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 *z*_{0} 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
*z*_{0}*, *and *h*_{1}*, h*_{2}*
*tend to zero as *D**r
*® 0. Replacement of increments
of the differentials then reduces to omission in (2) of the terms
*h*_{1}*D**r and**
h*_{2}*D**r
*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 *z*_{0}* = x*_{0}* + iy*_{0}*
*and *w*_{0}* = u*_{0}*+ iv*_{0}*
*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* = *x*_{0},
···, *v* = *v*_{0} and subtract from (4)
and (6) the equations obtained). Taking into consideration (8),
we may assert that the circle with centre at *z*_{0}:

for the transformation (4) becomes the ellipse with centre at
the point *w*_{0}:

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
*l*²* *=
1 or *l *= ±1.

The case *l *= 1 yields

Then, *D **=
ad - bc = a*²* + b*²* > *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 = -* a*²*
- b*²* < *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 - z*_{0}|= *r *becomes
a circle* C** such that the distance of any of its points
from the circle |*w* - *w*_{0}| = *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 *z*_{0} between curves *G*_{1}* *and *G*_{2}* *is equal to
the angle at the point *w*_{0 }between the images *G*_{1}**
*and *G*_{2}*
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
*G*_{k }to the tangent to *G*_{k}****.*

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* '(*z*_{0}) = 0,
then in the neighbourhood of the point z_{0 }the Taylor
expansion of the difference f(z) - w_{0} has the form

where *n* ³ 2
and *c*_{n} ¹
0 (cf. **20.**).
Hence, it follows that for small |*z* - *z*_{0}|
= *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 w_{0}
a branch point of order *n*, i.e., Mapping 20 is not
single-sheeted near the point* z*_{0}.
Consequently, also the mapping *w *= *f*(*z*)
is multi-sheeted near *z*_{0}. 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*
'(*z*_{0}) ¹ 0 ensures
single-sheetedness of the mapping in a sufficiently small
neighbourhood of the point *z*_{0} - this is
proved in the same way as the preceding statement. However, if
the condition *f *'(*z*_{0})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*=*z*^{4}
is not single-sheeted, but at every point of the semi-ring *dw/dz*
= 4*z*³ 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 *z*_{0 }of the domain *D*. This
means that there exists a branch *j*_{0}(*z*)
of the multi-valued function *j*(*z*)
such that *j*(*w*_{0})
= *z*_{0}. At the point *z*_{0},
the derivative* f* '(*z*_{0}) 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 *z*_{0} onto the
neighbourhood of the point *w*_{0}. 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* z*_{0}, for which*
f* '(*z*_{0}) 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 *=
(2*k* + 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

**F****undamental
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 *w*_{0}
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 z*_{0}* given in D and w*_{0}*
in D* and the real number a*_{0}*, 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 *z*_{0}* = w*_{0}*
= a*_{0}* = *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, *F*_{1}(*z*) º *F*_{2}(*z*), however, then also *f*_{1}(*z*) º *f*_{2}(*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 *x*_{0},
*y*_{0}, (*x*_{0}*+iy*_{0}=
*z*_{0}) 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)

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 *w*_{0}, which *f*(*z*)
does not assume on the boundary *C* of the domain *D*,
the number of *w*_{0}-points of teh function *f*(*z*)
inside *D* equals

where *D*_{C}arg
|*f*(*z*) - *w*_{0}| is the total
change of arg |*f*(*z*) - *w*_{0}|
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, D_{C}_{*}arg(*w* - *w*_{0})
equals 2*p* for all points *w*_{0},
lying inside *C**, and vanishes for all points outside*
*C*, whence for all points *w*_{0}*,*
lying inside *C*, N*(*w*_{0}) = 1, but for
all points lying outside *C*, N*(*w*_{0}) =
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=z*³ 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 3*p* ,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')

