**33. Examples**** **We consider examples
of conformal mappings which involve combinations of elementary
functions.

1) *Mapping of
a strip onto the unit circle *Let there be given in the *z*-plane
the strip *D*: -*p*/4
< Re *z* < *p*/4,
which is to be mapped onto the circle |*w*| < 1 with
correspondence of the three boundary points: *f*(±*p*/4) = ±1, *f*(*i* ¥) = *i *(*i
*¥ denotes the upper point at infinity of the strip).
First of all, we rotate by a right angle and double the width of
the strip:

next we use the fact that the exponential function

transforms the
strip -*p*/2 < Im *z*_{1} <*p* /2, onto which (2) maps *D,*
into the right half-plane Re *z*_{2} > 0
(really, whence changes from 0 to ¥, arg *z*_{2 }= *y*_{1 }from -*p*/2
to *p*/2). There remains to map this
half-plane onto the unit circle so that *z*_{2 }=
*i*, -*i*, 0,
corresponding to the points *z *= *p* /4, -*p*
/4, *i *¥ become the
points *w*
= 1, -1. *i.* Such a problem is solved with the aid
of (2) of **32.****
**(wehave changed the notation):

Substituting (2) and (1) into (3), we obtain the final solution of the problem

(cf. **9.**)
We will still explain the correspondence of the lines for this
mapping. The family of vertical lines Re(*z*) = const. for
Mapping (1) become the family of horizontal lines, which Mapping
(2) converts into the family of rays arg *z*_{2} =
const., i.e., into the circles passing through the points *z*_{2}
= 0 and *z*_{2} = ¥
.The bi-linear mapping transforms these lines into the points *w*
= *i *and *w* = *-i. *Consequently, the rays
considered become the family of circles passing through the
points *w* = ±*i. *Th orhogonal family of segments
Im *z* = const. become the family of circles with *w*
= ± *i *as points of symmetry (Fig. 55).

The inverse function

realizes the inverse mapping of a circle into a
strip. Replacing here *iz* by *z*_{1}* *and
*iw* by *p**w*_{1}*/*2*H*,
we obtain the mapping of the circle |*z*| < 1 onto the
strip of width *H*: -*H*/2 < Im* z* < *H*/2.
It has the form

(where we have changed to *z* and *w*).Mapping
(5) takes the points *z* = ±1 into the point *w* =
± ¥ and the point *z** *= *i *into *w=iH*/2..
Its derivative

becomes infinite at the points *z* = ±1.

2) *Mapping of half-plane* *with
removed cut onto half-plane. *Let there be excluded from the
half-plane Im *z *> 0 the segment (*a*. *a +
ih*). In order to obtain this mapping, we use the fact that
the mapping *w* = *z*² doubles the angles at the
origin and, consequently, one may "straighten" the
angle between the excluded segment and the *x*-axis.

Correspondingly, we move the half-plane *z*
by *a* to the left: *z*_{1}* = z - a *and
obtain with the mapping *z*_{2 }*= z*_{1}²
the plane with the removed ray -*h*² < Re *z*_{2}<
¥, Im *z*_{2} = 0. Next, we again shift the plane *z*_{2
}by *h*² to the right: *z*_{3 }*=
z*_{2} + *h*². Finally, applying the mapping *z*_{4
}*= *Ö*z*_{3},
we obtain the upper half-plane. Thus, the mapping becomes

Shifting again the *z*_{4 }*-*plane by *a*
to the right, in order to bring the point *z* = *a*
+ *ih* into the point *a*, we obtain finally

The derivative of Mapping (7)

vanishes at the points *B* and *D *(where
*z* = *a*) and becomes infinite at the point *C*
(where *z = a + ih*). The lines *v* = *v*_{0}*
*= const. correspond to the lines of fourth order

symmetric with respect to the line *x* = *a, *on
which their ordinates attain their maxima. For large values of *v*_{0},
the curves (9) differ little from straight lines (Fig. 56).

For *h = *0, Mapping (7) becomes the
identity mapping *w = z*. We will find the principal part
of (7) for small *h*. For this purpose, we transform (7),
neglecting powers of *h* larger than the second. Applying
the well -known formula for roots, we find

The approximate formula (10) ceases to be true for *z*
close to *a*, because then the quantity *h*²/(*z
- a*)ceases to be small.

3) *Mapping the circle with removed radial cut onto the
unit circle *(Fig. 57). Exclude form the circle |*z*|
< 1 the straight segment [(1 - *h*)*e*^{a}, *e*^{i}^{a}]. The transformation
of the region obtained onto the unit circle may be reduced with
the aid of the supplementary bi-linear mapping to (7). However,
it is simpler to employ the properties of the Joukowsky function **7.**
Turning the region in the *z-*plane by the angle *-a *and applying the Joukowsky
function *z *=* *½(*z/e*^{i}^{a }+ *e*^{i}^{a}/*z*), we
transform this region into the outside of the segment [-1, 1 + 2*h*_{1}],
where *h*_{1 }= *h*²/4(1 - *h*)*.
The analogous transformation *w** *=*
*½(*w/e*^{i}^{a }+ *e*^{i}^{a}/*w*)

converts the circle in the *w*-plane into the outside
of the segment [-1, 1]. It is readily seen that the linear
transformation

maps arcs into arcs of the obtained regions in
the *z *- and w- planes; replacing w and
*z *by their expressions, we
find the required mapping:

* In fact, images of the points *z
*= (1 - *h*)*e*^{i}^{a}* *are for the mapping under consideration are the points *z*_{0}* *= ½[1 - *h
+* 1/(1 - *h*)] = 1 + *h*²/1(1 - *h*).

For *h *= 0, we have *h*_{1}*
*= 0 and *w* º *z*.
We will find the principal part of Mapping (11) for small *h*.
For this purpose, we substitute *w* = *z* + *w* and neglect smaller than second
order terms of *h*_{1}, noting doing this that *w* and *h*_{1}are of
the same order (so that quantities of order *w* ² may be neglected). We obtain

or

whence

Thus, for small *h*_{1} and points *z*
not too close to the point e^{i}^{a} , our conformal mapping may be represented by the
approximate formula

Differentiating (12), we obtain the principal part of the derivative

Note yet a link between the arguments of the
points *z* = e^{i}^{j} and *w*
= e^{i}^{q} of the circles,
corresponding to each other by Mapping (11). Separating in (12),
after substituting *z* = e^{i}^{j} and *w*
= e^{i}^{q}, the real part and
setting

we obtain*

whence

* We take the first two terms of
the Taylor series for cos *q* at the point *q *= 0.

4) *Mapping of plane with
removed rays onto the strip *0* < v < H *(Fig.
58). For the sake of definiteness, we require that the left of
the ray becomes the lower shore , the right the upper shore *.

* These conditions determine only two of the real parameters of the mapping (they reduce to giving the corresponding two pairs of boundary points), the third remains arbitrary (cf. (15)),

We map by the bi-linear mapping

our rays into the single ray (0, ¥) and then onto the real axis by the transformation

The region under consideration then becomes the
upper half-plane. In order to obtain the lower correspondence of
the points, we subject this half-plane to a bi-linear mapping
onto itself so that the images of the points at infinity *A *and*
C* of the initial domain in the* z*-plane (i.e., the
points 1) become 0 and ¥ :

(*k *is an arbitrary positive constant).
There remains still to apply the logarithm, *w* = (*H*/*p*)ln *z*_{3}, in
order to obtain the mapping onto the strip with lower
correspondence of the boundaries

where *c* = (*H*/*p*)ln
*k/a* is an arbitrary real constant. There correspond to
the two families of lines *u *= const. and *v* =
const. in the *z*-plane for the mapping (15) families of
ellipses and hyperbolae with foci ±*a.*

5) *Mapping of the strip *0 <* y
< *2*H with the cut - *¥
£ *x* £
*a, y = H onto the strip* 0 < *v* < 2*H *(Fig.59).
The function

maps the strip with its cut onto the upper half-plane with the
removed segment (0, *bi*) of the imaginary axis, where *a*=*e*^{a}^{p}^{/2H}.
Function (7) of Example 2):

(in (7), one must set *a* = 0, *h*
= *b*) maps this last domain onto the half-plane. Applying
the logarithm, we find the required mapping

6) *Mapping of the strip* 0 < *y* < 1 *with
the cut* 0 £ *y* £ *h*, *x = a onto the strip *0*
< v* < 1 (Fig. 60). The function *z* = *e*^{p}^{(z-a)}*
*maps the strip with cut onto the upper half-plane with a cut
arc of the unit circle. The mapping

takes this arc into the segment of the imaginary axis (0, *bi*),
where *b *= 1/*i *tanh *p**hi*/2
= tan *p**h*/2.

