**21. Laurent
Series**** **Taylor series are a tool
which is convenient for the representation of functions, analytic
in circular domains. However, it is very important to have a tool
for the representation of functions in domains of a different
shape. For example, when studying functions, which are analytic
everywhere in some neighbourhood of a point *a *except the
point *a *itself, we come to study annular regions of the
form 0 < |*z *- a| < *R*. It turns out that
for functions, analytic in annular domains *r<*|*z-a*|<*R*,
where *r* ³ 0, *R *£ ¥, one may
construct expansions in positive and negative powers of (*z*
- a) of the form

a generalization of Taylor expansions. We will study such expansions next.

Thus, let the function *f*(*z*) be
analytic in some annulus *K*: *r* < |*z - a*|
< *R*, where *r* ³
0, *R *£ ¥.
We select arbitrarily the numbers *r' *and *R*',
so that *r* *< r' < R' < R, *and also the
number *k*, 0 < *k* < 1, and consider the
ring *r*'/*k < *|z - a| < *kR* '. At
an arbitrary internal point *z* of this ring, we may
represent *f*(*z*) using Cauchy's formula of **14.****
**which then assumes the form

where both the circles *C*:* |**z *- *a| = R' *and *c*:
*|**z *- *a| =r' *are
travelled in counter-clockwise directions.

We have for the first integral

whence the fraction here may be expanded in the
geometric progression, converging uniformly on *C*

Multiplying this expansion by *f*(z)/2*p**i*
and integrating it term by term with respect to z (this is possible due to the uniform
continuity), we obtain the expansion of the first term of (2) in
the power series

where

We note that it is impossible to represent (4),
as in **18.**,
in the form *f *^{(n)}(*a*)/*n*!,
since, generally speaking, *f*(*z*) is not analytic
at the point a.

For the second integral, we have

whence there converges uniformly the progression

As also above, we obtain the expansion of the
second term of (2) in a series, but now in terms of negative
powers of (*z - a*):

where

We note in (5) and (6) that the index *-n*
takes the values 1, 2, ···, the index *n* the values
-1, -2, ···; joining both the expansions, we obtain

Moreover, by **13.**,
we may replace in (4) and (6) the circles *C* and *c *by
any circle *g *: |*z - a*|
= *r*, where *r*' < *r *< *R* ', whence both
these formulae may be joined in

The expansion (7) of *f*(*z*) in
terms of positive and negative powers of (*z - a*) with
its coefficients given by (8) is called the *Laurent expansion
*of *f*(*z*) about the point *a*; the
series (3) is called *prope*r, the
series (5) the *smooth part *of
this expansion.

Since in our reasoning *r*' and *R*'
may be as close as we please to *r* and *R, *and *k
*may differ from 1 as little as we please, Expansion (7) may
be assumed to have been established for all points *z* of
the ring of analyticity of *f*(*z*).

By Abel's Theorem, the regular part of Laurent's
series converges everywhere in the circle |*z - a*| < *R*,
while it converges uniformly in any circle |*z-a*|<*kR*
(0 < *k *< 1). The principal part represents the
power series in terms of the variable *Z*=1/(*z-a*),
whence it converges by the same theorem for |Z| < 1/*r*,
i.e., everywhere outside the circle |*z - a*| > *r*,
where for* *|*z - a*| > *r*/*k, *0
< k < 1, its convergence is also uniform.

Thus, we have proved

**Theorem 1 **(P. Laurent, 1843
(1813 - 1854)) * *In any ring K*: *r < |z - a| <
R, in which the analytic function f*(*z*) *is
analytic, this function may be represented by its Laurent
expansion *(7) *which converges uniformly in any closed
domain belonging to the ring K*.

* This theorem was also obtained by K. Weierstrass in 1841, however, he only published his result in 1894. Series of the form (7) are also encountered in the work of L. Euler in 1748.

We obtain from (8) for the
coefficients of the Laurent series exactly in the same way as in **17.** the following *inequality of Cauchy*:
If the function *f*(*z*) is bounded on the circle |*z
- a*| =*r*, let |*f*(*z*)| £ *M*__,____ then__

Finally, we note that the region of convergence of an arbitrary series of the form

always contains a certain circular ring* *r *<
|*z - a*| < *R, *where 0 £
*r* £ ¥ , 0 £ *R *£ ¥
.

* This ring may turn out to be
empty, if *r* ³ *R*, but in the
case *r = R *any set on the circle may serve.

This is readily verified with the aid of Abel's
Theorem by subdividing the series into regular and principal
parts. For the case *r* < *R *holds

**Theorem 2 ***If the series *

*converges in the ring r *< |*z - a*|
< *R, then its sum f*(*z*) *is analytic in
the ring and the expansion* (10) *is the Laurent series
for f*(*z*).

In fact, the analyticity of *f*(*z*)
is displayed on the basis of the Theorem of Abel and Weierstrass
in the same way as in Theorem 4 of **21.**.
Moreover, on any circle *g *: |*z
- a*| = *r*, where *r*
< *r < **R*, the
series (10) converges uniformly and remains so after
multiplication by (*z - a*)^{-n+1 }(*n
*= 0, ±1, ±2, ···). In order to integrate the expansion

along the circle *g* and
employ the relations, which are readily proved for any integer *n*,

(cf. the derivation of (4) in **13.**),
then we obtain the expression for the coefficients of the series
(1):

which coincides with (8). Consequently, Series
(10) is the Laurent expansion of the function *f*(*z*)
and the theorem has been proved.

Theorem 2 is the uniqueness theorem of the
expansion in Laurent series, because it follows from it that an
expansion of an analytic function by any method in a series of
positive and negative powers of (*z* - *a*) is the
Laurent expansion of this function.

**22.
Singularities**** **The methods developed
in **21.** for
Laurent expansions allows us to study in depth the behaviour of
analytic functions in the neighbourhood of points of the simplest
type at which the analyticity of these functions is violated - so
called singularities. A point *a *is called an *isolated singularity**
*of the function *f*(*z*) if there exists a
neighbourhood 0 < |*z* - a| < *R *of this
point (excluding the point* a *itself) in which *f*(*z*)
is analytic. We emphasize that we speak here of points in the
neighbourhood of which the function is single-valued (the
condition of single-valuedness is included in the condition of
analyticity of the function (cf. **5.**).
We will discuss singularities of a multi-valued character in **25.**

We distinguish three types of isolated singular
points depending on the behaviour of the function *f*(*z*)
in their neighbourhood. A point is called

1) a *removable
singularity* if there exists a finite

2) a *pole *if *f*(*z*)
becomes infinitely large on approach to *a*, i.e., if
there exists

(this means that |*f*(*z*)| ® ¥ as *z*
® *a**,*

3) an *essential
singularity* if there does not exist

**.**

Consider the basic properties of functions at
their singularities. If *a* is an isolated singularity of *f*(*z*),
then, by Theorem 1 of **21.**,
this function may be expanded in a Laurent series in the ring of
its analyticity 0 < |*z - a*| < *R*:

This expansion has a different form depending on the character of the singular point. We present three relevant theorems:

**Theorem 1 ***In order for a to
be a removable singularity of f*(*z*) *it is
necessary and sufficient that its Laurent expansion does not have
in the neighbourhood of the point a a principal part.*

Clearly, if the Laurent expansion of *f*(*z*)
does not have a principal part, i.e., of *f*(*z*)
is represented by a power series

there exists the finite * and *a* is a
removable singularity.

