundary value problems, as we shall see in Chapter 18. The task is to get a good grasp of which conformal mappings map certain regions conformally onto each other, such as, say mapping a disk onto a half-plane (Sec. 17.3) and so forth. Indeed, the first step in the modeling process of solving boundary value problems is to identify the correct conformal mapping that is related to the “geometry” of the boundary value problem.

The following class of conformal mappings is very important. Linear fractional transformations (or Möbius transformations) are mappings

where a, b, c, d are complex or real numbers. Differentiation gives

This motivates our requirement ad − bc ≠ 0. It implies conformality for all z and excludes the totally uninteresting case w′ ≡ 0 once and for all. Special cases of (1) are

The proof in this example suggests the use of z and instead of x and y, a general principle that is often quite useful in practice.

Surprisingly, every linear fractional transformation has the property just proved:

Extended Complex Plane

The extended complex plane (the complex plane together with the point ∞ in Sec. 16.2) can now be motivated even more naturally by linear fractional transformations as follows.

To each z for which cz + d ≠ 0 there corresponds a unique w in (1). Now let c ≠ 0. Then for z = −d/c we have cz + d = 0, so that no w corresponds to this z. This suggests that we let w = ∞ be the image of z = −d/c.

Also, the inverse mapping of (1) is obtained by solving (1) for z; this gives again a linear fractional transformation

When c ≠ 0, then cw − a = 0 for w = a/c, and we let a/c be the image of z = ∞. With these settings, the linear fractional transformation (1) is now a one-to-one mapping of the extended z-plane onto the extended w-plane. We also say that every linear fractional transformation maps “the extended complex plane in a one-to-one manner onto itself.”

Our discussion suggests the following.

General Remark. If z = ∞, then the right side of (1) becomes the meaningless expression (a · ∞ + b)/(c · ∞ + d). We assign to it the value w = a/c if c ≠ 0 and w = ∞ if c = 0.

Fixed Points

Fixed points of a mapping w = f(z) are points that are mapped onto themselves, are “kept fixed” under the mapping. Thus they are obtained from

The identity mapping w = z has every point as a fixed point. The mapping w = has infinitely many fixed points, w = 1/z has two, a rotation has one, and a translation none in the finite plane. (Find them in each case.) For (1), the fixed-point condition w = z is

For c ≠ 0 this is a quadratic equation in z whose coefficients all vanish if and only if the mapping is the identity mapping w = z (in this case, a = d ≠ 0, b = c = 0). Hence we have

To make our present general discussion of linear fractional transformations even more useful from a practical point of view, we extend it by further facts and typical examples, in the problem set as well as in the next section.

17.3 Special Linear Fractional Transformations

We continue our study of linear fractional transformations. We shall identify linear fractional transformations

that map certain standard domains onto others. Theorem 1 (below) will give us a tool for constructing desired linear fractional transformations.

A mapping (1) is determined by a, b, c, d, actually by the ratios of three of these constants to the fourth because we can drop or introduce a common factor. This makes it plausible that three conditions determine a unique mapping (1):

Mapping of Standard Domains by Theorem 1

Using Theorem 1, we can now find linear fractional transformations of some practically useful domains (here called “standard domains”) according to the following principle.

Principle. Prescribe three boundary points z₁, z₂, z₃ of the domain D in the z-plane. Choose their images w₁, w₂, w₃ on the boundary of the image D* of D in the w-plane. Obtain the mapping from (2). Make sure that D is mapped onto not D*, onto its complement. In the latter case, interchange two w-points. (Why does this help?)

Fig. 388. Linear fractional transformation in Example 1

Mapping half-planes onto half-planes is another task of practical interest. For instance, we may wish to map the upper half-plane y 0 onto the upper half-plane v 0. Then the x-axis must be mapped onto the u-axis.

Mappings of disks onto disks is a third class of practical problems. We may readily verify that the unit disk in the z-plane is mapped onto the unit disk in the w-plane by the following function, which maps z₀ onto the center w = 0.

To see this, take |z| = 1, obtaining, with c = ₀ as in (3),

Hence

from (3), so that |z| = 1 maps onto |w| = 1, as claimed, with z₀ going onto 0, as the numerator in (3) shows.