Using again (7) of Example 2), we obtain the mapping of the given domain onto the upper half-plane:

The points *A* and *E *go for this
mapping to the points

by the bi-linear mapping

we take them to 0 and ¥ and then apply the logarithm

Thus, we obtain the mapping onto the lower strip.
However, obviously, there corresponds to the point *C* the
point *z*_{5} = 0; in order order to shift it to
the point *a* on the real axis, one must still move this
strip to *a*. Thus, the required mapping has the form

or, finally,

For small *h*, the expression in curly brackets, which
we will.denote by *z *, becomes

(we replace cos, tan and Ö
by their approximate expressions and neglect during
multiplication terms of higher order than *h*²). Using
elementary formulae for the hyperbolic functions, we findd:

We now obtain the approximate formula for the conformal
mapping (17). For this purpose, we replace on the right hand side
of (17) arctanh *z *by two
terms of its Taylor expansion with centre at the point *z*_{0}* **=*
tanh* **p*(*z - a*)/2:

Substituting on the right hand side the values *z *and *z*_{0},
we find finally

7) *Mapping of eccentric circular ring onto
concentric ring. *We consider first the case when each
neighbourhood of the ring lies in the outside of the other (Fig.
61). We construct on the common tangent to these circles, as on
the diameter, the semi-circle *G *;
it intersects the line of the centres of the circumference of the
ring at two points *a* and *b *which are symmetric
simultaneously with respect to the two circles *C*_{1}* *and *C*_{2},* *because there pass
through *a* and *b *the line of the centres and the
line *G*, orthogonal to both
circles. By the property of the bi-linear function

the circles *C*_{1}*
*and *C*_{2},
become the to circles *C**_{1}*
*and *C**_{2},
relative to which the points *w* = 0 and *w* = ¥ , corresponding to the points *z*
= *a* and *z* = *b, *are symmetric.
Consequently, the point *w* = 0 is the common centre of *C**_{1}* *and *C**_{2}. For this, the eccentric ring
between *C*_{1}* *and
*C*_{2} for this goes
to the concentric circular ring between *C**_{1}* *and *C**_{2}, Fig. 61 likewise shows the
correspondence of the lines for this transformation; the net in
the *z* -plane is a part of the net in Fig. 53.

The additional mapping

where *q* ranges from -¥ to +¥, gives the ring obtained onto the strip.
This fact does not contradict the statement in **28.****
**regarding the impossibility of mapping doubly-connected
domains onto simply-connected ones, because the mapping function
(20) is multi-valued. And what is more, the function (20)
generates a single-sheeted mapping onto the strip domain on its
Riemannian surface, lying on the ring, and , obviously, this
domain is simply-connected.

The case when one circle of the ring lies inside
the other leads to a study with the aid of an additional linear
mapping *z*=1/(*z-c*), where *c *is an
arbitrary point, lying between the circles.

**34.
Mapping of sickles****. **We will call a
circular sickle a domain bounded by two arcs of circles in the
entire plane (i.e., in particular, also straight lines). The
examples which we will consider here have an important role in
applications as well as the further development of the theory.

1) *Mapping of the outside of an arc onto the
outside of a circle. *(This is the degenerate case when the
two arcs, bounded by a sickle, coincide). We will assume that the
ends *AB *in the *x-*plane lie at the points ±*a
*and that the circles in the *w-*plane passes through
those points. Moreover, we will assume that the centre of the arc
lies at the point *z = hi*, the centre of the circle at
the point *w* = *hi* so that the tangent to the arc
at the point *a *forms with the negative *x*-axis
the angle *a *= 2arctan *h*/*a,
*and the tangent to the circle at the point *w = a - *angle
*b *= *p*/2 - *a*/2)* *with the positive *u*-axis
(Fig. 62).

We map with the aid of the bi-linear function

the outside of the arc *AB* onto the
outside of some ray. Since |*dz*_{1}/*dz*|_{z=a}
> 0, the angle of inclination of this ray to the negative axis
is likewise *a*. Moreover, we
will seek the mapping onto the outside of the obtained circle in
the *w*-plane. We employ for this purpose again the
bi-linear mapping

which maps the circle in the half-plane as well
as its neighbourhood into some line. Since |*dw*_{1}/*dw*|_{w=a}
> 0, the inclination of this line to the positive axis is *b*. Consequently, the mapping

takes our circle along the outside of the ray,
forming with the positive axis the angle 2*b
*=* p - a*. Thus, this ray coincides with the
one obtained for Mapping (1); eliminating *z*_{1}
between (1) and (2), we obtain the required mapping

We find from this equation

For the mapping under consideration, any circle *C*
', touching the circle *C* at the point *a*, is
mapped into a closed curve enveloping the arc *AB *and
having at the point *B* (*z = a*) a point of
return; this curve resembles the profile of an aerofoil (Fig.
63).

The function (3) realizes the conformal mapping
of the outside of this curve onto the outside of the circle,
bounded by the circle *C* '.

The method of obtaining classes of aerofoils, proposed by N.E.Joukowski, is based on this observation; it is especially simple for computations (Joukowski profile).

The shape of the Joukowski profile depends on three
parameters: *a - *characterizing its width, *h -*
its curvature and *d* the distance between the circles *C*
and *C* ' (Fig. 63).

2) *Mapping of half-plane with deformed
segment onto half-plane. *The function *z*_{1}*
= z*/(*a - z*) maps the given domain (Fig. 64) onto
the sector *a* < arg *z
< **p*.
Consequently, the function

maps this domain onto the upper half-plane. We give still the
normalization *w*(¥)=¥, *w'*(¥
) = 1; since for this mapping the point *z* = ¥ goes to the point *z*_{2}*
= -*1,* *one adds the linear transformation

where the constant *k* is found from the
second normalization condition. In fact,

whence, expanding the power inside the brackets
by the binomial formula, we find for large |*z*|

Consequently,

Finally, we have

where *C *is an arbitrary real constant.
In order to obtain the principal part of Mapping (4) for small *a*
and *a*, we use the first terms
of the Taylor series. We have

whence

and, multiplying this by

we find

*Here and hereafter multi-term denotes terms of 4-th order smallness.

We will now compute the area *s
*of the removed section. We have: *s
*= *ar*²* *- (*a*/2)*r *cos
*a *, where *r* = *a*/2sin
*a *is the radius of the
circle. Neglecting small quantities of higher order, we obtain: *s *= *(a*²/4)(*a*/sin² *a -
*cotan *a*) » *a*²*a*/6.
Thus, we may finally write (5) in the form

Resorting to a parallel displacement, we obtain the somewhat more general result: The function

generates a conformal mapping onto the
half-plane *v > *0 of the half-plane *y* > 0
with removed area *s*, bounded
by the segment (*b*, *b + a*) and another circle of
smaller curvature.