* By Theorem 4 of **19.**** **, the right hand side of (2) is
analytic at *z* = *a*, whence it is continuous and
its limit as *z* ® *a *equals the sum of the series at the point *a*,
i.e., *c*_{0}.

Conversely, let *a* be a removable
singularity of of *f*(*z*). Then, due to the fact
that
exists and is finite, the function *f*(*z*) is
bounded near *a*; let |*f*(*z*)| £ *M.*

We will employ Cauchy's inequality of **21.**

since here the number *r *may
be chosen as small as we please, it is clear that all the
coefficients *c*_{n}
with negative subscripts vanish and the Laurent expansion of *f*(*z*)
does not contain a principal part, whence the theorem has been
proved.

**Note**** **We
have proved essentially the stronger statement: *If the
function f*(*z*)* is bounded in the neighbourhood
of an isolated singularity a, then its is a removable singularity
of this function. *

Obviously, the term *removable singularity* justifies
"removal" of such a special point by setting

after this has been done, the function *f*(*z*)
will be analytic also at the point *a*, because it will be
represented in the entire circle |*z-a*|<*R'*,
where *R' *£ *R.* In
such a neighbourhood, the analytic function *g*(*z*)
= 1/*f*(*z*) for which obviously

Consequently, by the preceding theorem, *a* is a
removable singularity of *g*(*z*) and, setting *g*(*a*)
= 0, we find that *a* is a zero of the function *g*(*z*).
Conversely, if *g*(*z*) has a zero at the point *a*
( and is not identically zero), the function *g*(*z*)
= 1/*f*(*z*) is analytic by Theorem 1 of **20.****
**in some neighbourhood 0 < |*z - a*| < *R
*of the point *a*; obviously, *f*(*z*)
has a pole at the point *a*.

Thus, zeroes and poles of analytic functions are interrelated
simply. We will agree to call the *order*
*of a pole** a *of the function *f*(*z*)
the order of the zero *a *of the function *g*(*z*)
= 1/*f*(*z*).

**Theorem II ***For a point a to be a pole of
the function f*(*z*) it is necessary and sufficient
that the principal part of the Laurent expansion of *f*(*z*)
*in the neighbourhood of a contains only a finite number of
terms:*

*Then the number of the oldest negative term
of the series coincides with the order of the pole.*

Let *a *be a pole of order *n* of
the function *f*(*z*). Then the function *g*(*z*)
= 1/*f*(*z*), *g*(*a*) = 0, has at
the point *a* a zero of order *n *and, according to
**20.**,
it has near the point *a* the form

where *j*(*z*)
is analytic at and *j*(*a*) ¹ 0.In this neighbourhood,

However, the function 1/*j*(*z*)
is analytic in some neighbourhood |*z - a*| < *R *of
the point *a*, whence is has the Taylor expansion

where *c*_{-n}* = 1/**j*(*a*) ¹
0. Expanding this expansion in the form (4), we obtain the known
expansion (3), true in the neighbourhood 0 < |*z - a*|
< *R*.

Now, let conversely, in some neighbourhood 0 < |*z - a*|
< *R *of the point *a* apply Expansion (3),
where *c*_{-n}* *¹ 0. Then, the function *j*(*z*)
= (*z - a*)^{n}*f*(*z*), *j*(*a*) = *c*_{-n},
represents in the circle |*z - a*| < *R *the
Taylor series

i.e., it is analytic.Since

and the point *a* is a pole of the
function *f*(*z*). Obviously, the function *g*(*z*)
= 1.*f*(*z*) = (*z - a*)^{n}/*j*(*z*) has at the point *a*
a zero of order *n*, whence the order of the pole *a*
is equal to *n*m and the theorem has been proved.

This theorem yields directly

**Theorem 3 ***The point a is
then and only then an essential singularity of the function f*(*z*)*,
if the principal part of the Laurent expansion of the last at the
point a contains infinitely many terms.*

The behaviour of a function in the neighbourhood of an essential singularity is explained by

**Theorem 4 **(Yu. Sokhotskii* 1868)
*If a is an essential singular point of the function f*(*z*)*,
then* *there exists for any complex number A a sequence of
points z*_{k}* *®
*a such that*

** *This theorem is usually
ascribed to Weierstrass; however, it was proved in the
dissertation of the Russian mathematician Julian Vacilievich
Sokhotskii (1842 - 1929) and published 8 years later as an
addition to Weierstrass' work. Simultaneously with Sokhotskii,
the theorem was obtained by the Italian mathematician F.
Cazorati.

First of all, there exists a sequence* z*_{k}*
*® *a *for which* *,
because otherwise *f*(*z*) would be bounded in the
neighbourhood of *a* and the point *a* would be a
removable singular point (cf. Note to Theorem I).
Now, let there be given an arbitrary complex number *A. *There
can occur one of the two cases:

1) in any neighbourhood of the point *a *lies a point *z*
at which *f*(*z*) = *a, *when Sokhitskii's
Theorem has been proved, because one may construct out of such
points a sequence *z*_{k}®*a *so that *f*(*z*_{k})
= *A*, which means also

2) in some neighbourhood of the point *a* the function *f*(*z*)
does not have the value* A.*

In the second case, in the neighbourhood above,
the analytic function *g*(*z*) = 1/(*f*(*z*)
- *A*). Then, *a* may neither be a pole nor a
removable singularity, because in those cases there would exist
the finite or infinite limit

whence *a *is an essential singularity of
the function *g*(*z*) and there exists, by the
proved theorem, a sequence *z*_{k} ® *a *for which

Obviously, then

and Sokhotskii's theorem has been proved.

Sokhotskii's theorem and the theorems proved above allow to assert that in the neighbourhood of an isolated singularity an analytic function either tends to a definite (finite or infinite) limit or is completely undetermined, i.e., it tends by different sequences to any previously given limit. No other cases can arise.

We list still examples of elementary functions with singularities of different types:

1) The functions

have at the origin a removable singularity. This
is shown most simply using their known Taylor expansions (5) in **18.** and
Theorem I above. For example, we have for *z* ¹ 0

2) The function

has an infinite set of poles at the points

at which the denominator tends to zero (these
points are located on the two bisectors of the co-ordinate
angles), All the poles are of first order, since the function has on
them first order zeroes (with derivative 2*ze*^{z²}*
*different from zero).

3) The functions

have at the origin an essential singularity. This
is most simple verified by replacing 1/*z* by *z *in
the Taylor series (5) in **18.****
**and using Theorem III above (for example, for any *z*
¹ 0, we have

For example, we verify the truth of Sokhotskii's
Theorem for the first of these series. For *A = *¥, the sequence *z*_{k}
=1/*k* of this theorem will be *z*_{k}
= -1/*k, k = *1, 2, 3, ···, because, obviously,

for *A* = 0, one may take *z*_{k}
= -1/*k, k = *1, 2, 3, ···, because then

finally, for finite *A**
*¹ 0, we choose

(ln denotes here any value of the logarithm).

4) The function

has at the origin a non-isolated singularity, because its poles accumulate at the origin (cf. Example 2 above).

According to their character, singularities subdivide the following two simplest classes of single-valued analytic functions.

1) **Holomorphic
function****s: **Functions are said to be
integral (holomorphic, if they do not

have any special points*. By the theorem of **18,**,
one may assert that every holomorphic function represents a power
series converging in the entire plane (and, conversely,
every functions, converging in the entire plane, represents a
holomorphic function). Examples of holomorphic functions are all
polynomials. exponential functions, sin *z*, cos *z*,
etc. Obviously, a sum, difference of product of holomorphic
functions are again such functions.