We use again for the proof the argument principle. According
to this principle, for every point *w*_{0} not on *C*,*
the number* of w*_{0}*-*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., *D*_{C*}arg(*w - w*_{0}) equals -2*p*, if *w*_{0} lies inside *C**, and 0.* *if *w*_{0} lies outside *C** . Thus, *N*(*w*_{0}) = 1 for all internal points of the domain *D*
*and *N*(*w*_{0}) = 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* = *z*³ 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 *z*_{0},
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")

For the proof, we remove from the region *D*
the circle |*z - **z*_{0}| < *r *of small radius with centre at the
point *z*_{0}; the arc of this circle which belongs to *D* we
denote by *g*_{r}, and the part of the contour *C*, which remains
after removal of the circle, by *C*_{r}; we denote by the contour *C*_{r} + *g*_{r}. We apply already to the region , bounded by the curve , the
boundary correspondence principle.

Let *w*_{0}* *be an arbitrary point of the domain *D*. *Since
*w*_{0} is finite and *f*(*z*)
® ¥
as *z* ® *z*_{0}, the
radius *r *of the circle may be chosen so small that in
the reduced part of *D* there would be neither one *w*_{0}-point of the function *f*(*z*). Then, *N*(*w*_{0}) - the number of *w*_{0}-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 *C*_{r}*,*
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 *C*_{r}*** 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
*z*_{0 }in the
form

One may then assert that

because the quantity *w*_{0}(*z*
- *z*_{0})^{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*(*w*_{0})
= 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*(*w*_{0})
= 1.

We have proved that the function* f*(*z*)
assumes in *D *any value *w*_{0}* *of
the domain *D* *once and only once.If the point *w*_{0
}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*(*w*_{0})
< 1 and, consequently, *N*(*w*_{0}) = 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
*z*_{0 }

In particular, let *a = b *= 1, i.e.,
*z*_{0 }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 *z*_{0 }of
infinitely large order below the third. The example with the
function *w* = *z*³ . 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 **z*_{0 }*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
points*z*_{k}*,
*corresponding to the point *w = *¥,
by circles |*z - **z*_{k}|
< _{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 *w*_{0}, not lying on *C* *(and, consequently, finite),
we choose* r*_{k }so
small that the numbers of *w*_{0}-points of the function *f*(*z*) in the
domains *D and * are the same. We now make *r*_{0}* *tend to 0, keeping *r*_{1}, *r*_{2}*,*
··· , *r*_{n-1
}fixed. Here the proof of the
preceding theorem is completely applicable, whence *N*(*w*_{0}) = 1 for all points in *D* *and *N*(*w*_{0}) = 0 for *w*_{0},
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.

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 *z*_{0}* *= -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 *n**j **/2
*decreases from 1 to 0 on the arc *A*_{1}*B*_{1}*
*(we agree to assume that on this arc arg *w
*= 0); on the arc *B*_{1}*A*_{2},
it changes from 0 to -1 (on this arc, we agree to assume that arg
*w *= -*p*);
on the arc *B*_{2}*A*_{3},
from 0 to 1 (assuming arg *w* =
-2*p*), etc. (Fig. 42b).
Correspondingly, on the segment *A*_{1}*B*_{1}*
, *the modulus of the corresponding point* w *grows
from ¥ to 1, and the argument is
equal to zero; on the segment *B*_{1}*A*_{2}*
, *the modulus of *w* decreases from ¥ to 1 and the argument equals 2*p*/*n*;* *on the
segment *A*_{2}*B*_{2}*, *the
modulus of *w* grows from 1 to ¥ (as
before arg *w* = 2*p*/*n*); on *B*_{2}*A*_{3}*
, *the modulus of* w *decreases from ¥ to 1. arg *w* =
2·2*p*/*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 *B*_{k}*
*(*k *= 1, 2, ··· , *n*)*. *Since
each point *B*_{k }is a simple zero for *z*^{n}*
*+ 1, then in its neigbourhood *w *»
*C*_{k}*/*(*z - B*_{k})^{2/n},
where *C*_{k }is some constant. By the
boundary correspondence principle, the corresponding boundary
(Theorem 4 in which now *a*_{k}* *= 1*, b*_{k}* =* 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/*w*_{1}*
*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 *w*_{1} , 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 *z*_{2}*
= *z_{1}*/*(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 *a*²,
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 + *e^{x}* *cos*
*y, *v* = *y + *e^{x}* *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 - e*^{x}*
*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* - *a*_{1}), *y* = *k*(*x*
- *a*_{2}) (*k *¹
0, ¹ ¥ ) into the curves