3) *Mapping of a circle with removed circular
segment onto a circle.* Let the removed sickle be close to *e*^{i}^{a}* *and the area *s *of this sickle be small (Fig.
65). We execute the two bi-linear mappings

of the unit circles of the *z-* and *w-*planes
onto the upper half-planes *z*
and *w*.

The sickle *s *becomes*
*the sickle

adjacent to the point *z*
= 0. By (6), we then find:

or, reverting to the variables *z *and *w*,

(we disregard everywhere terms of smaller order
than *s*). For this mapping,
the point *z* = 0 becomes the point *w
*_{0}=*s **e*^{i}^{a}/8*p*;
executing the additional bi-linear mapping

of the circle onto itself, so that *w *_{0}*
*goes to 0, we obtain

After substituting for *w* its approximate
value (8) and simple transformations (in which we again neglect
small order terms above *s*),
we obtain finally

(we write again *w* instead of *w *_{1}). Mapping (9)
establishes the following correspondence of the points *z *=
*e*^{i}^{j}
and *w=e*^{i}^{q}:

or ( if one takes imaginary parts and neglects small higher order terms)

We find for the modulus of the derivative of the mapping on the boundary

and for the derivative at the origin

4)* Mapping of a strip with removed sickle
onto a strip. *Let there be removed from a strip 0 < *y*
< 1 the segment *s*, bounded
by the segment (0, *a*) of the real axis and the arc of a
circle of small curvature (Fig. 66). Performing the supplementary
mappings *z *= *e*^{p }^{z}*,
**w *= *e*^{p }^{w }of
the strip onto the upper half-plane and applying (7),

where *s*** = |d**z**/dz|*_{z=0}·*s *= *p*²*s *is the area of the segment*, we
find the required mapping

(we neglect everywhere small order terms of *s*). Shifting yet the *w*-plane
by *s*/2, we find finally

* We assume in (7) that *b *=
1 and const. = *s***/**p.*

By an additional shift of the *z*-plane,
we obtain the general result: The function

maps conformally the strip 0<*y*<1 with the
removed circular section *s, *located
on the segment (*b*,*b+a*) onto the strip 0<*v*<1.
It establishes the following correspondence of the points *y*
= 0, *y *= 1 and the lines *v = *0, *v = *1:

5) *Formula for the extension
for mapping of a sickle onto a strip. *Let *the sickle D*
be bounded by the arcs *C*_{1} and *C*_{2}
of circles intersecting at the points ±*a* and let -*ih*_{1}*
*and* -ih*_{2}* *be the points of
intersection of these arcs with the imaginary axis. Using (5) of
1) in **33.**, it is
simple to construct the conformal mapping of the sickle **D**
on to the strip 0 < *v* < *h*:

where *l *=
2arctan *h*_{k}/*a, k = *1, 2*.
Differentiating (16), we find

* It is sufficient for obtaining
(16) to note that (5) of **33.** , in which *z*
is replaced by *z*/*a*, yields the mapping of the
given sickle onto the horizontal strip the boundary of which
passes through the points

(cf. (11) of **9.**). The
width of this strip is *h* = (*H*/*p)*(*l*_{1}* - l*_{2}), whence one finds *H*. There remains to shift the
strip so that its lower shore coincides with the real axis; we
thus arrive at (16).

One can obtain from (17) an
approximate formula, which is important for applications. Let *z*_{0} be the point *C*_{2}, *n* the
segment of the normal to the circle *C*_{2 }at this point which
is inside the sickle *D*, the angle between the
tangents to *C*_{1
}and *C*_{2
}at the ends of the segment *n *and
*k*_{1 }, *k*_{2 }the curvatures of *C*_{1
}and *C*_{2
}(Fig. 67). We assume that *h*
and together with it *k*_{1 }- *k*_{2 }and *n are *infinitely small of first order and
the curvatures *k*_{1 }and *k*_{2 }are bounded. Then, we may assert that

where* *one may
represent *h *in the form of a homogeneous third degree polynomial in *n*,
, *k*_{1
}- *k*_{2
}with bounded coefficients.

In order to derive (18) from (17),
one must express in the latter *a*, *l*_{1}*, l*_{2}, z_{0}* *in
terms of *n, *, *k*_{1}, *k*_{2}
and then expand |*f *'(*z*_{0})|
in powers of *p*, , *k*_{1}- *k*_{2}
to second order inclusively. However, the actual realization of
this approach leads to most complicated manipulations. Hence we
will obtain (18) for the particular case when the circle *C*_{2 }coincides with the *Ox*-axis, i.e., when *k*_{2} = 0 (Fig. 68). In this case, (17)
yields

where *l*_{1}*= *2 arctan *ly*/*a* is half of the angle of
the arc *C*_{1} (Fig. 68). On the other hand, we have *l*_{1}=* *arcsin *a**k*_{1}+ *a*³*k*³_{1}/6 + ···, *x =* (sin )/*k*_{1}, *n *- cos /*k*_{1} - cosl_{1}/*k*_{1},
whence

Moreover, we have exact to fourth order:

and

Dividing the first of these equations by the second, we obtain, including second order quantities ,

Using to the same order of accuracy the relation

we obtain finally

which coincides with (18) for *k*_{2}
= 0.

We go for the transition to the general formula
when *k*_{2 }¹ 0 to the auxiliary *z-*plane
and in it the sickle *D**, bounded by the segment of the
real axis *C*_{2}*** and another circle *C*_{1}**
*with curvature *k*_{1}*;
we denote by *n* *the segment of the imaginary axis,
included in the sickle *D** and by the angle, formed by
the arc *C*_{1}** *and
the perpendicular to the imaginary axis at the point of their
intersection (Fig. 69). We map conformally the lower half-plane *z *onto the outside of the circle *C*_{2} with curvature *k*_{2}:

Then, the sickle *D* *becomes the sickle *D*,
bounded by the arc *C*_{2} and the other circle *C*_{1} with a certain curvature* k*_{1}, the i,imaginary axis becomes the
real axis and the segment *n* *the segment *n *of
the real axis, where

and the circle *C*_{1 }forms
with the perpendicular to the real axis at the point *z*_{1 }of their intersection the angle (Fig.
69).

We find now the link between *k*_{1} and the other parameters. For this
purpose, we denote by *ds* the length element of the arc *C*_{1}, by *a*(*s*)
the angle formed by the tangent to *C*_{1
}with the *x*-axis (*s* is measured
from the point *z*_{1} so
that *a*(0) = + *p*/2) and by *ds** the
element of the arc *C*_{1}*,
corresponding to *ds*. We have *k*_{1
}= *d**a*/*ds*.
However, the infinitesimal increment of the angle *a* may be given the form

where the second term *d*arg *dz*/*d**z *= Im d ln *dz*/*d**z *denotes the increment of arg *dz*/*d**z *on the arc *d**z* = *e*^{i}^{q}*ds** of the
circle *C*_{1}*, joining to
the point *z*_{1}* = **-n*i*. By (21), the
increment equals

and we find for the curvature

because by the same formula (21)

Hence also by the second formula (21), by which
1/(1 - *n*²) = 1 + *k*_{1}*n*/2,
we find

Let the function *w = f*(*z*)
realize the conformal mapping of the sickle, bounded by the arcs *C*_{1 }and *C*_{2 }onto the strip of width *h*. By (21), we have at
the point *z*_{0}* = *1/*k*_{2} , corresponding to the point *z *= 0,

The quantity |*dw*/*d**z*|_{z}_{
= 0} may be calculated with (20) by replacing in it *n*
and *k*_{1 }by *n**
and *k*_{1}*. Using the
found values (22) and (23) for *n** and *k*_{1}*, we obtain to terms of second order
inclusively

Since, by the conditions that the difference *k*_{1 }- *k*_{2 }is
small, the same degree of accuracy has *k*_{1}*k*_{2 }*n*² = *k*²_{2 }*n*² and we obtain the
required formula

**2.3
Symmetry principle and mapping of polygons**

We will consider now the methods** **of
great importance for the actual construction of conformal
mappings. The first of these methods** **operates on
the so-called symmetry principle, which was formulated by Riemann
and established by G. Schwarz. This method , as will be shown in **36.****, **allows
to construct in certain cases the solution of the problem of
finding conformal mappings.