*Here and in the sequel, we talk
of *finite *points of the plane; we do not emphasize this
in the text, because the concept of the point at infinity will be
introduced in **24.**

2) **Fractional
functions**** **A function is said to be*
fractional *(or *meromorphic*) if it does not have
other singularities but poles. This definition yields that in any
bounded region a meromorphic function may have only a finite
number of poles. In fact, if there were in such a region
infinitely many poles, there would exist a sequence of them,
converging to some point a, which would be an isolated
singularity, but not a pole. In the entire plane, there may also
be infinitely many poles. Examples of meromorphic functions are
all integral functions, fractional rational functions,
trigonometric functions etc. Obviously, the sum, difference,
product and quotient of two meromorphic functions and, generally
speaking, any fractional rational function *R*(*f*_{1},
*f*_{2}, ···, *f*_{n})
of meromorphic functions is again a meromorphic function.

For more details of integral and meromorphic functions, cf. Chapter V.

**23.
Residue Theorem. Argument principle** We will now
introduce the concept of a residue* of a function, very important
for further applications, and prove several special theorems
related to it; examples of the calculation of residues and
different applications are considered in the sequel (in a basic
manner, in Chapters V and VI).

* The concept of residue was introduced by A. Cauchy in his "Memoir regarding Integrals" (1814); in his "Exercises in Mathematics"(1826-1829), Cauchy gave also many applications of this concept in analysis. Cauchy stated in his work that he arrived at the concept of residue by developing an idea of Euler.

The *residue *of the function* f*(*z*)
at the isolated point *a *(denoted by res *f*(*a*))
is the number

where *g *is a
sufficiently small circle |*z *- *a*| = *r*. According to **13.**,
the magnitude of a residue does not depend on the quantity *r *for sufficiently small *r*.

Formulae (8) of **21.**
for the coefficients of the Laurent series yield directly for *n*
= -1

i.e., *the residue of a function f*(*z*)
*at a singular point a equals the coefficient of the first
term with negative subscript in the Laurent expansion of f*(z)
in the vicinity of *a*.

Thus, it follows that at a removable singular
point the residues of a function always vanishes. The residue of
at a pole of order *n* is given by

It is derived by multiplying the Laurent series

by (*z - a*)^{n},
differentiating the result obtained *n* - 1 times and then
going to the limit for *z* ® *a
*(direct substitution of *z* = *a *in the
expression for the derivative is impossible, because *a*
is a singular point of *f*(*z*)).

For a first order pole, Formula (3) becomes especially simple:

If for this in the neighbourhood of the point *a*
the function *f*(*z*) is determined by the fraction
of two functions, analytic at this point:

where *j*(*a*)
¹ 0, and *y*(*z*)
has at *a* a zero of first order (i.e., *y*(*a*) = 0, *y*'(*a*) ¹
0), then (4) may be replaced by

**Example **The meromorphic function
cotan *z*² has first order poles at the points

and a second order pole at *z* = 0 (this
is most simply verified by considering the zeroes of tan *z*²).
The residue at *z* = 0 is, by (3),

(this follows also from the fact that the Laurent series of
cotan *z*² with centre at *z* = 0 may only contain
even powers of *z*. By (5), the residue at the point with *j *= cos *z*², *Y *= sin *z*² is

* We have given the numerator and
denominator by the first terms of their Taylor series about the
origin *z* = 0.

Application of the theory of residues is based in
a fundamental manner on the following important *Residue
Theorem*:

**Theorem I **(A.Cauchy, 1825) *Let
the function f*(*x*) *be continuous on the boundary
C* of the region D and analytic everywhere inside this domain
except for a finite number of singular points a*_{1},*
a*_{2}, ··· , *a*_{n}, *then*

* Here and in the sequel, the
continuity of *f*(*z*) on the boundary of the
region is implied in the sense of continuity over the region,
i.e., in the sense that there exists at any point *z*_{0}
of the boundary

where *z* ®
*z*_{0}* *via points of the region *D.*

The
proof follows from Cauchy's Theorem for multi-connected domains (**13.**). We
surround every point *a*_{n }by a circle *g*_{k}:
|*z - a*_{k}| = *r*_{k}
so small that all of them lie in the region *D *and do not
intersect each other (Fig. 26). Since *f*(*z*) is
analytic in *D*^{*}, bounded by the curve *C *and
the union of the circles* **g*_{k},
and continuous in then, by Cauchy's theorem,

where all the are travelled in the clockwise sense. Changing the
direction of the travel of the circles *g*_{k}
and using the definition of residues (1), by which

we obtain the required result (6).

* Here and further on, continuity
of *f*(*z*) on the bounary of the domain is
understood in the esense of continuity in the domain, i.e., in
the sense that at any point *z*_{0} of the
boundary there exists

where *z *® *z*_{0}*
via* points* *of the domain *D*.

The basic importance of the residue theorem consists here of the fact that it allows to reduce the calculation of quantities "as a whole", such as the integral along a closed contour of a finite quantity, to the calculation "in the small", differential quantities . .like residues. In fact, residues are computed with the aid of integrals along infinitely small contours or even with the aid of a simple limiting process. (Formulae (3), (4) and (5)). The method of reduction of calculations of quantities "as a whole" to a calculation of differential quantities is standard in mathematical analysis (cf. the calculation of integrals with the aid of transformations, which are determined on the basis of known derivatives). We will apply residue theory especially in Chapter V.

We will still dwell on the concept of logarithmic
residue. We mean by the *logarithmic residue *of the
analytic function *f*(*z*) at the point *a*
the residue of its logarithmic derivative

Obviously, one can speak of logarithmic residues
not only at singularities, but also at zeroes of *f*(*z*).
If *a* is a zero of *f*(*z*) of order *n*,
then in the neighbourhood of this point

whence

The second factor here is a function, analytic at
*a*, because *c*_{n}* *¹ 0, whence it can be expanded in a Taylor
series with centre at *a* (the free term of this series
equals *n*) and

We have obtained the Laurent series of the
logarithmic derivative [ln*f*(*z*)]' in the
neighbourhood of the point *z* = *a*, from which it
is seen that the point *a* is its first order pole with
residue equal to *n*.

Now, let *a* be a pole of *f*(*z*)
of order *n*. Since the function *g*(*z*) =
1/*f*(*z*) has at *a* a zero of order *n*
and since

{ln* f*(*z*)}' = -[ln* g*(*z*)}',

then, only by what has been proved for the
logarithmic derivative {ln* f*(*z*)}', the point *a*
is a first order pole with residue *n*. Thus, we have
proved

**Theorem 2 ***At zeroes and
poles of the function f*(*z*),* its logarithmic
derivative f'*(*z*)/*f*(*z*) *has
first order poles, where at a zero the function's logarithmic
residue is of order zero and at a pole - of the order of the pole
with a minus sign.*

Theorem 2 and the Residue Theorem allow application of logarithmic residues for the evaluation of the number of zeroes and poles of analytic functions in given domains.

