II. CONFORMAL MAPPING

This chapter is devoted to the mappings, generated by analytic functions, so-called conformal mappings.

The concept of conformal mapping arises with a number of important concepts of mathematics. Arising from physical tasks, it finds many important applications in different fields of physics - the method of conformal mappings solves successfully practical problems of hydro- and aero-dynamics, the theory of elasticity, the theory of electro-magnetic and heat fields, etc.

Different problems, connected with conformal mappings, were solved by D'Alembert, Euler and Karl Friedrich Gauss (1777-1855). Based on their results, Bernhard Riemann laid in his dissertation "Basic general theory of functions of a complex variable" (1851) the foundations of the geometrical theory of functions and, in particular, proved (although not correctly) the basic theorem on the possibility of conformal mapping of arbitrary simply connected domains onto each other. In his studies, Riemann, following Euler, employed physical presentations, linked to conformal mappings.

Beginning with the Nineteenth Century, conformal mappings were applied as mathematical apparatus to the study of the mechanics of continuous media. Among the initiators of such applications, leaders are N.E.Joukowskii and S.A.Chaplygin (hydro- and aero-dynamics), G.V.Kolosov and N.I.Muskhelishvili (theory of elasticity).

2.1 General propositions. Examples

We will study in this section the concept of conformal mapping and the general principles of the theory of conformal mapping. We will not be able to prove many of these (proofs demand material beyond the framework of this book) and restrict ourselves to facts, which are essential principles, and illustrate them by a number of examples.

27. Concept of conformal mapping Let there be given the continuous and mutually single-valued mapping of the region D onto some region D*

Moreover, assume that the functions u(x, y) and v(x, y) are differentiable in those domains. We fix an arbitrary point z0 of D and in its neighbourhood replace the increments of the functions u and v by differentials. By definition, one may then write

where the partial derivatives relate to the point z0, and h1, h2 tend to zero as Dr ® 0. Replacement of increments of the differentials then reduces to omission in (2) of the terms h1Dr and h2Dr which are small of higher order than the remaining terms in these formulae ( we assume that

differ from zero).

Speaking geometrically, this is equivalent to the replacement of the mapping w = f(z) by the mapping

which is referred to as the principal linear part of the mapping (1). Mapping (3) may be given the form

do not depend on x and y. This represents the so-called linear transformation of the (x, y)-plane.

We will now note the basic properties of linear transformations. Every linear transformation (4) is defined single-valuedly in the entire z - plane; we will assume. that the determinant

is non-zero*; then the transformation, inverse to (4),

is likewise single-valued in the entire w-plane. Thus, for D ¹ 0, not only every z corresponds to one value of w, but also to every value of w there corresponds one value of z, i.e., the transformation (4) yields a mutually single-valued mapping of the entire z-plane onto the entire w-plane.

* In the case D = 0, the mapping is is said to degenerate.

Consider a bundle of parallel straight lines with the angular coefficient k = tan j, i.e., the straight lines y = kx + C. Replacing here x and y using (6), we see that there corresponds to this bundle likewise parallel straight lines

with the angular coefficient

Hence, Mapping (4) transforms squares on the z-plane into parallelograms in the w-plane.

Let z0 = x0 + iy0 and w0 = u0+ iv0 be a pair of points, corresponding to each other for the mapping (4). Then, this mapping may be given the form

and its inverse the form

(it is sufficient for the derivation of (7) and (8) to substitute in (4) and (6) x = x0, ···, v = v0 and subtract from (4) and (6) the equations obtained). Taking into consideration (8), we may assert that the circle with centre at z0:

for the transformation (4) becomes the ellipse with centre at the point w0:

We pose now the question: What conditions must be met by the coefficients of Transformation (4) in order that it maps circles again into circles? By (9), it is necessary and sufficient for this that

The first of these yields a/d = - b/c = l, whence a = lad, b = -l.c. Substituting these results into the second equation (10), we find l² = 1 or l = ±1.

The case l = 1 yields

Then, D = ad - bc = a² + b² > 0. We set now

which we may do because we have

The transformation (4) thus becomes

These relations may be given the complex form

and reduce to the linear function of a complex variable

where

whence it is seen that under conditions (11) the linear transformation (4) reduces to shearing of the z-plane by the vector B = l + IM, rotation by the angle a = Arg A and similarity extension with the coefficient Ö = |A| (cf. 4.)

In the case l = -1, we have

and D = - a² - b² < 0. Performing only those calculations, we see that Transformation (4) may be written in the form

Consequently, under Conditions (14), there is added to the above transformations the transition from z to i.e., symmetry with respect to the real axis (cf. 1).

From a geometric point of view, Transformations (12) and (15) are clear in that they preserve the similarity of the figure, in particular, the angle between two straight lines, transform squares in the z-plane into squares in the w-plane, etc. Linear transformations with these properties are said to be orthogonal. Thus, Condition (10) is the condition of orthogonality* of Transformation (4). Moreover, it is clear that Transformation (12) preserves the direction of travel along closed contours (i.e., preserves orientation) and (15) changes them to the opposite direction( changes orientation). Thus, Conditions (11) select orthogonal transformations, preserving orientation, and Condition (14) orthogonal transformations, changing orientation.

* Note that we arrive at the same conditions of orthogonality if we demand that the angle of rotation q - j of any ray arg z = j does not depend on the angle j.

We now proceed to arbitrary mappings. The mutually single-valued mapping

of the domain D onto the Domain D* is said to be conformal if, in the neighbourhood of any point of D, the principal linear part of this mapping is an orthogonal transformation maintaining orientation*. From this definition follow the two basic properties of conformal mappings:

* A mapping w = f(z) is said to be a conformal mapping of the second kind, if its principal linear part is an orthogonal mapping which changes rotation.

1) A conformal mapping maps infinitesimal circles into circles with an accuracy of small higher orders (circular family).
2) A conformal mapping preserves the angles between curves at the points of their intersection ( maintenance of an

gles).

The first property shows that for small r the circle C: |z - z0|= r becomes a circle C* such that the distance of any of its points from the circle |w - w0| = r, produced through any point of the curve C* for the mapping involved, is small of higher order in r. The second property states that the angle at the point z0 between curves G1 and G2 is equal to the angle at the point w0 between the images G1* and G2* of these curves (Fig. 38).
*It is sufficient for the proof of this property to note that there is understood by the angle between curves the angle between their tangents and that the principal linear part of a differentiable transformation moves the tangent to the curve Gk to the tangent to Gk*.