The second method is especially important for
applications, since it yields the possibility of writing (it is
true, generally speaking,** **only in the form of an
integral ) the function which realizes** **the
mapping of the upper half-plane onto an arbitrary domain, bounded
by a polygon.

**35.
Symmetry Principle**** **It states in one
particular case a simple sufficient condition for the existence
of the analytic continuation function, realizing a conformal
mapping:

**Theorem 1 **(B. Riemann, G.
Schwarz) *Let the boundary of the domain D*_{1}*
contain an arc of the circle*** ***C and the
function w = f*_{1}(*z*)** ***realize***
***a conformal mapping of this domain onto the domain D*_{1}** such that the arc C becomes the
segment C* of the boundary D*_{1}**
, which is also an arc of a circle. Under these*** ***conditions,
f*_{1}(*z*) *permits the analytic
continuation f*_{2}(*z*)** ***through
the arc C* *into the domain* *D*_{2},
*symmetric with D*_{1} *with
respect to C, where the function w=f*_{2}(*z*)*
realizes the conformal mapping of D*_{2 }*onto
the domain D*_{2}*, symmetric
with *D*_{1}** relative
to C*, and the function*

*realizes the conformal mapping * of the
domain D*_{1}* + C + D*_{2}* onto
the domain D*_{1}** + C* + D*_{2}**.*

For single-sheetedness of this
mapping, one must demand that the domains *D*_{1}***
and* D*_{2}**** *do not
intersect.

For the proof, we perform the bi-linear mappings

transforming *C* and *C**
into the segments *G** *and
*G **'** o*f the real axes of the *z*- and *w-*planes;
let the domains *D*_{1} and *D*_{1}**
*thus* *become the domains *D*_{1}
and *D*_{1}**, *and
the function *w = f*_{1}(*z*) the function *w **= l*_{*}*f*_{1}*l*^{-1}(z) = *j*(*z*),
realizing the conformal mapping of *D*_{1}
and *D*_{1}** *(Fig.
70)*. We denote by *D*_{2}
the domain, symmetric with respect to *D*_{1}
relative to *G*.

* Euler has formulated the symmetry principle for this case in 1777.

We now construct in *D*_{2}
the function

and show that it is by analytic continuation the
function *f*_{1}(*z*).
To start with, the function *j**
*_{2}(*z*) is
analytic in *D*_{2}. In
fact, we have for any points *z *and
*z *+ Dz of *D*_{2}:

where and + are
points in *D*_{1}. By
the strength of the analyticity of *j*_{1}(*z*) in *D*_{1},
the right hand side has a limit for

® 0, whence there exists also the
derivative

at any point of the domain *D*_{2},
i.e., *j** *_{2}(*z*) is analytic in *D*_{2}. By its design, there
exist the boundary values of *j**
*_{2}(*z*) on the
segment *G*:

because, by the boundary correspondence principle
for conformal mappings (**29.**)
there exists

Relation (3) assumes the form

but, since the value of *j**
*_{1}(*x*) is real (*G**,
by assumption, is a segment of the real axis), then on the
segment *G*

Hence, by the principle of continuous continuation (**25.**),
*j*_{1}(*z*) is the analytic continuation
through *G*. of *j*_{1}(*z*)

It also follows from the construction of the function *j** *_{2}(*z*) that it realizes the conformal
mapping of *D*_{2}* *onto *D*_{2}*,
symmetric to *D*_{q}
with respect to *G**. However,
the function *j*(*z*), constructed from *j** *_{1}(*z*) and its analytic continuation *j** *_{2}(*z*):

realizes the conformal mapping of *D*_{1 }+ *G
*+ *D*_{2}* *onto *D*_{1}*
+ *G***** *+ *D*_{2}*.

We return now to the old variables *z* and
*w *with the aid of the substitutions, inverse to (1). By
the strength of the properties of the bi-linear transformations,
we obtain in the domain *D*_{2}, symmetric with
respect to *D*_{1} relative to the arc *C*,
the function *f*_{2}(*z*), analytically
continued by *f*_{1}(*z*) through the arc *C*
and realizing the conformal mapping of *D*_{2}
onto *D*_{2}*, symmetric with *D*_{1}*
with respect to the arc *C*, *and the theorem has been
proved.

As an example of the application of the symmetry
principle, we will prove the uniqueness theorem of conformal
mapping for the given correspondence of three boundary points,
about which we have spoken in **29.**

**Theorem 2 ***There exists one
and only one conformal mapping w = f*(*z*) *of the
domain D onto the domain D*, which transfers three boundary
points z*_{k}* of D into three boundary
points w*_{k}* of the domain D*. The
points z*_{k}* and w*_{k}*
are arbitrary, but in the same order following the passage around
the borders of the domains.*

We consider first the case when *D *and *D* *are
unit circles. By (2) of **32.**,
one may construct the bi-linear mapping of the circle |*z*|
< 1onto the circle |*w*| < 1 with the given
normalization *f*(*z*_{k}) = *w*_{k}*.*
We will prove the uniqueness of this mapping. Let *w = g*(*z*),
*g*(*z*_{k}) = *w*_{k}*
*be another mapping of the circle |*z*| < 1 onto
the circle |*w*| < 1. The function *g*(*z*)
satisfies the conditions of the symmetry principle and,
consequently, continues analytically into the domain, symmetric
to the circle |*z*| < 1 with respect to the circle |*z*|
= 1, i.e., in the outside of the circle |*z*| > 1.
Together with its continuation, *w* = *g*(*z*)
realizes a single-sheeted mapping of the entire *z*-plane
and, by Theorem 1 of **31.**,
is a bi-linear function. However, then it may be asserted that *g*(*z*)º*f*(*z*), because, by the
bi-linear mapping, proved in **32.**, it is
completely determined by stating the correspondence of the three
points.

The general case is readily reduced to the one
considered. In fact, let *z *= *j*(*z*),
*z*_{k}* *= *j*(*z*_{k})
and *w *= *y*(*z*),
*w*_{k}* *= *y*(*z*_{k})be
any conformal mappings of the domains *D* and *D* *onto
the unit circles |*z*| < 1
and |*w*| < 1 and *w *= *F*(*z*),
*F*(*z*_{k})
= *w*_{k }the
mapping of the circle |*z*|
< 1 onto the circle |*w*|
< 1 (its existence and uniqueness were proved above). The
mapping of the domain *D* onto the domain *D* *with
given normalization will obviously be

where *y*^{-1}*
*is the inverse mapping of *y*.
The existence of the second mapping *w* = *g*(*z*)
of *D* onto *D* *with the same normalization would
reduce to the existence of the second mapping

of the circle |*z*|
< 1 onto the circle |*w*|
< 1 with normalization *G*(*z*_{k})
= *w*_{k},
which contradicts the statement above, and the theorem has been
proved.