Formula (3) is illustrated by the following example. Another interesting case will be given in Prob. 17 of Sec. 18.2.

Fig. 389. Mapping in Example 5

Fig. 390. Mapping in Example 6

This is the end of our discussion of linear fractional transformations. In the next section we turn to conformal mappings by other analytic functions (sine, cosine, etc.).

17.4 Conformal Mapping by Other Functions

We shall now cover mappings by trigonometric and hyperbolic analytic functions. So far we have covered the mappings by zⁿ and e^z (Sec. 17.1) as well as linear fractional transformations (Secs. 17.2 and 17.3).

Sine Function. Figure 391 shows the mapping by

Fig. 391. Mapping w = u + iv = sin z

Hence

Since sin z is periodic with period 2π, the mapping is certainly not one-to-one if we consider it in the full z-plane. We restrict z to the vertical strip in Fig. 391. Since f′(z) = cos z = 0 at the mapping is not conformal at these two critical points. We claim that the rectangular net of straight lines x = const and y = const in Fig. 391 is mapped onto a net in the w-plane consisting of hyperbolas (the images of the vertical lines x = const) and ellipses (the images of the horizontal lines y = const) intersecting the hyperbolas at right angles (conformality!). Corresponding calculations are simple. From (2) and the relations sin² x + cos² x = 1 and cosh² y − sinh² y = 1 we obtain

Exceptions are the vertical lines which are “folded” onto u −1 and u 1 (v = 0), respectively.

Figure 392 illustrates this further. The upper and lower sides of the rectangle are mapped onto semi-ellipses and the vertical sides onto −cosh 1 u −1 and 1 u cosh 1 (v = 0), respectively. An application to a potential problem will be given in Prob. 3 of Sec. 18.2.

Fig. 392. Mapping by w = sin z

Cosine Function. The mapping w = cos z could be discussed independently, but since

we see at once that this is the same mapping as sin z preceded by a translation to the right through π units.

Hyperbolic Sine. Since

the mapping is a counterclockwise rotation Z = iz through π (i.e., 90°), followed by the sine mapping Z* = sin Z, followed by a clockwise 90°-rotation w = −iZ*.

Hyperbolic Cosine. This function

defines a mapping that is a rotation Z = iz followed by the mapping w = cos Z.

Figure 393 shows the mapping of a semi-infinite strip onto a half-plane by w = cosh z. Since cosh 0 = 1, the point z = 0 is mapped onto w = 1. For real z = x 0, cosh z is real and increases with increasing x in a monotone fashion, starting from 1. Hence the positive x-axis is mapped onto the portion u 1 of the u-axis.

For pure imaginary z = iy we have cosh iy = cos y. Hence the left boundary of the strip is mapped onto the segment 1 u −1 of the u-axis, the point z = πi corresponding to

On the upper boundary of the strip, y = π, and since sin π = 0 and cos π = −1, it follows that this part of the boundary is mapped onto the portion u −1 of the u-axis. Hence the boundary of the strip is mapped onto the u-axis. It is not difficult to see that the interior of the strip is mapped onto the upper half of the w-plane, and the mapping is one-to-one.

This mapping in Fig. 393 has applications in potential theory, as we shall see in Prob. 12 of Sec. 18.3.

Fig. 393. Mapping by w = cosh z

Tangent Function. Figure 394 shows the mapping of a vertical infinite strip onto the unit circle by w = tan z, accomplished in three steps as suggested by the representation (Sec. 13.6)

Hence if we set Z = e^2iz and use 1/i = −i, we have

We now see that w = tan z is a linear fractional transformation preceded by an exponential mapping (see Sec. 17.1) and followed by a clockwise rotation through an angle π(90°).

The strip is and we show that it is mapped onto the unit disk in the w-plane. Since Z = e^2iz = e^{−2y + 2ix}, we see from (10) in Sec. 13.5 that |Z| = e^−2y, Arg Z = 2x. Hence the vertical lines x = −π/4, 0, π/4 are mapped onto the rays Arg Z = −π/2, 0, π/2 respectively. Hence S is mapped onto the right Z-half-plane. Also |Z| = e^−2y < 1 if y > 0 and |Z| > 1 if y < 0. Hence the upper half of S is mapped inside the unit circle |Z| = 1 and the lower half of S outside |z| = 1, as shown in Fig. 394.