Taking into consideration (5) and (11), we may write the conditions of the conformality of Mapping (1) in the form

where there must be

because for D = 0 the principal linear part of the mapping w = f(z) degenerates, which contradicts the conformality condition. Thus, the conditions of conformality coincide with the D'Alembert-Euler conditions of differentiability (cf. 5.) (analyticity) of the function f(z) in the domain D, where Inequality (17) shows that the derivative f '(z) must differ everywhere from zero.

Moreover, we have

whence we readily arrive at the geometric interpretation of the derivative of a function of a complex variable. We have

i.e., the modulus and argument of the derivative f '(z) stand for the extension and angle of rotation of the principal linear part of the mapping w = f(z) at the point z or, in other words, the coefficient of extension and the angle of rotation of the transformation w = f(z) at the point z.

The reasoning which we have given here leads to the conclusion:

In order that the function w = f(z) shall realize a conformal mapping of a region D, it is necessary and sufficient that in this region it will be 1) single-sheeted, 2) analytic and 3) everywhere in D the derivative f '(z) differs from zero.

We note that, if f '(z0) = 0, then in the neighbourhood of the point z0 the Taylor expansion of the difference f(z) - w0 has the form

where n ³ 2 and cn ¹ 0 (cf. 20.). Hence, it follows that for small |z - z0| = r, the mapping, generated by the function f(z), differs by small higher order terms from the mapping

However, the inverse mapping of (2) has at w0 a branch point of order n, i.e., Mapping 20 is not single-sheeted near the point z0. Consequently, also the mapping w = f(z) is multi-sheeted near z0. Thus, one may omit Condition 3) in only that given formulation, because it follows from 1) (single-sheetedness of the mapping).

Likewise note that, conversely, the condition f '(z0) ¹ 0 ensures single-sheetedness of the mapping in a sufficiently small neighbourhood of the point z0 - this is proved in the same way as the preceding statement. However, if the condition f '(z0)¹0 is fulfilled at every point of the domain D, then does not yet follow from this the single-sheetedness of the mapping throughout the domain even when it is single-valued. For example, obviously, in the semi-ring 1 < |z| < 2, 0 < arg z < p, the mapping w=z4 is not single-sheeted, but at every point of the semi-ring dw/dz = 4z³ ¹ 0.

In conclusion, we comment on mappings generated by single-valued functions which, however, are not single-sheeted in a domain D. We have stated in 26. that every such function w = f(z) realizes a mutually single-valued mapping of a domain D onto the Riemann surface R of the inverse function z = j(w). Let the point P of the surface R, lying below the point w, differ from the branch point and let it correspond to some point z0 of the domain D. This means that there exists a branch j0(z) of the multi-valued function j(z) such that j(w0) = z0. At the point z0, the derivative f '(z0) ¹ 0, because otherwise, as is seen from Expansion (19), P would be a branch-point of the surface R. Thus, the function f(z) yields a mutually single-valued mapping of a sufficiently small neighbourhood of the point z0 onto the neighbourhood of the point w0. Obviously, this mapping will be conformal.

Thus, the function w = f(z), single-valued, but not single-sheeted in the domain D, realizes a mapping, conformal in a sufficiently small neighbourhood of every point z0, for which f '(z0) ¹ 0. We will call points, where f '(z) = 0, and likewise their images on Riemann, surfaces branch points (for example, the Joukowsky function w = ½(z + 1/z) has branch points at z = ±1, w = sin z and at the points z = (2k + 1)p/2, etc.

28. Fundamental problem Given an arbitrary analytic function, we may consider different conformal mappings produced by it. Any domain D, in which this function is single-sheeted, is mapped with its aid conformally onto some domain D*. Thus, we may obtain different examples of conformal mappings, geometrically illustrating the given function. Generally speaking, we have already dealt with this problem in 1.3, where all the mappings under consideration were single-sheeted in corresponding domains, yielded analytic functions and were, consequently, conformal.

However, for practical purposes, a significantly more difficult inverse problem is of interest - the so-called

Fundamental Problem of the Theory of Conformal Mapping Given domains D and D*, construct the function which generates the conformal mapping from one of these domain onto the other.

Since there does not exist for the solution of this problem a sufficiently simple algorithm, the development of the theory of conformal mappings occurs in the following directions:

1) Clarify the general conditions of the existence of a conformal mapping and its uniqueness;
2) determine the different particular classes of domains, mappings of which may be effected with the aid of combinations of elementary functions;
3) with the aid of the general properties of analytic functions, study the different properties of conformal mappings in dependence on the form of the mapped domains;
4) develop approximate methods of conformal mapping.

We will first dwell on the first of these problems. It is clear, first of all, that in that general form - the above formulation - this problem cannot be solved. Thus, it is impossible to map mutually single-valuedly and continuously a multi-connected domain onto a simply connected domain. Without dwelling on the complete proof, we will state the principle of the impossibility of such a mapping. Let us assume that there does exist such a mapping of a multi-connected domain D onto a simply-connected domain D*. Select in D a closed curve C, which contains external or boundary points of D (such a curve always exists). The mapping under consideration maps C into a closed curve C*, lying in D*. If inside the domain D* the curve C* extends in a continuous manner to some point w0 of D*, then, by the continuity of the mapping, the curve C must, remaining inside the domain D, extend in a continuous manner to some point of D, which obviously is impossible, since there lie inside the contour C points which do not belong to D.

Moreover, for example, it is impossible to map conformally the complete or open z-plane onto a bounded domain D* of the w-plane. In fact, if such a mapping did exist, the function w = f(z) would be analytic in the entire open plane and at the same time bounded, because all values of this function were located in the domain D*; however, by the Cauchy-Liouville theorem in 17. , f(z) must be constant, which is impossible.

Nevertheless, two arbitrary simply-connected domains, the boundaries of which consist of more than one point, allow to map conformally on each other and by an infinite number of methods, namely may be allotted correspondence of any two fixed points and any two directions at these points. In other words, there holds the following, so-called basic theorem of the theory of conformal mapping.

Theorem (B. Riemann 1851) Whatever be the simply-connected domains D and D* (with boundaries consisting of more than one point) and whatever be the points z0 given in D and w0 in D* and the real number a0, there exists one and only one conformal mapping

of the domain D onto the domain onto the domain D* such that

The proof of the existence of conformal mappings requires preparation of a special apparatus beyond the scope of this book and will be omitted. Relying only on the existence, we will prove the existence of a conformal mapping under the given conditions of normalization (2).