We will still consider an application of the
principle to the problem of the existence of conformal mappings
of multi-connected domains. By the basic theorem of **28.**,
any two simply-connected domains may be transformed
singe-sheetedly and conformally onto each other. On the other
hand, we have seen that it is impossible to also map a
simply-connected domain onto a multi-connected one. There arises
the question whether it is possible to map one onto another
domain of the same order of connectivity. It turns out that also
the answer to this question, generally speaking, is negative. In
fact, even in the simplest case of concentric rings there holds

**Theorem 3 ***For the existence
of a conformal mapping w = f*(*z*) *of the ring r*_{1}*
< |z| < r*_{2}* onto the ring **r*_{1}* < |z| < **r*_{2}* , it is necessary
and sufficient to satisfy the condition*

In order to prove the necessity of Condition (5),
we note that the function *f*(*z*) satisfies the
conditions of the symmetry principle and on the basis of this
principle continues analytically in the domains

symmetric with the ring *r*_{1}*
< |z| < r*_{2} with respect to to the circles *|z|
= r*_{1}* *and* |z| = r*_{2}*,
*respectively. The ring under consideration *r*²_{1}*
/r*_{2 }*< |z| < r*²_{2}*/r*_{1
}is mapped by *f*(*z*) (together with its
continuation) onto the ring *r*²_{1}*
/**r*_{2 }*<
|z| < **r*²_{2}*/**r*_{1}. Hence we apply again
to the function *f*(*z*) the symmetry principle and
may continue it into the ring

Executing such a continuation unlimitedly, we
find that *f*(*z*) achieves a single-sheeted
mapping of the domain 0 < |*z*| < ¥ onto the domain 0 < |*w*| < ¥, where either

depending on whether the circle *|z| = r*_{1}*
*corresponds to the circle *|w| = **r*_{1}*
*or *|w| = **r*_{2}.
Hence we conclude that *f*(*z*) is a bi-linear
function of one of the two forms

where *a* is a complex constant. In both
cases, obviously, (5) is fulfilled. The sufficiency of (5)
follows from the fact that, where it is fulfilled, the rings are
similar and may be mapped onto each other by simple stretching.

In addition to the proved theorem, we state that
any doubly-connected domain may be all the same mapped onto some
ring *r*_{1 }*<
|w| < **r*_{2},
where for a given radius *r*_{1
}the radius *r*_{2}*
*is determined uniquely for a given domain. In the same
manner, an arbitrary *n*-ply-connected domain may be
mapped onto some domain, obtained by omission from it of *n*
circles. The proof of these results can be found in M.V.Keldysh's
paper [7] or R.Courant's book [5].

In conclusion, we present a generalization of the
symmetry principle to the case when the boundary of the mapped
domains contains analytic arcs. We call an arc *C* *analytic*,
if it may be given by the parametric equations

in which *x*(*t*) and *y*(*t*)
are *analytic functions of the real variable t *on an
interval (*a, b*), i.e.,
functions which can be expanded in the neighbourhood of every
point *t*_{0}* *of this interval in series
of power of *t* - *t*_{0}*. *For
this, it is assumed that nowhere in the interval *x*'(*t*)
and *y*'(*t*) vanish simultaneously (i.e., there
are no special points of *C*). The curve *C* may
also be closed if *x*(*a*)
= *x*(*b*), *y*(*a*) = *y*(*b*).

Here applies the so-called *Principal of
Analytic Continuation *

**Theorem 3 (G. Schwarz) ***Let
the function w = f*(*x*) *yield a conformal mapping
of the domain D, the boundary of which contains the analytic arc
C, onto some domain D*, where there corresponds to the arc C an
analytic arc C* of the boundary of the domain C*. Under those
conditions, the function w = f*(*x*) *may be
analytically continued through the arc C.*

In fact, let the point *z*_{0}*
*be an arbitrary point on the arc *C, z = z*(*t*)
= *x*(*t*) + *iy*(*t*) be the
equation of this curve and *z*_{0}* = z*(*t*_{0})
. Since, by assumption, *C* is an analytic arc, then in
some neighbourhood |*t* - t_{0}| < *d *the function *z = z*(*t*)
may be expanded in a series of powers of *t* - t_{0};
by Abel's Theorem, this will converge also for complex values of *t*,
which we will denote by *z*, in
the circle |*z* - t_{0}|
< *d*. Consequently, in this
circle, the analytic function *z = z*(*z*)
is defined. Since *z'*(t_{0}) ¹
0, then, restricting in the case of need to *d*,
one may assume that the mapping *z = z*(z) is conformal. By its design, z = z(z) maps the diameter of the circle |z - t_{0}| < d
onto some segment of the circle C, containing the point z_{0};
we will assume that the upper semi-circle is mapped for this
inside the domain D, the lower one into its outside.

The function *w* =* f*(*z*)
yields the conformal mapping

(*w* = *w*(*w*)
is the inverse function of *w* = *w*(*w*)) of the upper semi-circle |*z* - *t*_{0}| < *d* onto some part of the upper
semi-circle |*w-t*_{0}|
< *d*_{1} , where
the diameter of the first semi-circle becomes part of the
diameter of the second circle. By the symmetry principle of
Theorem 1, the function *j*(*z*)
admits analytic continuation into the lower semi-circle
and (together with its continuation) yields the conformal mapping
of the entire circle |*z* - *t*_{0}|
< *d* onto some part of the
circle |*w **- **t*_{0}| < *d*_{1} , containing a
segment of the diameter.

The constructed continuation *w**
= **j*(*z*)
yields the analytic continuation of *f*(*z*)
through the segment of the curve *C*. In fact, in a part
of the outside of the domain *D*, the corresponding lower
semi-circle |*z - **t*_{0}|
< *d*, defined by the
analytic function *w=w*[*j*(z(*z*))] (z(*z*) is the function, inverse to *z*(*z*)) the boundary value of which on
the segment of the arc *C* coincides with the boundary
values of *f*(*z*). By the principle of continuous
continuation, this function is the analytic continuation of the
function *f*(*z*). Since *z*_{0} is
an arbitrary point of the curve *C, *it may be asserted
that *f*(*z*) is the analytic continuation through
the entire arc *C, *and the theorem has been proved.

In particular, if the entire boundaries of *C*
and *C* *of the given domains are analytic curves, then *f*(*z*)
is analytically continued through the entire boundary of the
domain *D* (and, consequently, is analytic in the closed
domain ).

The next section presents several examples of the application of the symmetry principle in the execution of conformal mappings.

**36. Examples**
1) *Mapping of the outside of a cross onto the outside of
the unit circle *(Fig. 71). We construct the auxiliary
(dotted) cut *FAB *along the imaginary axis and consider
in the right half of the figure the mapping *z*_{1}*
= z*²; it maps this half onto the *z*_{1}-plane
with removed piece from *A *(-¥)
do *D*(*a*²) along the real axis. Moreover, we
apply the mapping

which maps the region obtained onto the right
half-plane. The auxiliary cut then goes to the segment of the
imaginary axis containing ¥ from *F*(*-f*i),
where *f = *Ö (*a*² +*
*c²), to the point *B*(*gi*), where *g = *Ö (*a*² +* *b²) (Fig. 71).

The function *z*_{2}* = *Ö(*z*²* - a*²) satisfies
the conditions of the symmetry principle, which yields the
analytic continuation through *FAB *into the left
half-plane and together with its analytic continuation , which we
have denoted by again by *z*_{2}* = *Ö(*z*²* - a*²) yields a
mapping of the outside of the given cross onto the outside of the
segment *BF *of the imaginary axis of the *z*_{2}-plane.

There remains to map the last domain onto the outside of the unit circle. For this purpose, we apply the linear mapping

mapping the outside of *BF* onto the
outside of the unit circle, and then the inverse Joukowsky
mapping (cf. **7.****):**

In particular, for *b* = *c* = a,
we find

whence

(cf. **30.****,
**Example 3).