Let *f*(*z*) be analytic everywhere
inside the region *D *except at a finite number of poles *b*_{1},
*b*_{2}, ··· , *b*_{m}*
*of multiplicity *p*_{1}, *p*_{2},
··· , *p*_{m}* *, respectively,
continuous on the boundary *C* of this region and not
vanish on *C*; moreover, let *f* '(*z*) be
continuous on *C*. Then, the function *f*(*z*)
has in *D* only a finite number of zeroes, because
otherwise there would exist an infinite sequence of zeroes,
converging to an internal or boundary point of *D*;
however, this sequence cannot converge neither to an internal
point (by the uniqueness theorem of **20.**)
nor to a boundary point (because *f*(*z*) ¹ 0 and continuous on *C*). We
denote the zeroes of *f*(*z*) in *D* by *a*_{1},
*a*_{2}, ··· , *a*_{l}*
*and their multiplicities by *n*_{1}, *n*_{2},
··· , n_{l }, respectively. Applying to the
logarithmic derivative of *f*(*z*) the Residue
Theorem and Theorem 2, we find

where *N* and *P *denote the total
number of zeroes and poles, respectively, each of them occurring
only as often as its order. We will next explain the meaning of
the left hand side of (8). We have

where ln and arg denote any branches of these
functions, continuous along *C*. Since for a passage along
a closed contour *C *the function ln |*f*(*z*)|
returns to its initial value, the first integral on the right
hand side of (9) is zero. On the other hand, if the point *w*
= 0 lies inside the contour, described by the point *w* = *f*(*z*)
when *z* travels around *C*, then the finite value
of arg *f*(*z*) may differ from the initial one
(Fig. 27) and then the second term will not be zero. The quantity

- the total change of the argument of *f*(*z*)
during the passage along *C*, divided by 2*p -* represents geometrically itself
the *number of cycles *around the origin *w* = 0 of
the vector *w* for passage of the curve *G, *corresponding to *C,* for
the mapping *w* = *f*(*z*) (In Fig. 27, this
number is 1). Relations (8) and (9) express the so called *Argument
Principle*:

**Theorem 3 ***Let the function f*(*z*)
*be analytic everywhere inside the domain D except at a finite
number of poles, continuous on the boundary C of this domain and
not vanish on C*; *moreover, let f *'(*z*) *be
continuous on C. Then, the difference between the total number of
poles and the poles of this function inside D equals the number
of cycles of the vector w during transition of the curve **G **, corresponding to C for the
mapping w = f*(*z*) *or, what is the same thing,
the sum of the logarithmic residues of f*(*z*) *in
the region D*:

The following results, relating to the computed
number of zeroes and poles of the function and their important
applications, will be explained in **75.** We state
here only one modification of (8), relating not only to the
number of zeroes and poles, but also to their positions.

Consider side by side with the function *f*(*z*),
satisfying the conditions of the argument principle, the function
*j*(*z*), analytic in *D
*and continuous in Obviously, there may only correspond to
special points of the function *g*(*z*) = *j*(*z*)*f '*(*z*)/*f*(*z*)
the zeroes and poles of *f*(*z*), where in the
neighbourhood of each such point* c *has an expansion of
the form

(cf. Theorem 2), where the (+)-sign occurs when
the point *c* is a zero of *f*(*z*) and the
(-)-sign when it is a pole, whence the residue of *g*(*z*)
at the point *c* equals ±*j*(*c*)*n
*and, replacing in (8) the function *f*(*z*)'/*f*(*z*)
by *g*(*z*), we find

In particular, setting *j*(*z*)
º *z*, we find

Here, the right hand side represents the
difference between the sums of all zeroes and poles of *f*(*z*)
in the domain *D *where every zero and pole enters the sum
as often as corresponds to its order.

Consider now together with the function *f*(*z*)
the function *g*(*z*) = *f*(*z*) - *a*,
where *a* is a fixed complex number; the poles of *g*(*z*)
coincide with the poles of *f*(*z*) and the zeroes
are *a-points* of *f*(*z*)*,* i.e.,
points at which* f*(*z*) assumes the value *a*.
If *f*(*z*) is analytic everywhere in *D*,
except at a finite number of poles, but is on the boundary *C *of
this region continuous and assumes the value *a*, where *f*
'(*z*) is continuous on C, Formulae (10) and (12) are
applicable to the function *g*(*z*) = *f*(*z*)
- *a*.

Thus, we obtain

where *a*_{1} *a*_{2},*
···, a*_{n} are *a*-points of the
function *f*(*z*) in the region *D *of
orders* *n*_{1}*,* *n*_{2},*
···, n*_{l} and poles *b*_{1}*,*
*b*_{2},* ···, b*_{l }*,
*of orders *p*_{1}*,* *p*_{2},*
···, p*_{l}, respectively.

* The order of *a*-points
of *f*(*z*) are called the orders of the
corresponding zeroes of the function *f*(*z*).

These formulae will be used in Chapter VII.

**24. Point at Infinity**
We have considered so far only finite points of the complex
variable plane; however, it is useful for a study of certain
problems to introduce the point at infinity. This is most clearly
done with the aid of the so called *spherical
projection** *of the *z*-plane onto the
sphere, touching the plane at its south pole. Such a projection
relates to each point *z* of the complex plane the point *Z*
of the sphere, which is obtained is obtained by intersection of
the sphere by the ray joining *z* to the north pole of the
sphere (Fig. 28). The stereographic projection establishes a
mutually single valued correspondence between the complex plane
and the sphere with the north pole above. The points *Z*
yield spherical images of the complex numbers *z* - their *numericals*.

In order to extend the correspondence to the
entire sphere, one introduces on the plane the conditional *point
at infinity *(the complex number *z* = ¥) and assumes it to correspond to the
sphere's north pole. The number *z* = ¥
does not enter into arithmetic operations like the
ordinary complex numbers. However, for example, one says that the
sequence |*z*_{n}| converges to the point
at infinity, if there is for any *M*>0 a number *n*_{0},
beginning with which |*z*_{n}| > *M
*(we have done this above in similar cases). This terminology
is justified in that, in fact, the stereographic projection *Z*_{n}
maps the point *z*_{n }of our sequence
into a sequence which converges to the North Pole of the sphere.

The plane of the complex variable augmented by
its point at infinity is called the *complete complex plane**
*(the plane without such a point is set to be *open*).
As we have seen, the complete complex plane is equivalent to the
sphere and, for geometric representations of the concept, it is
very convenient to resort to spherical illustration of complex
numbers.

We understand by the *neighbourhood* of
the point at infinity the circle on the sphere with centre at the
north pole or, in other words, the set of points *z* which
satisfy the inequality |*z*| > *R *(with
addition of the point at infinity). After introduction of this
concept, we may consider a region, augmented by the point at
infinity inside or on its boundary, i.e., an unbounded region.
The definition of the order of connectivity, given in **3.**
for bounded domains, extends without any changes to unbounded
domains (for example, the neighbourhood of the point *z* =
¥ with inclusion of the last turns
out to be a simply connected region, while the same neighbourhood
with exclusion of the point *z* = ¥ is
doubly connected).

Likewise, with out any changes, there is extended
to infinite *z*_{0} and *w*_{0} the
definition in **5. **of the limit of a function with
the aid of neighbourhood. Then, the function tending to the limit
*w*_{0} = ¥ is said to
be infinitely large (cf. **22.**,
definition of a pole).

Let the function *f*(*z*) be
analytic in some neighbourhood of the point at infinity (except
at the point *z* = ¥ itself;
the concept of analyticity at this point has not yet been
defined). The definition of special points of **22.** extends to
such points without change; one says that *z *= ¥ is a special point at infinity, a pole or
a an essentially special point of *f*(*z*)
depending on, naturally, whether it is infinite or there does not
exist at all

However, the criterion of type of special point,
connected with the Laurent expansion (**22.**** **,
Theorems 1 - 3) changes as is shown by the following reasoning.
Let *z* = 1/*z *and

then *j*(*z*) will be analytic in some
neighbourhood of the point *z*
= 0. The last will be for *j*(*z*) a special point of the same type
as also *z* = ¥ for *f*(*z*),
because

Obviously, the Laurent expansion of *f*(*z*)
in the neighbourhood of *z* = ¥ may
be obtained simply by setting *z *= 1/z
in the Laurent expansion of *j*(*z*) in the vicinity of *z* = 0. However, for such a
replacement, the true term is replaced by the principal one and
conversely.