To start with, consider the particular case when the domains D and D* are the unit circles |z| < 1, |w| < 1, and z0 = w0 = a0 = 0. We must then prove that, if w = f(z) maps conformally the circle |z| < 1 onto the circle |w| < 1, where f(0) = 0 and f'(0)>0, then

The proof is based on Schwarz's Theorem of 15. Since we have |f(z)| < 1 for z < 1, because w = f(z) maps the circle |z| < 1 onto the circle |w| < 1, then by this Lemma

Applying the same reasoning to the inverse function of f(z). we obtain

Consequently, |f(z)| º |z| and by the same Lemma

Since by assumption f '(0) > 0, then a = 0 and f(z) º z.

We now proceed to the general case. Let there exist two conformal mappings D and D*:

which satisfy the conditions

We map conformally the circle |z| < 1onto the domain D with the aid of the function

and the domain D* onto the circle |w| < 1 with the aid of the function

Obviously, the functions

yield conformal mappings of the circle |z| < 1 onto the circle |w| < 1 with the normalization

By what has been proved above, F1(z) º F2(z), however, then also f1(z) º f2(z), and the uniqueness of the mapping has been proved.

In conclusion, we note a generalization of the Cauchy-Liouville Theorem of 17. which is a direct consequence of Riemann's Theorem.

If the function w = f(z) is analytic in the open plane and does not assume values lying on some arc g, then it is constant.

In fact, let w = j(w) be a function realizing a conformal mapping of the outside of the curve g onto the inside of the unit circle (it exists by Riemann's Theorem and, of course, is not constant). Consider the compound function w = j[f(z)] = g(z); it is analytic in the open plane and all its values lie inside the unit circle, whence, by the Cauchy-Liouville Theorem of 17. this function is constant. However, if g(z) is constant, then also f(z) is constant, as had to be proved.

For example, in particular, f(z) is constant if it is analytic in the open plane and all its values lie in a certain half-plane (then it does not assume values lying on any arc in the supplementary half plane).

29. Correspondence of boundaries We consider basic facts, relating to the correspondence of boundaries, which can be established for conformal mappings of domains. For the sake of convenience, we introduceon the boundary C of the domain D the real parameter s - the length of the arc measured from some fixed point of C, so that on C we will have z = z(s). If any function f(z) is continuous in a closed domain D, we will set on the boundary C of this domain

and call j(s) the boundary function for the function f(z).

We present without proof the theorem regarding the correspondence of boundaries.

Theorem 1 Let the function w = f(z) create conformal mappings of the domains D and D*. Then,

1) if the boundary of D* does not have infinite branches, f(z) is continuous on the boundary of the domain D and the boundary function w = f(z) = j(s) realizes a continuous and mutually single-valued correspondence of the boundaries of the domains D and D*.

2) If the boundaries of D and D* do not contain infinite branches and have at each point continuous (and consequently also bounded) curvature, then the boundary function j(s) is continuously differentiable.

In this context, it is always assumed that multiple points of the boundary are as their multiplicity; thus, in Fig. 39, the points of the two shores of the cut cd and de are assumed to be different (and to these shores correspond the different segments c*d* and d*e*), the points b* and f* are likewise (to them there correspond even different points b and f). If we omit in the first part of the theorem the condition of absence of infinite branches of the boundary D*, then the function j(s) remains continuous at all points of the boundary D, which correspond to finite points. However, at points which correspond to to the point at infinity of the boundary D* ( there may several of them, it this point is multiple), the function 1/j(s) is continuous.

We present yet, likewise without proof, certain more exact results, realting to the existence of the derivative ofa conformal mapping on the boundary of the domain. The first of these was obtained by K.Karatheodory in 1929:

1) If the function w = f(z), f(0) = 0 yields a conformal mapping of the upper half-plane onto the domain D, the boundary C of which in the vicinity of the point w = 0 represents itself a continuous curve, then there exist neighbourhoods , passing through w = 0, one of which lies entirely in D and another which lies entirely outside D, then there exist for z ® 0 by points of the upper half-plane

This result was obtained in 1931 by M.A.Lavrentiev and P.A. Becconov:

If in the neighbourhood of the point w = 0 the boundary C is conjugate *, lies between the curves v = ± |u|1=a , 0 < a < 1 and where s is the abcissa of the points of C, the distance of which along C to the point w = 0 is s, then there exists

* i.e., each of its segments has a definite length.

For practical purposes, the result of O. Kellog et al. is sufficient. In order to formulate it, we agree to call a certain arc a Liapunov arc, if it is conjugate, has at every point a tangent and the angle , the inclination of this tangent to the x-axis as a function of the arc length s, satisfies the Hölder condition

where K is some constant and 0 < a £ 1. One then has the

Theorem If the function w = f(z) realizes a conformal mapping of the region D, the boundary of which contains a Liapunov arc c, onto the domain D*, where c is also mapped into a Liapunov arc c*, then on c the derivative f '(z) exists, does not vanish and satisfies a Hölder condition.

The proof of Kellog theorem may be found in G.M.Goluzin [6], p. 468.

We yet note that the normalization condition (2) in the basic theorem in 28. containing the three real parameters x0, y0, (x0+iy0= z0) may be replaced by the condition of correspondence of three boundary points of the domains d and D ':

selected arbitrarily, but in agreement with the order of sequence for passage along the boundary *. This assertion will be proved in 35.

* Condition (1) as well as Conditions (2) involve three real parameters, because teh positions of points on the boundary of the region are determined by a single parameter.

In the practice of conformal mapping is important the following, in a known sense inverse of Theorem 1, the boundary correspondence principle:

Theorem 2. Let there be given two simply-connected domains D and D* with boundaries C and C*, where D* is bounded. If the function w = f(z)
1) is analytic in D, continuous in and
2) there exists a mutually single-valued mapping of C onto C* with conservation of the direction of travel,
then it yields also a (single-sheeted) mapping of D onto D*.

We employ for its proof the argument principle of 23. For any complex value of w0, which f(z) does not assume on the boundary C of the domain D, the number of w0-points of teh function f(z) inside D equals

where DCarg |f(z) - w0| is the total change of arg |f(z) - w0| as z travels along C (cf. Formula (13) in 23.; the number of poles of f(z) in the domain d is zero, because f(z) is continuous.)