2) *Mapping of the outside of the unit circle
with the excluded segment *1 £ |*z*|
£ 1 + *a,* arg *z = *2*k**p**/n *(*k = *0, 1,
··· , *n*-1) *onto the outside of the unit circle *(Fig.
72). We insert auxiliary cuts from the points *B*_{1}*
*and *B*_{2 }to infinity by continuation of
the radii of the circle and construct the conformal mapping of
the obtained sector onto the same sector, but in sucha manner
that the points *B*_{1}* *and *B*_{2
}go to infinity by continuation of the radii of the circle
and construct the conformal mapping of the segment obtained onto
such a sector, but in such a manner that the points *B*_{1}*
*and *B*_{2 }move to *A*_{1}*
*and *A*_{2 }. This may be realized by the
following example: With the aid of the transformation *z*_{1}*
*= *z*^{n/2}* *, we map the
sector onto the upper half-plane with omitted semi-circle and
then with the aid of the Joukowsky transformation *z*_{2}*
= *½(*z*_{1} + 1/*z*_{1}) onto
the upper upper half-plane. Then

*B*_{1}* *and *B*_{2
}go for this to the points

Moreover, we compress the half-plane *z*_{3}
= *z*_{2}/(1 + *a*_{1}) and apply
the inverse Joukowsky mapping *z*_{4} = *z*_{3}
+ Ö (*z*_{3}² - 1).
As a result, we obtain again the upper half-plane with removed
unit semi-circle, but the points *B*_{1}* *and
*B*_{2 }go now to the points ±1. There remains to
apply the mapping *w = z*_{4} ^{2/n},
in order to obtain the lower mapping of the sector onto the
sector:

The function (5) satisfies the conditions of the
symmetry principle; applying this principle, we find that this
function continues through the ray *B*_{2 }*C*
and together with its continuation yields the mapping of the set
of the first and second sectors of the *z*-plane onto the
set of the first and second sectors of the *w*-plane (Fig.
72). The continuation obtained again continues through the ray *B*_{2}*C*
and the new continuation maps the third sector of the *z*-plane
onto the third sector of the *w*-plane (so that the point *B*_{3
}falls onto the circle). Repeating this argument, we find
that the function (3) together with its analytic continuations
yields the required mapping.

Fig. 72 shows parts of the images of the circles
|*w*| = *r *for this
mapping for *n = *8 and *a* = 0.25; it is seen that
for *r *= 1.8 and larger
infusion of excluded segments there is effectively no difference
- practically, these images do not differ from circles.

For *a* = 0, we have also *a*_{1}*
= *0, whence (5) becomes *w *º
*z*. We will now find the principal part of (5) for small *a*.
We find in correspondence with (4): *a*_{1 }= *n*²*a*²/8..
Neglecting small terms of order higher than *a*², we
obtain from (5) the approximate formula for our mapping:

.

or, finally,

(compare with (10) of Example 4 in **30.**). Formula
(6) applies for points which are not too close to the *n*th*-*degree
roots of 1.

3) *Mapping of upper half-plane with excluded
sections *0 £ *y *£ *h*, *x* = *ka *(*k*
= 0, ±1, ±2, ···) *of the upper half-plane. *We
introduce the additional cuts *A*_{-1}*C *and
*A*_{0}*C *from the ends of the cuts to
infinity (broken lines in Fig. 73) and map the resulting
half-strips *CB*_{-1}*B*_{0}*C *onto
such a half-strip, however in such a manner that the points *A*_{-1
}and *A*_{0 }become the vertices of this
half-strip. For this purpose, we map first our half-strip onto
the half-plane: *z*_{1}* *= cos *z*/*a
*(cf. **9.**)
and repeat like the last: *z*_{2}= *z*_{1}/cosh
*p**h*/*a* (so
that the points *A*_{-1 }and *A*_{0 }become
the points *z*_{2}* *= 1) and use the
mapping inverse to the first:

*w* = (*a*/*p*)* *arccos
*z*_{2}* *. Thus, we obtain the required
mapping of a half-strip onto a half-strip:

Applying to this function an unlimited number of times the symmetry principle, we find that it yields the required mapping pattern of Fig. 53 on the half-plane.

Fig 73 displays the images of the lines *u*
= const. and *v* = const. for the mapping under
consideration for *h* = 0.5 and *a* = 2; it is seen
that for *v *= 2 and a large effectce of the excluded cuts
is practically not noticed - in effect, the prototypes under
consideration do not differ from straight lines.

4) *Mapping of the plane with the removed
segment -a *£ *x* £ *y = hH *(*k* = 0, ±1,
±2, ···) *onto the plane with excluded segments of the
real axis *(Fig. 74, both domains infinitely connected).

We produce additional (broken lines)
cuts along the imaginary axis and map one of the two mapping
domains, for example the right one, onto the upper half-plane.
For this purpose, we turn the *z-*plane by 90º and use
the results of the preceding problem: The function

yields the conformal mapping of the right half of the domain
onto the upper half-plane. Then, the point *A*_{0}
(*z* = *a*) becomes the point *w* = arccos 1
= 0, the point *B*_{-1} (z = 0) the point

the point *C*_{-1} (*z = -iH*)
the point

the point *A*_{-1}* *becomes
-*p**, B *becomes *p** + b *and, in general, the
points *A*_{k }become the points *w =
-k**p*, but he segments *B*_{k}*C*_{k}*
*become the segments [-(*k* + 1)*p
*+* **b*, -*k**p - **b*] (*k = *0,*
±*1*, ±*2, ···). According to the symmetry
principle, we may continue the function (8) through the set of
cuts *B*_{k}*C*_{k}*
*and discover then that this function together with its
continuations maps conformally the given domain onto the *w*-plane
with the cuts (*k**p**
- b, k**p** + b*) (*k
= *0,* ±*1*, ±*2, ···) and the problem has
been solved.

We will show how to obtain a single-valued, but
not single-sheeted mapping of a given (infinitely connected)
domain onto the outside of the unit circle (i.e., onto a
simply-connected domain). First of all, we map the introduced
supplementary cuts along the real axis and line *y* = *H
*( the broken lines in Fig. 74) and the strip obtained onto
the *z*_{1}-plane with cut real semi-axis: *z*_{1}=*e*^{2}^{p }^{z/H};
the given segments *A'*_{0}*A*_{0}*
*and *A'*_{1}*A*_{1}* *then
become the segment (*e*^{-2}^{pa}^{/H};*
e*^{2}^{p }^{a/H})
on the upper and lower shores of the cut. Moreover, by the linear
transformation *z* = (1/sinh(2*p**a*/*H*){*z*_{1
}- cosh(2*p**a*/*H*)},
we move these segments by the single and inverse Joukowsky
transformations *w* = *z + *Ö(*z*_{2}²-1) into the
upper and lower halves of the unit circle. Then, there correspond
to the rays *EA*_{0}*' *and *EA*_{1}*'
*in the*w*-plane the
segments (-cotanh *p**a*/*h,
*-1)* and the rays *A*_{0}*F* and *A*_{1}*F
*to the rays (1, ¥ ) on the upper
and lower shores of the cut along the real axis.By the symmetry
principle, the obtained function

admits analytic continuation through the segments *EA*_{0}*'
*and *A*_{0}*F*, where this
continuation will map the strip, symmetric to the first one, onto
the same domain (the function (9) has the imaginary period *Hi*).
Applying such a continuation an unlimited number of times, we
find that (9) yields a single-valued mapping of the given domain
onto the outside of the unit circle. However, this mapping is not
single-sheeted, because the inverse to (9) is infinitely-valued.

* In fact, the point *E *(*z = - *¥) becomes, respectively, the
point *z* = 0, *z* = coth 2*a *(*a = pa/H*)
and *w = **-*cotanh (*a = pa*/*H*) and *w*=coth 2*a* - Ö(cotanh² 2*u*-1) = - cotanh *a *(in the
formula of the inverse Joukowsky transformation, one must place
the minus sign in front of the root ).