Thus, one has

**Theorem 1 ***In the case of a
removable singularity at the point at infinity, the Laurent
expansion of f*(*z*) *in the neighbourhood of this
point does not at all contain positive powers of z, in the case
of a pole, it contains a finite number of them, in the case of an
essential singularity, an infinite number of terms.*

If *f*(*z*) has at the point *z*
= ¥ a removable singularity, one
usually says that it is *analytic at infinity* and sets

Obviously, in this case, the function is bounded
in some neighbourhood of the point *z* = ¥.

Let the function *f*(*z*) be
analytic throughout the entire plane. There follows from the
analyticity of the function at infinity its boundedness near this
point; let |*f*(*z*)| < *M*_{1}*
*for* *|*z*| > *R. *On the other
hand, there follows from the analyticity (and, consequently,
continuity) of *f*(*z*) in the closed circle |*z*|
£ *R *its boundedness in this
circle; let in it |*f*(*z*)| < *M*_{2}*
*. However, then the function *f*(*z*) is
bounded in the entire plane: For all *z*, one has |*f*(*z*)|
< *M *= max(*M*_{1}, *M*_{2}).

Thus, the Cauchy-Liouville Theorem (**17.**)
becomes

**Theorem 2 ***If the function f*(*z*)
*is analytic in the entire z-plane, it is constant.*

In conclusion, we present still the concept of
residue at the point at infinity. Let the function *f*(*z*)
be analytic in some neighbourhood of the point at infinity *z
= *¥ (except, may be, at this
point itself); we will understand by the *residue of the
function at infinity *

where *g*^{
- }is the sufficiently large circle |*z*| = *r*, travelled clockwise (so that the
neighbourhood of the point *z = *¥
remains on the left hand side as in the case of a finite point).
It follows directly from this definition that the *residue of
a function at infinity equals the coefficient of z*^{ -1}*
in the Laurent expansion in the vicinity of the point z = *¥*, taken with opposite sign.*

Finally, one obtains readily

**Theorem 3 ***If the function f*(*z*)
*has in the entire plane a finite number of singular points,
then the sum of all its residues, in cluding that at infinity, is
equal to zero.*

In fact, let *a*_{1}, *a*_{2},
··· , *a*_{n}, be finite special
points of *f*(*z*) and *g *the
circle |*z*| = *r*,
containing all of them. By the property of the integral, the
residue theorem and the definition of residue at infinity, we
have

**25.
Analytic Continuation. Generalization of the concept of analytic
function**** **We will now study the
problem of analytical continuation and introduce the concept of a
multi-valued analytic function - a generalization of the concept
of analyticity in **5****.**

Let the two domains *D*_{0} and *D*_{1}
without common points have a commen boundary section *g* and in these regions be given
(single-valued) analytic functions *f*_{0}(*z*)
and *f*_{1}(*z*), respectively. We will say
that the function *f*_{1}(*z*) is a *direct
analytic continuation *of the function *f*_{0}(*z*)
in the region *D*_{1}, if there exists a function *f*(*z*)
in *D*_{0} + *g *+ *D*_{1}
which equals *f*_{0}(*z*) at all points of *D*_{0}
and *f*_{1}(*z*) at all points of *D*_{1}:

By the uniqueness theorem of **20.**,
there is given for the domains *D*_{0} and *D*_{1}
and the section of the boundary *g *the
single valued analytic continuation of the given function *f*_{0}(*z*)
(if it is possible).

We present one simple sufficient condition for
analytical continuation, the so called *Principle of
continuous continuation*:

**Theorem 1 ***Let there be given two simply
connected regions D*_{0} *and* *D*_{1}
*without common points such that their boundaries have one
common section **g **and
in these regions analytic functions f*_{0}(*z*)*
and* *f*_{1}(*z*), *respectively.
Moreover, if these functions are continuous in D*_{0}
+ *g **and* *D*_{1}
+ *g** and coincide at all
points of the curve **g**,
then the function f*_{1}(*z*) *is the
direct analytic continuation of the function* *f*_{0}(*z*)
*in the region D*_{1}.

The proof is based on Morera's Theorem of **17.**
and Cauchy's Theorem of **12.**.
In fact, by our conditions, the function

is continuous in *D* = *D*_{0}
+ g + *D*_{1}; we will
show that its integral along any closed contour *C*, lying
in *D*, is zero. However, if *C *belongs completely
to one of the domains *D*_{0} or *D*_{1},
this is a direct consequence of Cauchy's Theorem. However, if *C*
belongs to *D*_{0} and *D*_{1},
then, denoting by *C*_{0} and *C*_{1}
the sections of the contour *C* lying in *D*_{0}
and *D*_{1}, respectively, and by *c* the
section of the curve *g*, lying
inside *C *(Fig. 29),. by Cauchy's Theorem (in the
generalized form of Theorem 5 of **12.**),
we find

Addition of these equations yields

Hence we conclude by Morera's Theorem that the function *f*(*z*)
is analytic in *D*, but this also means that is an
analytic continuation *f*_{0}(*z*), whence
the theorem has been proved.

In a few words, the theorem proved means that, if an analytic
function is a *continuous *continuation of an analytic
function *f*_{0}(*z*) through an arc g, that it is also an analytic continuation
of this function*.

* An analogous theorem in the real domain is
untrue: If two functions *f*_{0}(*x*) and *f*_{1}(*x*),
differentiable in adjacent intervals (*a*, *b*) and
(*b*, c) are continuous and coincide at the common
point *b *of these intervals, then the function *f*(*x*), obtained by their addition, may
not be differentiable on the interval (*a*, *b*) -
its graph may have at *b* an angular point.

On the basis of Theorem 1, we will now somewhat generalize the
above concept of direct analytic continuation. In fact, we will
assume that the domains *D*_{0} and *D*_{1}*
*have in common beside the boundary segment* **g* an internal point (Fig. 30) and
that there are given in these domains the analytic functions *f*_{0}(*z*)
and *f*_{1}(*z*). We will then say that *f*_{1}(*z*)
is the *direct analytic continuation *of *f*_{0}(*z*)
through the arc *g* if *f*_{0}(*z*)
and *f*_{1}(*z*) are continuous in *D*_{0
}+ *g *and *D*_{1}*
*+ *g *and their values on *g *coincide.

If *D*_{0}* *and *D*_{1}*
*do not have common internal points, this definition
coincides with the initial one. However, if *D*_{0}*
*and *D*_{1}* *have common points (for
example, the common section *d*
' in Fig. 30), the function *f*(*z*) determined by
(2) may be double-valued at the points *d*
', because at the point *z *of *d*
' the values of *f*_{0}(*z*) and *f*_{1}(*z*)
must coincide. Thus, the second definition of the extension is
more general than the first one.