By the strength of the mutual single-valuedness and continuity of the correspondence between the points of the contours C and C*, we have

However, obviously, DC*arg(w - w0) equals 2p for all points w0, lying inside C*, and vanishes for all points outside C*, whence for all points w0, lying inside C*, N(w0) = 1, but for all points lying outside C*, N(w0) = 0. Thus, the function w=f(z) assumes in D once and only once any value from D* and does not take any other values, i.e., it realizes a single-sheeted mapping of D onto D*, and the theorem has been proved.

Nowhere in the proof, there has been used the boundedness of the domain D. However, if the domain D* is unbounded, i.e., contains inside it* or on its boundary the point at infinity, then the principle requires a more precise definition. First of all, we must for its definition omit the requirement of the continuity of f(z) in , because f(z) stops to be continuous there, which corresponds to w = ¥. However, then without additional restrictions this principle remains true. For example, the function w=z³ yields continuous (except at the point z = ¥) and mutually single-valued correspondence of the points of the x-axis and u-axis with preservation of of the direction of travel along the contour and, however, is not single-sheeted in the upper half-plane. In fact, during this mapping, the upper half-plane, i.e., the angle of spreading p becomes the angle 3p ,which twice describes the upper half-plane (and yet once the lower one).

* If the domain D contains inside it the point z = ¥, then one must define the concept of conformity at this point. Such a definition may be obtained by going with the aid of the stereographic projection on the sphere of complex numbers. Cf. 31., where the problem is developed in detail.

The case when the domain D* is unbounded is practically important; we immediately will consider it. There apply two theorems (we retain the above notation and Condition 2), imposed on the function. f(z).

Theorem 3 Let the domain D* contain the point at infinity inside, then the boundary correspondence principle remains in force, if Condition 1) is replaced by
1') f(z) is continuous in
and analytic everywhere in D except at some internal point z0 at which it has a first order pole.

We use again for the proof the argument principle. According to this principle, for every point w0 not on C*, the number of w0-points of the function f(z) satisfies the condition

(it has inside the contour C* exactly one first order pole).

Since the domain D* contains the point at infinity, C* travels clockwise, i.e., DC*arg(w - w0) equals -2p, if w0 lies inside C*, and 0. if w0 lies outside C* . Thus, N(w0) = 1 for all internal points of the domain D* and N(w0) = 0 for all external points, as had to be proved.

The following theorem concerns the case of regions containing the point at infinity on their boundary. As the above example w = z³ shows, one must for maintenance of the boundary correspondence principle introduce additional restrictions.

We assume, first of all, that the domain D* has only one such point, i.e., that w = ¥ is a simple point of the boundary C*; moreover, we assume that the branch C*, going to infinity, has asymptotes*. We denote by bp ( 0 £ b £ 2) the angle between these asymptotes as shown in Fig. 40 (where the cases b = 0 and b = 2 are shown; as always, the shading indicates the complements of the domains). Let the point w = ¥ correspond to the point z on the contour C; we denote by ap (0<a£2) the angle between the tangents to the contour C at this point. Moreover, we assume that the function f(z) in the neighbourhood of the point z is of infinitely large order m, i.e., that there exists a constant A ¹ 0, ¹ ¥ such that, as z tends to z0, by points of the domain D, there exists the limit

* This condition is for the point at infinity the analogue to conditional piecewise smoothness.

In the case under consideration, the boundary correspondence principle is not directly applicable, because f(z) becomes infinite on the contour C. It turns out that, in order to maintain the principle, one must introduce a restriction on the order m of the growth of the mapping function. In fact, with the above conditions and notation, one has

Theorem 4 The boundary correspondence principle is preserved if Condition 1) is replaced by
1") The function f(z) is analytic everywhere in D and continuous in
except at the point z0, but in the part of the neighbourhood belonging to D this point is of infinitely large order m, where

For the proof, we remove from the region D the circle |z - z0| < r of small radius with centre at the point z0; the arc of this circle which belongs to D we denote by gr, and the part of the contour C, which remains after removal of the circle, by Cr; we denote by the contour Cr + gr. We apply already to the region , bounded by the curve , the boundary correspondence principle.

Let w0 be an arbitrary point of the domain D*. Since w0 is finite and f(z) ® ¥ as z ® z0, the radius r of the circle may be chosen so small that in the reduced part of D there would be neither one w0-point of the function f(z). Then, N(w0) - the number of w0-points of the function f(z) in the region D - will be equal to the number of such points in the region . and we obtain by the argument principle

For the calculation of the first term, we note that, as the point z moves along Cr, the corresponding point w travels in the positive direction along the entire curve C*, excluding certain part of it lying within a small neighbourhood of the point w = ¥, whence

where Cr* is image of the arc C and O(r) denotes a quanity which tends to zero with r (in the sequel, when required, we will use this symbol when it may denote a rweal as well as a complex quantity).

For the calculation of the second term Condition (2) is used, yielding f(z) in the neighbourhood of the point z0 in the form

One may then assert that

because the quantity w0(z - z0)m, infinitely small for r ® 0, may be included in the symbol O(r). Since A ¹ 0 and constant, the first term here tends to 0 for r ® 0; moreover, it is clear from Fig.40 above that

whence

Sunstituting (5) and (6) into (4), we find

whence for for r ® 0

According to introduced bound (3) on the growth of f(z), we have am - b > 2 and obtain by (7) N(w0) = ½(2 - b + am)>0, since now am > 0 and b £ 2. However, in the open inteval (0, 2) lies the only integer 1, whence N(w0) = 1.

We have proved that the function f(z) assumes in D any value w0 of the domain D* once and only once.If the point w0 lies outside the domain D*, our reasoning is preserved, if we place in (5) -bp instead of (2 - b) and then obtain instead of (7)

Hence, in this case, -1 < N(w0) < 1 and, consequently, N(w0) = 0, i.e., the function f(z) does not take in the domain D a value which does not belong to D*. Thus, the function f(z) yields a single-sheeted mapping of D onto D*, and the theorem has been proved.

Note that we have under the conditions of the proved theorem m = a/b (this is seen from (7) or (8)), i.e., in the neighbourhood of the point z0

In particular, let a = b = 1, i.e., z0 and w = ¥ are not angular points. Then, Inequality (3) becomes m < 3. Thus, the function f(z) must be in the neighbourhood of the point z0 of infinitely large order below the third. The example with the function w = z³ . introduced above shows that in the case under consideration the theorem is precise, i.e., infinity of order 3 already does not secure single-sheetedness of a mapping.