Introducing in the *w*-plane polar
co-ordinates, *w = **r*e^{i}^{q},we obtain *r** = *e^{x}*, q = **k*(*x - a*_{n}) or

where (*n *=* *1,2).
These are two similar logarithmic spirals. If *k*(*a*_{2}*
- a*_{1}) < 2*p*, then
the length of the vertical segment between the straight lines in
the *z* -plane is less than 2*p *and
the mapping is single-sheeted into the strip between them (cf. **8.**)
Changing *a* from *a*_{1}* *to* a*_{2},
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*(*a*_{2}* - a*_{1})
= 2*p*, both spirals coincide
and we obtain a mapping onto the plane with a removed spiral.

For *k*(*a*_{2}* - a*_{1})
> 2*p, *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 *= Ö*r, **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 2*q *(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* = *re*^{i}^{j}*, w* = *r**e*^{i}^{q}*, *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 C*_{0}: |*z - z*_{0}|
= *R*_{0}, if

1) they lie on one ray through *z*_{0}:

2) |*z - z*_{0}|·|*z* - z*_{0}| = *R*²_{0}.

(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 *C*_{0
}, 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 C*_{0 }*if
and only if they are the vertices of the rays of the circle
orthogonal to the circle C*_{0}.

In fact, let the points *z* and *z** be
symmetric with respect to *C*_{0 }and *G *be an arbitrary circle, passing
though *z and z* *(Fig. 50). Draw through the point *z*_{0}
the tangent to the circle *G**.*
By a known theorem, the square of the length of this tangent |*z*'-*z*_{0}|²
equals the product of the secant |*z** - *z*_{0}|
by its external part *z = *|*z* - *z*_{0}|,
i.e.,

Since *z* and *z** are symmetric
with respect to *C*_{0}, this product equals *R*²_{0
}and, consequently, |*z*' - *z*_{0}| =
*R*_{0}. Thus, the tangent to *G*
is the radius of the circle *C*_{0}, i.e., *G* is orthogonal to *C*_{0}.

Conversely, if *z* and *z** are
vertices of rays of the circle {*G*},
orthogonal to the circle*R*_{0}, then they lie on
one ray, passing through *z*_{0}, because this ray
belongs to the bundle*. Moreover, the tangent *z*_{0}*z'
*to any circle *G *is a
radius of the circle *C*_{0 }and by the same
theorem |*z* - *z*_{0}|·|*z** - *z*_{0}|
= *R*_{0}², i.e., *z *and *z* *are
symmetric with respect to* C*_{0}. 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 *C*_{0
}is a conformal mapping of the second kind (change of orientation). In fact,
let *z*_{0} and *R*_{0 }- the
centre and radius of the circle *C*_{0}, when the
point *z**, symmetric with the point *z* with
respect to *C*_{0}, becomes

whence

|*z* - *z*_{0}|·|*z**
- *z*_{0}| = *R*_{0}² and arg|*z**
- *z*_{0}| = arg |*z* - *z*_{0}|.

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 *z*_{0 }of the circle *C*_{0},
with respect to which there occurs inversion (Fig. 51). Construct
the straight line *C*, *perpendicular to the line of the
centres of the circles *C*_{0 }and *C*, at
the distance *R*_{0}²/2*R *from* z*_{0
}(*R*_{0} and *R *are the radii of*
C*_{0 }and *C*). The similarity of the
triangles *z*_{0}*z***z*
*and *z*_{0}*z**z**
*(Fig. 51) yields

Consequently, the points *z* and *z* *are
symmetric with respect to *C*_{0}. 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 *z*_{0},
then the inversion of this line obviously coincides with itself.