We will yet generalize somewhat our definition.
Let there be given a chain of simply-connected domains *D*_{0},
*D*_{1}, ··· , *D*_{n}*
*such that each pair of adjoining domains *D*_{k}
and *D*_{k+1}* *have a common
boundary segment *g*_{k,k+1}* *(Fig. 31) and let there be given
in each domain *D*_{k}* *the
single-valued analytic function *f*_{k}(*z*).
We will say that *f*_{n}(*z*) is
the *analytic continuation** of the function *f*_{0}(*z*)
in the domain *D*_{n}* *through
the chain of domains, if for any *k* = 0, 1, ··· , *n*
-1 the functions *f*_{k}(*z*) and *f*_{k+1}(*z*),
respectively, are continuous in *D*_{k}*
*+ *g*_{k,k+1}* *and *D*_{k+1}*
*+ *g*_{k,k+1}* *and their values on *g*_{k,k+1}* *coincide. For *n* = 1, we
obtain the preceding definition.

*Already without the word "direct".

Note that also here for fixed *D*_{0},
*D*_{1}, ··· , *D*_{n}*
*and *g*_{01}, *g*_{12}, ··· , *g*_{n-1,n}*
*the analytic continuation of the function *f*_{0}(*z*)
in the domain *D*_{n}* *(if it is
possible) is determined single-valuedly. Already for a change of
intermediate links of the chain *D*_{k}
or even for a replacement of any arc *g*_{k,k+1}* *by another common arc of the
boundary of *D*_{k} and *D*_{k+1},
the value of the analytic continuation may change.

We present a related simple example. Let *D*_{0}*
*and *D*_{1}* *represent the upper and
lower halves of a ring 1 < |*z*| < 2, for which, Im *z
*> 0 and Im *z *< 0, respectively, and *f*_{0}(*z*)
is the branch of Ö*z, *characterized by the condition 0 < arg *z
<* *p*. If we take for *g*_{01}
the segment (-2, -1), then the analytic continuation *f*_{0}(*z*)
in *D*_{1} is completely determined as the branch of Ö*z*, for which *p* < arg *z
<* 2*p* (on *g*_{01}
the value of the argument must change continuously). Now, if
retaining without change *f*_{0}(*z*) and
the domains *D*_{0}* *and *D*_{1}*,
*we replace the segment *g*_{01}by
the segment then the analytic continuation *f*_{0}(*z*)
in *D*_{1} is determined as the branch of Ö*z* for which -*p < *arg* **z *<
*p* (now, the value of the argument must change
continuously on the segment ). These continuations *f*_{1}(*z*)
and
differ, for example,

(they only differ in sign). Thus, the value of the analytic continuation actually may change even only for a change of the links of the chain of arcs.

Now, let *f*_{0}(*z*), *D*_{0},
*D*_{1}* *and* g*_{0}
have the previous meaning; we select yet one link of the chain *D*_{2}
= *D*_{0}, linked to *D*_{1}* *by
:(1,
2); then the analytic continuation *f*_{2}(*z*)
is determined as the branch of Ö*z* for which 2*p* < arg *z <* 3*p* and at any point *z
*of the upper half of the ring the values of *f*_{0}(*z*)
and *f*_{2}(*z*) will differ (differ in
sign). We see that when the domains *D*_{0}* *and
*D*_{k}* *overlay each other, then
at their common points the values of the function and its
analytic continuation may differ.

The concept of analytic continuation, introduced
above, allows to introduce the concept of a complete analytic
function (generally speaking, a multi-valued one). Let there be
given in some domain *D*_{0}* *a
single-valued analytic function *f*_{0}(*z*).
It may happen that *f*_{0}(*z*) is *not
continuable* either through one or another arc of its
boundary *C *of the region *D*_{0}. For
example, let *D*_{0} be the circle |*z*|
< 1 and

The function *f*_{0}(*z*)
has a singularity at the point *z* = 1, because for real *z
= x*, as is easily seen, In fact,

Consequently, for any *n*, one may find *d **> *0* *such
that for *x *> 1 - *d, *one
has

but this means, all the more, that

Moreover, we have

whence also at the points *z = *Ö1(i.e. *z* = ±1)
one has singularities. Analogously, for any natural number *n*,

whence also at the points , which lie at the
vertices of the regular 2*n*-polygon, inscribed in the
circle |*z*| = 1, the function *f*_{0}(*z*)
has singularities. Thus, the set of the singularities of the
function *f*_{0}(*z*) is everywhere dense
on the circle |*z*| = 1 and *f*_{0}(*z*)
is really uncontinuable through any arc of this circle.

In such cases, we will say that the contour *C* is a *natural border** of the
function f*_{0}(*z*); we will call *D*_{0}
an *essential domain *of
this function and the function itself a *complete analytic
function.*

Now, let *f*_{0}(*z*) be continueable
beyond the limit* D*_{0}. We will consider all
possible of its analytic continuations along all possible chains
of regions. We will view the values of all such continuations as
well as at the values of one function *f*(*z*). We
will call such a function a *complete analytic function*,
but single-valued analytic functions one of which is composite
(continuation of the function *f*_{0}(*z*))
- its *branches*. The domain *D*, obtained by
joining all domains, comprising the chain, along which occurs the
analytic continuation, and the arcs, along which these domains
are closed, we will call the *domain of the existence of f*(*z*).

One may study the function *f*(*z*) not in the
entire domain of its existence but only in a part of it; then we
will simply call *f*(*z*) an *analytic function*.
Such a definition of an analytic functions is a generalization of
that in **5.**,**
**because it obviously admits also multi-valued functions.
In the sequel, when speaking of analytic functions, we will
understand analyticity in this more general sense. However, if we
must emphasize that we are concerned with analyticity in the
sense of **5.** , we will speak of *single-valued
*analytic functions.

We close the study of the general concept of analytic
functions by a description of the points at which its analyticity
is violated - the singularities of the function. There is not
need to think that such points represent themselves something
exclusive, pathological, whence it is of little interest for
applications. In the contrary, singularities are of the greatest
interest for the study of analytic function, because in them,
descriptively speaking, lies all the basic information concerning
the functions. The reader will assess better the truth of this
statement, when he reaches, for example, Chapter V and sees
singularities so to say at "work". For the present, we
advise readers to remember that, by the Cauchy-Liouville Theorem
(in the formulation of **24.**),
all analytic functions, with the exclusion of constants, have
singularities, and likewise to remember the already frequently
places in this book, where there have been drawn on the basis of
the study of singularities important conclusions regarding* *analytic
functions (Theorems in **23.****
**Note at end of **19****.**.
etc.). We now proceed to exact defintions.

A point *a, *which belongs to the domain of existence
of an analytic functio and its boundary, will be called a *special
point *of the function *f*(*z*) if in it
analyticity is violated although itb is a branch of *f*(*z*).

As in **22.**.,
we limit consideration to specail points of the simplest kind -
so-called isolated points. A point is called an *isolated** *special point of *f*(*z*)
if there exists a neighbourhoof 0 < |*z - a*| < *R
*that *f*(*z*) is continued along any chain of
domains belonging to this neighbourhood.