Finally, we note the fact:

If w = ¥ is a multiple point of the contour C*, then for preservation of the truth of the boundary correspondence principle it is sufficient to require that Condition (3) be fullfilled even for one* point z0 of the contour C, corresponding to the point w = ¥ .

* There are as many such points as the multiplicity of the point w = ¥ of the domain D*.

In fact, we exclude the neighbourhoods of all pointszk, corresponding to the point w = ¥, by circles |z - zk| < k (k = 0, 1, ··· , n-1), where n is the multiplicity of the point w = ¥) and we denote the remaining domain by . For any value w0, not lying on C* (and, consequently, finite), we choose rk so small that the numbers of w0-points of the function f(z) in the domains D and are the same. We now make r0 tend to 0, keeping r1, r2, ··· , rn-1 fixed. Here the proof of the preceding theorem is completely applicable, whence N(w0) = 1 for all points in D* and N(w0) = 0 for w0, not belonging to D*, as was required to prove.

In the following section, we present a number of examples of applications of the boundary correspondence principle.

30. Examples

1) The function

assumes on the boundary of the unit circle the values

Hence it establishes a mutually single-valued correspondence between the unit circle and the points of the arc u > 1, v = 0. This correspondence is continuous everywhere, except at the point z0 = -1, in the neighbourhood of which

Here Theorem 4 of 29. is applicable, because a = 1, b = 2 and Condition (3) is met. Consequently, (1) realizes a single-sheeted conformal mapping of the circle |z| < 1 onto the domain, obtained from the w-plane by removal of the ray u>1, v = 0. Fig. 41 shows the net of the curves, corresponding to each other for the mapping under consideration; the net of curves in the w-plane are orthogonal, because they conform to the orthogonal net in the z-plane (formed by the circles |z|=const, and their radii).

2) We consider next the more general case - the function

which for n = 1 coincides with the function of the last example. On the unit circle, it takes the values

Note on the unit circle the vertices of the regular n-polygon

and the central arcs forming them (Fig. 42):

The quantity w = cos nj /2 decreases from 1 to 0 on the arc A1B1 (we agree to assume that on this arc arg w = 0); on the arc B1A2, it changes from 0 to -1 (on this arc, we agree to assume that arg w = -p); on the arc B2A3, from 0 to 1 (assuming arg w = -2p), etc. (Fig. 42b). Correspondingly, on the segment A1B1 , the modulus of the corresponding point w grows from ¥ to 1, and the argument is equal to zero; on the segment B1A2 , the modulus of w decreases from ¥ to 1 and the argument equals 2p/n; on the segment A2B2, the modulus of w grows from 1 to ¥ (as before arg w = 2p/n); on B2A3 , the modulus of w decreases from ¥ to 1. arg w = 2·2p/n, etc. Thus, the point w travels consecutively along n rays

each ray twice in the opposite directions (Fig. 42,b). The image of the circle |z| = 1has during this mapping w = ¥ as an n-fold point, to which correspond the points Bk (k = 1, 2, ··· , n). Since each point Bk is a simple zero for zn + 1, then in its neigbourhood w » Ck/(z - Bk)2/n, where Ck is some constant. By the boundary correspondence principle, the corresponding boundary (Theorem 4 in which now ak = 1, bk = 2/n) the function (2) realizes a single-sheeted conformal mapping of the circle |z| < 1onto the w-plane with n excluded rays (3).

3) In accomplishing the supplementary mapping w = 1/w1 of the w-plane, we obtain the mapping

of the unit circle onto the outside of a "star" consisting of n rays

(Fig. 43, instead of w1 , we write again w). For n = 2, we obtain w = ½(z + 1/z), i.e., the Joukowsky transformation (cf. 7.)

4) We now consider the unit circle |z| < 1 with removed segments of length a:

(Fig. 44, a). By the same method as above, we readily verify that the function

of the preceding example maps this domain onto the outside of the "star" with rays of length

which is similar to the star (5) (Fig. 44, b). By the similarity transformation z2 = z1/(1 + a), we create the rays of the star of unit length and then the function

the inverse of the function (4), maps the outside of this star onto the inside of the unit circle. Thus, the function

realizes the conformal mapping of the unit circle with n removed segments (6) onto the inside of the unit circle.

Since (9) in its expanded form is very complicated, it is of interest to obtain approximate formulae which are convenient for computations. We will assume that a in (7) is small; then (7) yields, neglecting terms of of order above a², by Taylor's formula

In an analogous manner, neglecting smaller terms in a, we find

and, from (8),

Substituting the above value of a, we finally find

This formula is suitable for points not too close to the points For a = 0, we have w º z as must be.

5) We consider the function

Setting z = x + iy, w = u + iv, we have: u = x + ex cos y, v = y + ex sin y, whence on the straight lines y = ±p, bounding the strip -p < y < p, there hold the relations u = x - e, v = ±p, i.e., these lines are transformed into double rays -¥<u<-1, v= ± p (the function u = x - ex attains its maximum u = -1 for x = 0). The boundary correspondence principle is not applicable, but direct analysis of the function shows that it yields a conformal mapping of the strip -p < y < p onto the domain, obtained from the w-plane by exclusion of the two strips - ¥ < u < -1, v = ±p. Fig. 48 displays the correspondence of the lines for this mapping (shown is the upper half of the domain, the mapping of the lower halves symmetrically).

6) The exponential function

transforms the parallel inclined lines y = k(x - a1), y = k(x - a2) (k ¹ 0, ¹ ¥ ) into the curves

Introducing in the w-plane polar co-ordinates, w = reiq,we obtain r = ex, q = k(x - an) or

where (n = 1,2). These are two similar logarithmic spirals. If k(a2 - a1) < 2p, then the length of the vertical segment between the straight lines in the z -plane is less than 2p and the mapping is single-sheeted into the strip between them (cf. 8.) Changing a from a1 to a2, we verify the fact that the exponential function realizes a mapping of this strip onto the strip between two logarithmic spirals (Fig. 46). If k(a2 - a1) = 2p, both spirals coincide and we obtain a mapping onto the plane with a removed spiral.

For k(a2 - a1) > 2p, the mapping is not single-sheeted.