If we choose for the analytic continuation function all new
and new examples of the sheets of the *w*-plane,
then in the result the continuation obtains an infinitely-sheeted
Riemann surface, lying on the outside of the unit circle in the *w*-plane. At the points *E* (*w* = -coth *p**a*/*H*)
and *w* =
¥, it has logarithmic branch points. This surface is
obtained, if one removes from the Riemann surface the functions,
inverse to the function (9), the part lying inside the cylinder
projected into the unit circle |*w*|=1.
The function (9) yields a single-sheeted mapping of the given
domain onto this surface.

5) *Mapping of a domain, bounded by second order curves. *

a) *Parabola. *Let the origin of the co-ordinates
initially lie at the focus of the parabola *y*² =2*p*(*x
+ p*/2) (Fig. 75). By the function *w *=
Ö*z, *the outside of this parabola is mapped onto
the half-plane Im *w* > Ö*p*/2. In fact, setting *z=x+iy,* *w*=*u+iv,
*we obtain

whence one sees that the lines *v* = *c*
become the parabolas *y*² = 4*c*²(*x* + *c*²);
for *c* = Ö*p*/2, we obtain the given parabola. Thus, the
function

yields the conformal mapping of the outside of the parabola onto the upper half-plane. Inside the parabola, the function (10) has a branch point.

In order to obtain the mapping of the inside of
the parabola, we place the cut along the ray *BFG* (Fig,
75) and note that the upper half of the parabola is mapped with
the aid of the function *z*_{1} = Ö*z* onto the half-strip 0 < *y*_{1}
< Ö*p*/2,
0 < *x*_{1} < ¥.
With the aid of the function .

this half strip is mapped onto the upper
half-plane, where the cut *BFG *corresponds to the ray -1
< *x* < ¥. Applying the
symmetry principle, and then the transformation *w =* iÖ(1 + *z*_{2}), we obtain
the known mapping of the inside of the parabola. onto the upper
half-plane

b) *Hyperbola* In order to find the
conformal mapping onto the upper half-plane of the domain,
included between the branches of the hyperbola

(Fig. 76), we make the cut *BD *along the real axis and
note that the function

maps the upper half-plane of the given domain onto the sector

(cf. **7.**)
However, by the symmetry principle, this function yields a
conformal mapping of the entire given domain onto all of the
sector

Thus, the function

yields the mapping of the domain, included between the branches of the hyperbola, onto the upper half-plane.

In order to obtain the mapping of the inside of
the right branch of the hyperbola, we makes a cut along the ray *DFG
*and notes that the function

yields the conformal mapping of the upper half domain onto the sector

The function

maps this sector onto the upper half-plane, where
there corresponds to the ray *DEG* the ray (-1, ¥ ) of the real axis. Applying the symmetry
principle and then the supplementary mapping *w* = *i*Ö(1 + *z*_{2}), we obtain
the required mapping of the inside of the right branches of the
hyperbola onto the upper half-plane

c) *Ellipse *The conformal mapping of the
outside of the ellipse

onto the outside of the unit circle yields the function

where *c* = Ö(*a*²
+ *b*²) (cf. Fig. 77).

In order to obtain the mapping of the outside of the ellipse, we make a cut along its larger axis and employ the function

We then obtain the mapping of the upper half of
the ellipse onto the upper half ring 1 < |*z*_{1}|
< (*a* + *b*)/*c, *Im *z*_{1 }>
0, where the cut enters the segment *AF*_{1}, *F*_{2}*C*
of the real axis and the unit semi-circle. The function *z*_{2}
= ln *z*_{1}* *maps.this half ring onto the
rectangle 0 < Re *z*_{2}* *< *d, *0
< Im *z*_{2} < *p, *where
*d* = ln[(*a* + *b*)/2]. The symmetry
principle cannot yet be applied, because this domain becomes a
single segment. The mapping of the rectangle onto the plane
cannot be obtained with the aid of a combination of elementary
functions - is exists as a so-called elliptic function (cf. **39.**, Example
1) - whence also the mapping of the inside of the ellipse onto
the half-plane is not described in terms of elementary functions.

**37.
Mapping of polygons****. **Prior to
deriving the formulae for the mapping of the half-plane onto a
polygon * , we will explain the problem of the behaviour of the
conformal mapping at angular points of a domain. For the sake of
simplicity, we assume that the boundary of the domain *D *in the neighbourhood of the
angular point *w*_{0} consists of straight
segments; we will denote the angle between these segments by *ap*, assuming that 0 < *a* £ 2
(Fig. 78). Let the function *w* = *f*(*z*)
yield the conformal mapping of the upper half-plane onto the
domain *D*, where the angular
point *w*_{0} corresponds to the point *z*_{0}
of the real axis.

* cf. **44. f**or
another , more constructive derivation of this formula

In order to explain the character of the function
*f*(*z*) in the neighbourhood of the point *z*_{0},
we introduce the auxiliary variable *w = *(*w
- w*_{0})^{1/}^{a}.
The compound function

realizes the conformal mapping of a part of the
neighbourhood of the point *z*_{0}, belonging to
the upper half-plane *z*, onto the neighbourhood of
the point *w *= 0, belonging to
one of the half-planes, whereas the segment of the real axis of
the *z*-plane corresponds to segment of a line (Fig. 78).
By the symmetry principle, the function *w*(*z*)
admits analytical continuation in the complete neighbourhood of
the point *z*_{0 }and can, consequently, be
represented by the Taylor series

This series does not contain a free term, because
*c*_{0}*' = **w*'(*z*_{0})
¹ 0, since the function yields a
conformal mapping. Turning with the aid of (1) to the function *f*(*z*),
we find that it can be represented in the neigbourhood of the
point *z*_{0 }by

Since the expression in the curly brackets is
non-zero for *z* = *z*_{0}, then onr may
separate in some neighbourhood of the point *z*_{0 }a
single-valued analytical branch of the function

Expanding this branch in a Taylor series, we
obtain the final representation of the function *f*(*z*)
in the neighbourhood of the point *z*_{0}

_{}

Whence we see that for *a *>
1 the derivative *f* '(*z*_{0}) = 0, for *a *< 1 *f* '(*z*_{0})
= ¥. For the inverse mapping *z = **j*(*w*), conversely, *j* '(*w*)
= 0 for *a *< 1 and *j* '(*w*_{0}) = ¥ for *a *> 1.Hence it is seen from (2)
that *z*_{0} for *a *¹
1, ¹2 is a branch point of the
function *f*(*z*) .

We note that in a more general case, when the
boundary of the domain *D *in
the neighbourhood of the point *w*_{0} consists of
smooth or even analytic arcs, intersecting in *w*_{0 }at
an angle *ap,* the above
derivation, generally speaking, is untrue. In this case, the
mapping function does not have the form (2): In the principal
term, its expansion may exhibit a factor which tends to zero and
infinity more slowly than any power of *z - z*_{0}
*.*

For example, we consider in the upper semi-circle
|*z*| < *r*, *y *³
0 the function

where the branch of the logarithm is
characterized by the condition 0 < arg *z* £ *p*. It
is readily seen that it transforms, for sufficiently small *r*,
the segment (0, *r*) of the *x*-axis into the
segment (0, *r* ln 1/*r*) of the *u*-axis,
the segment (-*r*, 0) into the ray *g
*:* u *= *x* ln (1/-*x*)*, v =
-x*/*p*, and the
semi-circle |*z*| = *r*, 0 £
*j *£ *p
*into an arc, close to the semi-circle (Fig. 79). By
the boundary correspondence principle, the function (3) is
single-sheeted for small *r* and conformally maps the
semi-circle onto the domain *D*, shown in Fig. 79.