We consider the chain consisting of the domains *D*_{k}
(*k *= 0, ±1, ±2, ···), represnting themselves the
ring 0 < |*z* - *a*| < *R cur along some
radius ,* for example, arg* |z - a| = *0*. *Let*
f*_{0}(*z*) be a branch of *f*(*z*),
single-valued and analytic in any ring *D*_{0}
with cut *g*_{0}. If
the values of *f*_{0}(*z*) on both shores
of the cut *g*_{0 }coincide,
we will say that *a* is a soecial point witha *single-valued
character *for the given branch (in this case, by the
uniqueness theorem of 20.
the analytic continuations of the branch *f*_{0}(*z*)
in the other rings *D*_{k} coincide with *f*_{0}(*z*)
). We have cinsidered such special points in **22.**

However, if the values of *f*_{0}(*z*)
on the shores of the cuts do not agree, then *a* is called
a special point of a *multi-valued character *or *branch-point**. *There may
arise two cases:

1) There exists the chain *D*_{ 0 }, *D*_{1}*,*
··· , *D*_{n-1}* *of rings,
joimed consecutively, for example, counter-clockwise (i.e., the
lower shore of the cut on *D*_{0}* , *joins
the upper shore of the cut at *D*_{1}*, *the
lower at* D*_{1 }the upper shore at *D*_{2}*,*
etc.) such that on the remaining free shores of the cuts *D*_{
0 }and *D*_{n-1}(the upper shore of *D*_{
0 }and the lower one of *D*_{n-1})
the values of *f*_{0}(*z*) and *f*_{n-1}(*z*)
coincide. The, by the uniqueness theorem of **20.****,
***f*_{n-1}(*z*) º *f*_{1}(*z*),
··· , *f*_{2n-1}(*z*) º *f*_{n-1}(*z*)
and, in general, the values *f*_{k}(*z*)
for *k *, changing from -¥ to
+¥ repeat periodically the values *f*_{0}(*z*),
*f*_{1}(*z*), ··· , *f*_{n-1}(*z*).
In this case, we will say that *a* is a *branch-point of finite
order** n.*.

If in the case under consideration all branches *f*_{k}(*z*)
tend for *z* ® *a *to
one finite or infinite limit, one says that *a* is an *algebraic branch poin*t.
For example, such are the points *z* = 0 and *z* = ¥ for the function

However, if the limit *f*_{k}(*z*)
for *z* ® *a *does not
exists, then *a *is called a *transcendental branch
point*. For example, aucha point is *z* = *a *for
the function

(*z =*¥ is for
it an algebraic branch point).

2. In all rings *D*_{k }the
chain values of the function differ. In this case, *a* are
called *logarithmic branch
points**.* For example, such are the points *z*
= 0 and *z* = ¥ for the
multi-valued function *w* = Ln *z.* Logarithmic
branch points belong to a number of transcendental functions.

In the neighbourhood of a branch point *a*
of finite order *n*, the function *f*(*z*)
admits expansion into the *generalized
power series*

In fact, we set *z - a = **z*^{ n}; then the chain of domains *D*_{n}* *(*n *= 0, 1, ··· , *n -*1)
in the *z*-plane will
correspond to the chain of adjacent sectors *D
*of the ring

.

with central angles 2*p**/n*.
Consider the function *j*(*z*)
= *f*(*a +**z*^{n})
and select in each sector *D*_{n}*
*a corresponding branch *f*_{n
}of the function *f*. Obviously, the
function *j*(*z*) is
continuously continued from *D*_{0}
to *D*_{1}*, D*_{2},
··· ,* D*_{n-}_{1} and its values on the free
shores of the cuts *D*_{0 }and* D*_{n-}_{1 }coincide. Hence, the point *z *= 0 is for this function an
isolated special point with a single-valued character and,
consequently, *j*(*z*) is represented in the neighbourhood of *z *= 0 by the Laurent series

Substituting here *z*
= (*z - a*)^{1/n}, we obtain the
known expansion (4).

In the case of an algebraic point *a*, the
expansion (4) contains a finite number of terms with negative *k
*(may be, these terms are not present) and in the case of a
transcendental point there are an infinite number of them.

**26.
Riemann Surfaces**** **Concluding this
chapter, we dwell on the topic of Riemann surfaces for
multi-valued functions. These surfaces yield a geometrical
description of the above process of analytic continuation and the
concept of multi-valued
analytic functions. Let there be given a (multi-valued)
analytic function *f*(*z*), defined in the domain *D*
of the plane of the complex variable *z. *We will consider
the regions *D*_{k}*,* out of
which in the process of analytic continuation is constructed the
domain *D*, as separate sheets, constituting so many
members as the given function has values in the domain *D*.

Consider in this *z*-plane some chain of
domains *D*_{0}, *D*_{1}, ··· , *D*_{n}*
*with common boundary segments *g*_{01}*, g*_{12}*,
*··· , *g*_{n-1,n}*. *Let the domains *D*_{0}
and *D*_{1} have common sections, where in one of
them the values *f*_{0}(*z*) and *f*_{1}(*z*)
coincide and in others differ. We will choose sheets,
corresponding to *D*_{0} and *D*_{1}
and join them along the lines, corresponding to *g*_{01}*.
*We place these sheets on *D*_{0} + *g*_{01 }*+ D*_{1}
so that each sheet lies on the corresponding domain and join
their sections, spread over those common parts *D*_{0}
and* D*_{1} in which *f*_{0}(*z*)
and *f*_{1}(*z*) coincide; we will consider
the joined parts as a single layer. However, over the common
sections of the domains *D*_{0} and *D*_{1},
in which the values of *f*_{0}(*z*) and *f*_{1}(*z*)
differ, we arrange the corresponding parts of the sheets one
above the other so that there will lie over such parts two
layers. We now agree to relate the value of *f*_{0}(*z*)
to the point of the first sheet, located above *z*, and
the value of *f*_{1}(*z*) to the same point
of the second sheet; then the function

will be *single-valued *in the set of
joined sheets.

We perform exactly the same operation on the
sheets corresponding to the domain *D*_{2} *, etc.
In this process, it may happen that the proposed joining of the
sheets becomes impossible without their intersection (cf. Fig. 32
which shows the neighbourhood of a third order branch point,
joined out of three rings 0 < |*z - a*| < *R *with
cuts; we do not consider the intersections, which arise during
the joining of the rings *D*_{0} and *D*_{2}).
As a result, we obtain a pile of, generally speaking,
multi-sheeted surfaces distributed over the domain *D*_{0}*
*+ *g*_{01} + *D*_{1}*
*+ ··· + *g*_{n-1,1}
+ *D*_{n}. If we execute the proposed
operation for all possible chains of domains, determining the
analytic function *f*(*z*), we obtain , generally
speaking, a multi-sheeted surface *R*, located over a
domain *D*. We will call this surface the *Riemann
surface of the function f*(*z*).

* For this, there may appear
points *z* for which there are distributed three layers of
sheets - this is the present case - if *D*_{2}*
*overlays those common parts of *D*_{0} and *D*_{1}in
which the values *f*_{0}(*z*) and *f*_{1}(*z*)
differ and if those values of *f*_{2}(*z*)
also differ from *f*_{0}(*z*) and *f*_{1}(*z*).

It is essential to note that *any analytic
function may be viewed as a single-valued function on its Riemann
surface. *For this, it is sufficient to relate different
values, assumed by the function at any point *z*, to
different sheets of the Riemann surface, located over this point.
For example, we will allot the three values of the root at a
point *z *¹ *z*_{0}* *in the neighbourhood of *z*_{0 }to three points on the surface of
Fig. 32, lying above the point *z. *

If the function *w = f*(*z*),
inverse to the single-valued function *z = **j*(*w*) ( as in all the
examples, which we have considered in **3.**,),
then it realizes obviously a mutually single-valued mapping of
our Riemann surface onto the complete *w*-plane or some
part of it. However, in the general case, *w = f*(*z*)
maps one Riemann surface onto another.

We will now present several examples of simplest Riemann surfaces*:

* We recommend the readers to join models of the Riemann surfaces considered below on paper and follow on these models the following arguments.