Now comes the linear fractional transformation in (6), which we denote by g(Z):

For real Z this is real. Hence the real Z-axis is mapped onto the real W-axis. Furthermore, the imaginary Z-axis is mapped onto the unit circle |W| = 1 because for pure imaginary Z = iY we get from (7)

The right Z-half-plane is mapped inside this unit circle |W| = 1, not outside, because Z = 1 has its image g(1) = 0 inside that circle. Finally, the unit circle |z| = 1 is mapped onto the imaginary W-axis, because this circle is Z = e^iø, so that (7) gives a pure imaginary expression, namely,

From the W-plane we get to the w-plane simply by a clockwise rotation through π/2; see (6).

Together we have shown that w = tan z maps S: −π/4 < Re z < π/4 onto the unit disk |w| < 1, with the four quarters of S mapped as indicated in Fig. 394. This mapping is conformal and one-to-one.

Fig. 394. Mapping by w = tan z

17.5 Riemann Surfaces. Optional

One of the simplest but most ingeneous ideas in complex analysis is that of Riemann surfaces. They allow multivalued relations, such as or w = ln z, to become single-valued and therefore functions in the usual sense. This works because the Riemann surfaces consist of several sheets that are connected at certain points (called branch points). Thus will need two sheets, being single-valued on each sheet. How many sheets do you think w = ln z needs? Can you guess, by recalling Sec. 13.7? (The answer will be given at the end of this section). Let us start our systematic discussion.

The mapping given by

is conformal, except at z = 0, where w′ = 2z = 0. At z = 0, angles are doubled under the mapping. Thus the right z-half-plane (including the positive y-axis) is mapped onto the full w-plane, cut along the negative half of the u-axis; this mapping is one-to-one. Similarly for the left z-half-plane (including the negative y-axis). Hence the image of the full z-plane under w = z² “covers the w-plane twice” in the sense that every w ≠ 0 is the image of two z-points; if z₁ is one, the other is −z₁. For example, z = i and −i are both mapped onto w = −1.

Now comes the crucial idea. We place those two copies of the cut w-plane upon each other so that the upper sheet is the image of the right half z-plane R and the lower sheet is the image of the left half z-plane L. We join the two sheets crosswise along the cuts (along the negative u-axis) so that if z moves from R to L, its image can move from the upper to the lower sheet. The two origins are fastened together because w = 0 is the image of just one z-point, z = 0. The surface obtained is called a Riemann surface (Fig. 395a). w = 0 is called a “winding point” or branch point. w = z² maps the full z-plane onto this surface in a one-to-one manner.

By interchanging the roles of the variables z and w it follows that the double-valued relation

becomes single-valued on the Riemann surface in Fig. 395a, that is, a function in the usual sense. We can let the upper sheet correspond to the principal value of Its image is the right w-half-plane. The other sheet is then mapped onto the left w-half-plane.

Fig. 395. Riemann surfaces

Similarly, the triple-valued relation becomes single-valued on the three-sheeted Riemann surface in Fig. 395b, which also has a branch point at z = 0.

The infinitely many-valued natural logarithm (Sec. 13.7)

becomes single-valued on a Riemann surface consisting of infinitely many sheets, w = Ln z corresponds to one of them. This sheet is cut along the negative x-axis and the upper edge of the slit is joined to the lower edge of the next sheet, which corresponds to the argument π < θ 3π, that is, to

The principal value Ln z maps its sheet onto the horizontal strip −π < v π. The function w = Ln z + 2πi maps its sheet onto the neighboring strip π < v 3π, and so on. The mapping of the points z ≠ 0 of the Riemann surface onto the points of the w-plane is one-to-one. See also Example 5 in Sec. 17.1.