7) The function

assumes on the boundary of the unit circle the values

setting w + u + iv and eliminating the parameter j, we obtain the equation of the image of the unit circle

This is the chain line of equal resistance (Fig. 47). By the boundary correspondence principle, we find that the function (14) realizes the conformal mapping of the unit circle onto the inside of this curve.

8) The function

or, in polar co-ordinates,

maps the circle r = cos j into the cardioid

(Fig. 48). By the boundary correspondence principle, the function (16) realizes the conformal mapping of the inside of this circle onto the inside of the cardioid.

9) The function

or r = Ör, q = j/2, where j = arg z changes from -p/2 to p/2, maps the same circle into the branch of the lemniscate r=Öcos 2q (Fig. 49). By the boundary correspondence principle, the function (18) realizes a mapping of the inside of this circle onto the inside of the right branch of the lemniscate.

2.2 The simplest conformal mappings

We present the simplest methods of solution of the basic problems of the theory of conformal mappings - the problems of finding the function which realizes a conformal mapping of a given domain onto another domain. We will give sufficiently many examples by which the reader will become familiar with them and may solve this problem by means of combinations of elementary functions (if this can be done). Such a selection requires a knowledge of the geometry of the elementary functions, and we advise the reader to study first 1.3 where the mappings are presented which these functions can realize before reading Sections 33. and 34. We also present methods for obtaining approximate formulae of conformal mappings which are especially important in practice.

For work with the simplest conformal mappings, one often employs fractional linear functions - to the study of which we proceed immediately. Note that the presentation, realized by such functions, is closely linked to the geometry of N.I.Lobachevski; however, we cannot explain this link here *.

* Cf. Markushevich [2], pp. 111 - 118.

31. Bi-linear Mappings We will use this term for the mapping, generated by the fractional linear function

where a, b, c and d are complex constants and ad - bc ¹ 0**. The function (1) is define in the entire z-plane (its value at the point z = -d/c is ¥ and at the point z = ¥ is equal to

** For ad - bc = 0, we have a/c = b/d, and the function (1) reduces to a constant.

Since the derivative

exists everywhere for z ¹ -d/c, the function (1) is analytic everywhere in the z - plane except at the point z = -d/c, where it has a first order pole. Equation (1) has the single-valued solution with respect to z

where the function (3) likewise is defined in the entire w-plane (its value at w = a/c is ¥ and the point ¥ is equal to - d/c). Hence the bi-linear function realizes a single-sheeted mapping of the entire z -plane onto the entire w-plane.

It is readily seen that the bi-linear function is the only function with such properties. In fact, we have the

Theorem 1 If a function f(z) is everywhere single-sheeted and analytic everywhere in the entire z plane, except at the point C, then it is bi-linear.

In fact, C may not be an essential special point of the function f(z), because then, by Sokhotskii's Theorem (cf. 22.) f(z) would be deliberately not single-sheeted. By the Cauchy-Liouville Theorem (in the form of 24.), C may not be also a removable special point. Consequently, the point c is a pole of first order, because in the neighbourhood of a higher order pole a function is again not single-sheeted. If C ¹ ¥, then the principal part of the function f(z) has in the neighbourhood of the point C the form B/(z - C); computing this part from f(z), we obtain the function j(z) = f(z) - B/(z - C), which does not have a singularity in the entire plane (as the only special point for j(z) may serve the point C, but it is a removable special point, because we have calculated from j(z) the principal part). Consequently, j(z) º A, a constant, and the function

is bi-linear. If C = ¥. the principal part of f(z)has the form Az and it is proved in the same manner that f(z) = Az + B, i.e., it is an integral linear function. Thus, the theorem has been proved.

Formula (3) shows that the function, inverse to the bi-linear function, is again bi-linear. It is readily shown that a compound function involving bi-linear functions is likewise bi-linear.

We explain next the geometric properties of bi-linear functions.

If c = 0, then (1) reduces to to a linear function the geometric properties of which have already been discussed in 4. In order to study the geometric properties of the function (1) for c ¹ 0, we write it in the form

where A, B and C are some constants * and study this mapping as a compound one consisting of the mappings

* In order to represent (1) in the form (4), it is sufficient to divide in (1) the numerator by the denominator using the rule for division of bi-nomials.

Mapping (a) is a displacement, (b) a shear and rotation with extension. There remains to study (c), which, with changed notation, becomes

In polar co-ordinates z = reij, w = reiq, the mapping (6) becomes

It is convenient to consider (7) as consisting of the two geometrically speaking more comfortable mappings

Mapping (b) is a symmetry transformation with respect to the real axis. Mapping (a) is an inversion, a symmetry transformation symmetric with respect to the unit circle. (cf. 2.).

In general, we will call points z and z* symmetric with respect to a circle C0: |z - z0| = R0, if

1) they lie on one ray through z0:
2) |z - z0|·|z* - z0| = R²0.
(The method of construction of symmetric points of 2. remains valid also in the general case).

A transformation which moves every point of the z-plane to the point z* , symmetric with respect to a circle C0 , is called symmetry with respect to this circle or an inversion.

We will prove the basic property of symmetric points: Points z and z* are symmetric with respect to the circle C0 if and only if they are the vertices of the rays of the circle orthogonal to the circle C0.

In fact, let the points z and z* be symmetric with respect to C0 and G be an arbitrary circle, passing though z and z* (Fig. 50). Draw through the point z0 the tangent to the circle G. By a known theorem, the square of the length of this tangent |z'-z0|² equals the product of the secant |z* - z0| by its external part z = |z - z0|, i.e.,

Since z and z* are symmetric with respect to C0, this product equals R²0 and, consequently, |z' - z0| = R0. Thus, the tangent to G is the radius of the circle C0, i.e., G is orthogonal to C0.

Conversely, if z and z* are vertices of rays of the circle {G}, orthogonal to the circleR0, then they lie on one ray, passing through z0, because this ray belongs to the bundle*. Moreover, the tangent z0z' to any circle G is a radius of the circle C0 and by the same theorem |z - z0|·|z* - z0| = R0², i.e., z and z* are symmetric with respect to C0. The property has been proved completely.

* We consider in the entire plane straight lines as particular cases of circles - such circles pass through the point at infinity.

There follows from this property, among others, that in the case when the circle degenerates into a straight line, the symmetry with respect to the circle becomes ordinary symmetry.

Inversion with respect to an arbitrary circle C0 is a conformal mapping of the second kind (change of orientation). In fact, let z0 and R0 - the centre and radius of the circle C0, when the point z*, symmetric with the point z with respect to C0, becomes

whence

|z - z0|·|z* - z0| = R0² and arg|z* - z0| = arg |z - z0|.