Now, let the circle (or straight line) *C *not pass
through *z*_{0}*. *Construct the point *z*_{1},
symmetric to *z*_{0}* *with respect *C*
and consider the bundle of circles {*G*}
with vertices at *z*_{0}* *and *z*_{1}.
Since all circles *G *pass
through *z*_{0}, by what has been proved above for
inversion with respect to *C*_{0}, the bundle {*G*} becomes the bundle of straight
lines {*G**} Obviously, the
vertex of this bundle will lie at the point *z*_{1}*,
symmetric to *z*_{0 }with respect to *C*_{0}.
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 *z_{1}* and z*_{2 },*
symmetric with respect to an arbitrary circle* *C*,_{
}*into a pair of points *z_{1}** and
z*_{2}**, 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 z_{1}*
and z*_{2}. On inversion, it becomes the bundle of
circles {*G****}with
vertices at z_{1}** and z*_{2}**. *Since
the circles *G* are orthogonal
to *C, *the circles *G***
*are also orthogonal to *C*, *whence z_{1}**
*and* z*_{2}** *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)

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(*l**z*) = *a*,
which does not pass through the point *z = -d*/*c
*(*a *¹ -Re(*l**d*/*c*))
the circles |*w* - w_{0}| = *r*,
where

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

c) The circle |*z - z*_{0}| = *r*, not
passing through the point *z = -d*/*c*(*r *¹ |*z + d*/*c*|) is the
circle |*w* - *w*_{0}| = *r*, where

d) The circle |*z - z*_{0}| = |*z*_{0
}*+ 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* z*_{1}*, *z_{2}*, z*_{3}
and *w*_{1}, *w*_{2},*w*_{3},
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 *z*_{1}*,
*z_{2}*, *and *z*_{3} into *w*_{1},
*w*_{2} and *w*_{3}, 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 = l*_{1}(*z*)
and *w = l*_{2}(*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 *z*_{k}
into 0, 1 and ¥. Consider now the
transformation

where *L*_{1}^{-1}
is the inverse mapping of *L*_{1}. 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 **" = L*_{2}[*L*^{-1}(*z** '*)] º
*z** '*, i.e., *L*_{1}^{-1
}is the inverse of *L*_{2 }and *L*_{1
}º *L*_{2}.
However, then also *l*_{1}* *º *l*_{2}, 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 *z*_{k}*
*or *w*_{k }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 *w*_{3 }*=*
¥ , *z*_{2 }*=*
¥, then (2) becomes

or *w* = *w*_{1}* + *(*w*_{2}*
- w*_{1})(*z - z*_{1})/(*z* - z_{3})
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 z*_{k}*
into three arbitrary points w*_{k}*.*

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 *z*_{k
}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 *w*_{k }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 *w*_{k} are distributed
with respect to *K* *as the points *z*_{k}*
*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 *z*_{1}*, *z_{2}*
*and *w*_{1}, *w*_{2} and *w*_{3},
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 *z*_{3 }=
*z*_{2 }+ *h *and *w*_{3 }= *w*_{2
}+ *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 = e*^{i}^{a}* *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* = *e*^{i}^{q} 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* = *e*^{i}^{a} , 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 *k*_{1}* *are
constants. We select *k*_{1 }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: |*k*_{1}(1 - *a*)/(1
- )|
= |*k*_{1}| = 1. Consequently, one may take *k*_{1
}= *e*^{i}^{a},
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 = e*^{i}^{j}* *and *w = e*^{i}^{q}* *(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 *z*_{k}
= *x*_{k}*, w*_{k}*
= u*_{k}* *of the real *x-*
and *u*-axes. Since the numbers *z*_{k}
and *w*_{k}* *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.