CHAPTER 17 REVIEW QUESTIONS AND PROBLEMS

1. What is a conformal mapping? Why does it occur in complex analysis?
2. At what points are w = z⁵ − z and w = cos (πz²) not conformal?
3. What happens to angles at z₀ under a mapping w = f(z) if f′(z₀) = 0, f″(z₀) = 0, f′″(z₀) ≠ 0?
4. What is a linear fractional transformation? What can you do with it? List special cases.
5. What is the extended complex plane? Ways of introducing it?
6. What is a fixed point of a mapping? Its role in this chapter? Give examples.
7. How would you find the image of x = Re z = 1 under w = iz, z², e^z, 1/z?
8. Can you remember mapping properties of w = ln z?
9. What mapping gave the Joukowski airfoil? Explain details.
10. What is a Riemann surface? Its motivation? Its simplest example.

11–16 MAPPING w = z²

Find and sketch the image of the given region or curve under w = z².

• 11. 1 < |z| < 2, |arg z| < π/8
• 12.
• 13. −4 < xy < 4
• 14. 0 < y < 2
• 15. x = −1, 1
• 16. y = −2, 2

17–22 MAPPING w = 1/z

Find and sketch the image of the given region or curve under w = 1/z.

• 17. |z| < 1
• 18. |z| < 1, 0 < arg z < π/2
• 19. 2 < |z| < 3, y > 0
• 20. 0 arg z π/4
• 21.
• 22. z = 1 + iy (−∞ < y < ∞)

23–38 LINEAR FRACTIONAL TRANSFORMATIONS (LFTs)

Find the LFT that maps

• 23. −1, 0, 1 onto 4 + 3i, 5i/2, 4 − 3i, respectively
• 24. 0, 2, 4 onto respectively
• 25. 1, i, −i onto i, −1, 1, respectively
• 26. 0, 1, 2 onto 2i, 1 + 2i, 2 + 2i, respectively
• 27. 0, 1, ∞ onto ∞, 1, 0, respectively
• 28. −1, −i, i onto 1 −i, 2, 0, respectively

29–34 FIXED POINTS

Find the fixed points of the mapping

• 29. w = (2 + i)z
• 30. w = z⁴ + z − 64
• 31. w = (3z + 2)/(z − 1)
• 32. (2iz − 1)/(z + 2i)
• 33. w = z⁵ + 10z³ + 10z
• 34. w = (iz + 5)/(5z + i)

35–40 GIVEN REGIONS

Find an analytic function w = f(z) that maps

• 35. The infinite strip 0 < y < π/4 onto the upper half-plane v > 0.
• 36. The quarter-disk |z| < 1, x > 0, y > 0 onto the exterior of the unit circle |w| = 1.
• 37. The sector 0 < arg z < π/2 onto the region u < 1.
• 38. The interior of the unit circle |z| = 1 onto the exterior of the circle |w + 2| = 2.
• 39. The region x > 0, y > 0, xy < c onto the strip 0 < v < 1.
• 40. The semi-disk |z| < 2, y > 0 onto the exterior of the circle |w − π| = π.

SUMMARY OF CHAPTER 17 Conformal Mapping

A complex function w = f(z) gives a mapping of its domain of definition in the complex z-plane onto its range of values in the complex w-plane. If f(z) is analytic, this mapping is conformal, that is, angle-preserving: the images of any two intersecting curves make the same angle of intersection, in both magnitude and sense, as the curves themselves (Sec. 17.1). Exceptions are the points at which f′(z) = 0 (“critical points,” e.g. z = 0 for w = z²).

For mapping properties of e^z, cos z, sin z etc. see Secs. 17.1 and 17.4.

Linear fractional transformations, also called Möbius transformations

(ad − bc ≠ 0) map the extended complex plane (Sec. 17.2) onto itself. They solve the problems of mapping half-planes onto half-planes or disks, and disks onto disks or half-planes. Prescribing the images of three points determines (1) uniquely.

Riemann surfaces (Sec. 17.5) consist of several sheets connected at certain points called branch points. On them, multivalued relations become single-valued, that is, functions in the usual sense. Examples. For we need two sheets (with branch point 0) since this relation is doubly-valued. For w = ln z we need infinitely many sheets since this relation is infinitely many-valued (see Sec. 13.7).