Consequently, inversion differs from the conformal mapping

only by an additional symmetry with respect to the real axis of the w-plane, i.e., it is a conformal mapping of the second kind.

Moreover, we see that inversion transforms any circle C of the entire plane again into a circle (circular property).

In fact, let again the circle C pass through the centre z0 of the circle C0, with respect to which there occurs inversion (Fig. 51). Construct the straight line C*, perpendicular to the line of the centres of the circles C0 and C, at the distance R0²/2R from z0 (R0 and R are the radii of C0 and C). The similarity of the triangles z0z*z* and z0zz (Fig. 51) yields

Consequently, the points z and z* are symmetric with respect to C0. We have proved that the point, symmetric to an arbitrary point z of the circle C, lies on the straight line C*, i.e., that C* is the inverse of the circle C. In particular, if C is a straight line passing through z0, then the inversion of this line obviously coincides with itself.

Now, let the circle (or straight line) C not pass through z0. Construct the point z1, symmetric to z0 with respect C and consider the bundle of circles {G} with vertices at z0 and z1. Since all circles G pass through z0, by what has been proved above for inversion with respect to C0, the bundle {G} becomes the bundle of straight lines {G*} Obviously, the vertex of this bundle will lie at the point z1*, symmetric to z0 with respect to C0. By the property of symmetric points, all circles {G} are orthogonal to C, and likewise inversion preserves angles (we have proved above that it is a conformal mapping of the second kind). whence the image C* of the circle C will be orthogonal to the bundle of straight lines {G*}, whence it follows that C* is a circle and the property has been proved.

In exactly the same way is proved yet another important property of inversion: Inversion transforms any pair of points z1 and z2 , symmetric with respect to an arbitrary circle C, into a pair of points z1* and z2*, symmetric with respect to the circle C* - the image of the circle C (property of conservation of symmetrical points).

In fact, we construct the bundle of circles {G}with vertices at z1 and z2. On inversion, it becomes the bundle of circles {G*}with vertices at z1* and z2*. Since the circles G are orthogonal to C, the circles G* are also orthogonal to C*, whence z1* and z2* are symmetric with respect to C*, and the property has been proved.

Since the mapping w = 1/z consists of two symmetries (Symmetry (a) with respect to the unit circle and symmetry (b) with respect to the straight line), it possesses also circular property and the property of preservation of symmetric points. Since the remaining transformations, consisting of arbitrary-linear fractional mappings (the mappings (a) and (b) of (5), i.e., shear and rotation with extension), obviously have these properties, these properties are preserved also for the arbitrary fractional linear mapping.

We shall prove that the arbitrary-linear mapping (1) preserves angles in the entire z-plane.

This is obvious for all points z, except z = -d/c and z = ¥, because for such points there exists dw/dz ¹ 0 (cf. (2)). In order to speak about conservation of angles at the points z = -d/c and z = ¥, one must introduce the concept of the angle at the point at infinity, where one may obviously limit consideration to the angle between two straight lines. We understand by the angle at the point at infinity between two straight lines the angle, taken with opposite sign, at the second (finite) point of intersection of two straight lines (in Fig. 52, a) the angle at infinity between the lines l and ll is negative). Clearly, the mappings (a) and (b) preserve angles everywhere.

There remains to show that the mapping (b) or what is the same thing, the mapping w = 1/z preserves angles at the points z = 0 and z = ¥. However, this is seen directly from Fig. 52 and our definition (for the mapping w = 1/z the line arg z = j becomes the line arg w = -j).

We formulate the basic properties of the bilinear mapping , proved above, in the form of the two theorems:

Theorem 2 The arbitrary bi-linear function

realizes the single-sheeted conformal mapping of the entire z-plane onto the entire w-plane. This mapping
1) transforms any circle of the entire z-plane into a circle of the complete w-plane (circular property)
2) maps any pair of points, symmetric with respect to the circle C, into a pair of points, symmetric with respect to the image of the circle C (property of conservation of symmetric points).

In conclusion, we present without derivation formulae by which one may compute images of straight lines and circles for the arbitrary bi-linear mapping (1):

a) There correspond to the straight line Re(lz) = a, which does not pass through the point z = -d/c (a ¹ -Re(ld/c)) the circles |w - w0| = r, where

b) The line Re(lz) = -Re(ld/c) through the point z = -d/c is

c) The circle |z - z0| = r, not passing through the point z = -d/c(r ¹ |z + d/c|) is the circle |w - w0| = r, where

d) The circle |z - z0| = |z0 + d/c| is the line

These formulae are obtained directly.

Example We will find the image of the line y = x + 2 for the mapping w = (z + 1)/(z - 1); since the line does not pass through the point z = 1, it is mapped into a circle with centre and radius given by (9):

(we have a=b=c=1, d=-1 and since the equation of the line has the form Re{(-i-1)(x+iy)}= 2, we have l=-i-1 and a=2).

32. Particular cases We will first explain the conditions which determine the bi-linear mapping. By the definition (1) of 31., such transformations involve the four parameters a, b, c and d. Since one of these coefficients would differ from 0 and may be assumed to be 1, dividing by this coefficient the numerator and denominator of the fraction, in fact, the fractional linear transformation depends on three complex or six real parameters.

Thus, it is clear that this transformation is determined by conditions, which reduce to six independent relations between the real and imaginary parts of the coefficients. The simplest form of such conditions reduces to giving in the z- and w-planes arbitrary three points z1, z2, z3 and w1, w2,w3, corresponding to each other in the mapping under consideration.

For the construction of the mapping, satisfying this condition, we consider the auxiliary z-plane and construct the bi-linear mappings of the z- and w- planes onto this plane, reducing the given three pointsin to 0, 1 and ¥. Such a mapping is readily written down:

Eliminating z from this system, we find the bi-linear mapping of the z-plane onto the w-plane, transforming the points z1, z2, and z3 into w1, w2 and w3, respectively; this mapping will be written in the form

We will show now that the mapping (2) is the unique bi-linear mapping which satisfies the conditions imposed. In fact, if there exist two different such mappings w = l1(z) and w = l2(z), then, taking still the second of the mappings (1), which we now denote by z = l(w), we obtain the two different bi-linear transformations

which transform the points zk into 0, 1 and ¥. Consider now the transformation

where L1-1 is the inverse mapping of L1. It is bi-linear, whence it may be given the form

Obviously, Mapping (3) leaves in place the points 0, 1 and ¥. From the correspondence of the points at infinity, we find that c = 0 and, consequently, z " = (a/d)z ' = b/d.