The arc *g *smoothly
joins the segment (0, *r* ln 1/*r*) at the point *w*
= 0 so that the angle *a *= 1;
nevertheless, the principal term of the expansion (*r*) is
"spoiled" by the factor ln 1/*z. *An analogous
effect is found for the function

We now proceed to the mapping of polygons. Let
there be given in the *w*-plane the closed polygon *A*_{1}*A*_{2}*
*··· *A*_{n}* *without
points of intersection which does not contain the point at
infinity (we will remove this restriction in **38.**)

According to Riemann's Theorem (**28.**),
there exists a function *w* = *f*(*z*) which
realizes a conformal mapping of the upper *z-*half-plane
onto the inside *D* of this
polygon. For the sake of definiteness, we give the correspondence
of three points of the real axis (for example, *a*_{1},
*a*_{2} and *a*_{3}) and three
points of the boundary of *D *(for
example, the vertices *A*_{1}, *A*_{2}
and *A*_{3}); then, by Theorem 2 of **35.**** **,
the function *f*(*z*) is uniquely determined. To
start with, we assume that this function is known, in particular,
that the points *a*_{4}, ··· , *a*_{n}
on the *x-*axis are known, which become the points *A*_{4},
··· , *A*_{n} of the polygon, and pose
the problem of finding its analytic expression.

Since the function *w = f*(*z*)
assumes on any segment (*a*_{k}, *a*_{k+1})
of the real axis values which lie on a straight segment *A*_{k},
*A*_{k+1}, then, by the symmetry
principle, it is analytically continued through this segment into
the lower half-plane. The analytic continuation of this function
yields the conformal mapping of the lower half-plane onto the
polygon *D* ', symmetric to the
polygon *D *with respect to the
segment *A*_{k}, *A*_{k+1}.
This analytic continuation may again be continued through any
segment (*a*_{k'}, *a*_{k'+1})
into the upper *z*-half-plane, where a new analytic
continuation will realize a conformal mapping of the upper *z-*half-plane
onto the polygon *D*",
symmetric with the polygon *D*'
with respect to the segment *A*_{k'} *A*_{k'+1},

We will assume that we have completed all
possible analytic continuations of the form just described. As a
result, we obtain, generally speaking, an infinitely-valued
analytic function *w* = *F*(*z*) for which
the initial function *f*(*z*) is in the upper
half-plane one of the single-valued branches.

Let *w* = *f*_{*}(*z*)
and *w* = *f*_{**}(*z*) be
two arbitrary branches of the function *F*(*z*) in
the upper half-plane. According to our design, these branches
yield a conformal mapping of the upper half-plane onto two
polygons *D*^{*}* *and *D*^{**},
which differ from each other by an even number symmetry
respecting the sides . However, since any pair of symmetry with
respect to two arbitrary lines reduces to a certain shift and
rotation, then everywhere in the upper half-plane *f*_{**}(*z*)
= *e*^{i}^{a}*
f*_{*}(*z*) + *a*, where *a
*and *a *are constants. The
same is also true for any of the branches of the function *F*(*z*)
in the lower half-plane.

Moreover, the function

is analytic in the upper half-plane, because *f*
'(*z*), as the derivative of a function yielding a
conformal mapping, vanishes nowhere. This function *g*(*z*)
remains single-valued for all possible analytic continuations of *f*(*z*)
by virtue of the above remark relating to the branches of the
function *F*(*z*) (we have

Thus, we may assert that *g*(*z*)
is a single-valued function, analytic everywhere in the entire *z*-plane
except at the points *z* = *a*_{k},
corresponding to the vertices of the polygon. The analyticity of *g*(*z*)
at infinity follows from the fact that *z* = ¥ becomes some point on a side of the
polygon, and not at its vertex. .

In order to explain the character of the function
*g*(*z*) at the point *z* = *a*_{k}*,
*we select any branch *f*(*z*) and employ (2).
We find:

whence we readily obtain the Laurent expansion of *g*(*z*)
in the neighbourhood of the point *z* = *a*_{k}:

This expansion shows that the point *z* = *a*_{k}
is for *g*(*z*) a first order pole with residue *a*_{k}
- 1.

Thus, the function *g*(*z*) has in
the entire plane only *n* special points. Computing from *g*(*z*)
the sum of the principal parts of its expansion at these points,
we find the function

which is regular in the entire plane and,
consequently, constant (cf. the Cauchy-Liouville Theorem in the
formulation of **24.**).
Since the function *f*(*z*) holds at the point *z*
= ¥, in the neighbourhood of this
point

and

Consequently, *g*(¥),
and that means also *G*(¥ ),
vanishes, whence

Integrating (5) along any path in the upper half-plane and then taking the exponential, we find

By a further integration, we obtain the required
expression for *f*(*z*) and we have proved

**Theorem 2 **(Schwarz -
Christoffel*, 1867 - 69) *If the function w = f*(*z*)
*yields the conformal mapping of the half-plane* Im(*z*)
> 0 onto the inside of a bounded polygon *D
**with angle **a*_{k}*p *(0 < *a*_{k
}£ 2, *k* = 1, 2,
··· , *n*) *at the vertices*, *where the
points* *a*_{k}* of the real axis *(-¥ < *a*_{1} < *a*_{2}< ··· < *a*_{n}
< ¥ *are known*) *are*
*the corresponding vertices of this polygon , then f*(*z*)
*is represented by the integral*

*where z*_{0}*, C and C*_{1}*
are certain constants.*

*Erwin Christoffel (1829 - 1900).

The Schwarz-Christoffel integral is obtained
under the assumption that the points *a*_{k},
corresponding to the vertices of the polygon, are known. However,
in problems of conformal mapping, there are only given the
vertices *A*_{k} of the polygon and the
points *a*_{k}
remain unknown. According to what has been stated in **29.**,
three of them (for example, *a*_{1}, *a*_{2}
and *a*_{3}) may be given arbitrarily, while the
remaining ones as well as the constants *C* and *C ' *must
be determined from the conditions of the problem*. This
circumstance creates a principal difficulty for the practical use
of the Schwarz-Christoffel integral.

* The constant z_{0 }may be fixed for good; for example, one may set z_{0
}= 0, whence in the sequel we will not assume
it to be an unknown parameter of (7).

Methods for the determination of
the constants *a*_{k}, *C* and *C*_{1}* *will
be stated below in concrete examples. Principally, the
possibility of finding them follows in essence from the proof of
Theorem 2, given above. In fact, let there be given a polygon *D**.*. On the basis of the
theorem, we may assert that there exists a unique conformal
mapping *w* = *f*(*z*) of the half-plane Im *z*
> 0 onto the polygon *D*,
mapping the three given points *a*_{1}, *a*_{2}
and *a*_{3} of the real axis into three vertices
of *D**,*.for example,
into *A*_{1}, *A*_{2} and *A*_{3}
. By what has been proved above, the function for this will be
given by (7) for a proper choice of the constants *a*_{4},
··· *a*_{n}, *C and* *C*_{1}*.
*Thus, when three *a*_{k }are given,
the remaining constants are determined, and besides in a unique
manner. We note yet that, according to (6), we have for *z*
= *x* > *a*_{n} arg(*x*
- *a*_{k}) on the real axis of the *z*-plane
and, consequently, arg *f* '(*z*) = arg *C, *and
since the segment (*a*_{n}, *a*_{1})
(containing *z* = ¥ ) becomes
for the mapping *w* = *f* (*z*) the segment *A*_{n}*A*_{1},
then arg *C *equals the angle *q*
which this segment forms with the *u*-axis (in Fig. 80, *a *= *q - p*). The constant *C*_{1
}is determined by stating the position of one of the
vertices. For the determination of the constants *a*_{k
}and *C, *one may employ the known lengths of
the sides of the polygon

although we do not use this method in by far not
all applications. In applications, one often employs approximate
methods for the determination of the constants *a*_{k
}and *C*; the reader can find out about this
from the book by P. F. Filchakov [10],
the work of G,N. Polozhego [12] or
the paper by N.P. Stenin in the collection [8].