The correspondence of the two other points yields the conditions b/d = 0, a/d = 1. Thus, z " = L2[L-1(z ')] º z ', i.e., L1-1 is the inverse of L2 and L1 º L2. However, then also l1 º l2, and this also proves our assertion regarding the uniqueness of Mapping (2).

It is not difficult to verify that (2) preserves the sense also in the case when one of the points zk or wk is the point at infinity, if one only replaces in this formula the one in the numerator and denominator by the ratio, in which this point participates (in (2), every point participates once in the numerator and once in the denominator). In fact, for example, let w3 = ¥ , z2 = ¥, then (2) becomes

or w = w1 + (w2 - w1)(z - z1)/(z - z3) and it seen immediately that the mapping obtained solves the problem. Thus, we have proved

Theorem 1 There exists one and only one bi-linear transformation of the entire z-plane onto the entire w-plane, which maps three arbitrary points zk into three arbitrary points wk.

From this theorem follows

Theorem 2 Any circle of the complete z-plane may be mapped with the aid of the bi-linear function into any circle in the complete w-plane.

In fact, we select on the boundary C of the circle in the z-plane three points zk numbered in the positive direction of this circle. If we take on the boundary C* of the circle in the w-plane three arbitrary points wk and construct by (2) the bi-linear mapping, then this transformation, due to the circular property, will convert the circle C into C*. Then, by the boundary correspondence principle, it transforms the circle K, bounded by the circle C, into one of two circles bounded by the circle C*..Now, if we construct by (2) the bi-linear transformation, it will, according to the circular property, take the circle C into C* and, according to the boundary correspondence principle, the circle K into one of two circles, bounded by C*. However, since a conformal mapping preserves orientation (cf. 27.) and the points wk are distributed with respect to K* as the points zk with respect to K, then K maps exactly into K*, and the theorem has been proved.

Note one limiting case of (2). We pose its problem of the construction of the bi-linear mapping between two pairs of corresponding points z1, z2 and w1, w2 and w3, and by the given derivative at the point z. For the solution of this problem, we replace the last condition by the condition of correspondence of the points z3 = z2 + h and w3 = w2 + ah; then the mapping is, by (2),

Cancelling -h on both sides and going to the limit h ® 0, we obtain the known mapping

We will now consider several important examples of bi-linear mappings.

1) Reflection of upper half-plane in the unit circle. Take the point a of the upper half-plane, shift it to the centre of the circle w = 0 (Fig. 53). By the property of preservation of conjugate points, the point symmetric to the point a with respect to the real axis, must go to the point w = ¥, symmetric to the point w = 0 with respect to the unit circle. Hence, the required mapping must have the form

where k is a constant factor. For any k, this function maps the upper half-plane onto some circle with centre at w = 0, because the point w = ¥ must be symmetric to the point w = 0 with respect to the circumference of this circle. We select k so that the circle will be a unit circle. It is sufficient for this to require the point z = 0 to move to the point of the unit circle:

Thus, one may set k = eia and our problem is solved by the function

where a is any real number (changing a means a rotation of the circle about its centre w = 0).

By the properties of the bi-linear mapping, there correspond to the arcs of the circle the bundle of the radii of the circle |w|<1 (i.e., the arc of the circle passing through the point w = 0 and w = ¥ ) the arcs (belonging to the upper half plane) of the circle passing through the points a and . There correspond to the family of circles with centre at the point w = 0 the circles having a and as symmetric points (cf. Fig. 53).

We note yet the inverse of (6), the mapping of the unit circle onto the upper half-plane. Setting for the sake of simplicity a=ih, we obtain from the formula

Seeing here w = eiq and multiplying the numerator and denominator by e-i(a+q)/2, we find the correspondence between the points of the unit circle and the x-axis, which determines Mapping (7):

The boundary derivative

is continuous everywhere on the circle except at the point w = eia , which corresponds to the point z = ¥ (cf. Theorem 1 in 29.).

2) Mapping of the unit circle onto the unit circle. We make a point of the circle |z| < 1 pass into the centre of the circle |w| < 1. The point a* = 1/, symmetric to a with respect to the unit circle, must move to the point w = ¥, whence the required mapping must have the form

where k and k1 are constants. We select k1 so that the circle in the w-plane is the unit circle. To achieve this it is sufficient to demand that the point z = 1 move to the point of the unit circle: |k1(1 - a)/(1 - )| = |k1| = 1. Consequently, one may take k1 = eia, and our problem is solved by the function

where a is any real number. Since

and |a| < 1, then a is geometrically the angle of rotation of the mapping (10) at the point a:

We note that the extension of Mapping (10) at the point a

tends to infinity as the point a approaches the boundary of the unit circle.

Fig. 54 shows the lines, corresponding to each other for this mapping. In the z-plane, the pattern is a part of that in Fig. 53.

Note yet the relation, linking the arguments of corresponding points of the unit circles z = eij and w = eiq (we assume, for the sake of simplicity, a = 0 and set

* In order to obtain (13), one need only substitute in (1) expressions for z, w and a, multiply both sides by and separate the real part.

Going to the more general case, we note that, if the radius of the circle in the z-plane equals R, the function w = f(z), mapping this circle onto the circle |w| < 1 for the conditions f(a) = 0, f '(a) = a, has the form

This formula is obtained from (1) by replacing z by z/R and, correspondingly, a by a/R.

3) Mapping of upper half-plane onto upper half-plane. We will find the general form of such mappings. One may obtain for every bi-linear function w = l(z), realizing a mapping of the upper z-half-plane onto the upper w-half-plane, from (2) by giving two sets of three corresponding points zk = xk, wk = uk of the real x- and u-axes. Since the numbers zk and wk are real, (2) assumes after transformation the form

where a, b, c, d are real. On the contrary, any function (15) with real coefficients transforms the x-axis into the u-axis and, consequently, the upper z-half-plane into the upper or lower w-half-plane. We obtain the upper half-plane, if we demand that the derivative dw/dz on the real axis be positive:

whence ad - bc > 0. Thus, (15) yields for real coefficients satisfying the condition ad - bc > 0 the general form of the linear mappings of the upper half-plane onto the upper half-plane.