1) *Riemann surface of the root*
:

We select as region *D*_{k} the plane
with the cut positive semi-axis: *D*_{k }is
characterized by the inequalities 2*k**p
*<* *arg *z* < 2(*k + *1)*p *(*k = *0, ±1, ±2,
···). In the initial domain *D*_{0}, we choose
the branch *f*_{0}(*z*), determined by the
conditions 0 < arg *z* < 2*p,**
*continue it into *D*_{1}*,* *D*_{2},
··· , *D*_{n-1}.Correspondingly, we
store *n* copies of the sheets, having the same form as *D*_{k},
and join the lower shore of the cut of the domain *D*_{0}*
*to the upper shore of the cut of the domain *D*_{1}*,*
the lower shore of the cut of the domain *D*_{1}*
*to the upper shore of the cut of the region *D*_{2},
etc. The values of *f*_{0}(*z*) and *f*_{n}(*z*)
on the positive semi-axis (and in the entire domain *D*_{n}*
= D*_{0}) coincide. Consequently, we must join
(without taking into consideration intersections which then
arise) the remaining free upper shores of the cut on sheet *D*_{0}*
*with the lower shore of the cut on *D*_{n-1}*.
*The values of in the remaining domains *D*_{k
}only repeat the selected values *f*_{0},
*f*_{1}, ··· , *f*_{n-1},
whence the n-sheeted surface is also the Riemann surface of the
function At the points *z* = 0 and *z*
= ¥ , it has algebraic branch
points of order *n* (cf. Fig. 33, where *n *= 4).

2) *Riemann surface of the logarithm w = *Ln *z*:
The domain D_{k }is the same as in the preceding example.
We select in *D*_{0 }the branch *w = *ln |*z*|
+ *i*arg *z*, where 0 < arg *z* < 2*p*, and this branch is continued
without bound in the domains *D*_{k}* *for*
k = *0, ±1, ±2, ···. This corresponds to the fact* *that
an innumerable multitude of samples of sheets of the same form as
*D*_{k}* *are joined following the
law: The lower shore of the cut of each sheet *D*_{k
}is joined to the upper shore of the sheet *D*_{k+1}.
The resulting Riemann surface of the logarithm has the form shown
in Fig. 34. At *z* = 0 and *z *= ¥, it has branch points of the logarithmic
kind.

3) *Riemann surface of the function *the inverse of the
Joukowski function.We select as *D*_{k }the
plane with the removed cut [1, -1], denote by *f*_{0}(*z*)
and *f*_{1}(*z*) those branches of the
function which map *D*_{0 }and *D*_{1
}, respectively, onto the inside and the outside of the
unit circle (cf. **7.**).
Since *f*_{0}(*z*) maps the lower edge of
the segment [-1, 1] onto the upper semi-circle, *f*_{1}(*z*)
onto the upper edge, we must join the lower edge of the cut in
sheet *D*_{0 }and the upper edge of the cut in
sheet *D*_{1}. The same must be done with the
upper shore of the cut in *D*_{0 }and the lower
one in *D*_{1 }, which map onto the lower
semi-circle. The double-sheeted surface obtained is the Riemann
surface of our function; it has branch points of the second order
at the points ±1 (Fig. 35)*. The surface differs from the
surface of Ö*z *only by the
additional linear fractional mapping; in fact, the
transformations *z* = (*z*+1)/(*z-*1) and *w*
= (*w+*1)/(*w-*1) translate the function into
*w = *Ö*z *(cf. **31.**).

* The surface has at ¥ two branching sheets.

4)* The *Riemann surface of
arcsin w = Arcsin z:* *We have seen in **9.****
**that the function *z* = sin *w *maps the
half-strip Im* *w* *> 0. -*p*/2
< Re *w* < *p*/2
onto the upper half-plane*, where the rays (1) and (4) in the
lower Fig. 36 become the rays *x* < -1 and *x>*1;
it follows from the oddness of sin *w* that the half strip
Im *w* < 0, -*p*/2
< Re *w* < *p*/2
becomes by it the lower half-plane, where the rays (2) and (3)
correspond to the rays *x* < -1 and *x* > 1
(Fig. 36). Thus, one of the branches of the function *w* =
Arcsin *z *(we denote it by *f*_{0}(*z*)
) maps the plane *z* with cuts along (- ¥ - 1) and (1, ¥
) (we denote it by *D*_{0 }) onto the strip *D*_{0 }: -*p*/2 < Re *w* < *p*/2 with corresponding boundaries.
listed in Fig. 36. Since sin (*w + **p*)*
= - *sin *w*, then the strip *D*_{1
}: *p*/2 < Re *w*
< 3*p*/2 becomes through
mapping *z* = sin *w *the same domain of the *z*-plane;
we denote this domain by *D*_{1}* *and by*
f*_{1}(*z*) the function, realizing the
inverse mapping. The corresponding boundaries *D*_{1}and
*D*_{1}are shown in Fig. 36.

Obviously, the branch *f*_{1}(*z*) is
the analytic continuation of *f*_{0}(*z*)
in *D*_{1}* *and for such a continuation
the function *z = *sin *w *remains continuous on
the straight line Re *w* = *p*/2.
Correspondingly, we must join across the shores of the cuts of
the sheets *D*_{0}* *and *D*_{1}*
*: (4) to (*1*_{1}) and (3) to (*2*_{1}).
There results a two-sheet surface with first order branch points
at *z* = 1 and with a cut along the ray (-¥, -1), on which are located the four
shores of the cuts L (*1*), (*2*), (*3*_{1}),
(*4*_{1}). Due to the periodicity, of the function
*z* = sin *w*, the set* *of the strips *D*_{2}and*
D*_{3}* *maps onto such a two-sheet
surface, consisting of the sheets *D*_{2}and* **D*_{3}*
*and having the four free shores of the cuts (*1*_{2}),
(*2*_{2}), (*3*_{3}) and (*4*_{3}).
We must join the two constructed surfaces, across the free shores
of the cuts to the sheets *D*_{1}* *and *D*_{2}:
(*4*_{1}) to (*1*_{2}) and (*3*_{1})
to (*2*_{2}) - this corresponds to a continuous
continuation of the function *z = *sin *w *through
the straight line Re *w* = 3*p*/2
(Fig. 36). Then, there appears under the point *z* = -1
the branch point, joining the sheets *D*_{1}* *and
*D*_{2}. Continuing such a design without a break
to the right and the left from the basic strip *D*_{0}, we obtain the
infinity of sheets of Riemann surfaces of arcsin. It has an
infinite set of branch points of the second kind at *z* =
±1 and a logarithmic branch point at *z* = ¥ (Fig. 37).

As is shown by this scheme, the function *z = *sin
*w *realizes a mutually single-valued and continuous
mapping of the entire finite *w*-plane onto out Riemann
surface. The inverse function *w* = Arcsin *z *is
single-valued on this surface.

**REFERENCES OF CHAPTER 1**

[1] I.I.Privalov: Introduction to the theory of functions of a
complex variable. Fizmatgiz 1950

[2] A.I.Markushevich: Theory of analytic functions. Gostechizday
1950

[3] A.I.Markushevich: Short course of the theory of analytic
functions, Fismatgiz, 1961

[4] S.Stoilov, Theory of functions of a complex variable , Vols.
1 and 2, translated from Roumanian, IL, 1962

[5] R. Courant: Geometric theory of the function of a complex
variable, translated from German, ONTI, 1934

[6] A.Hurwitz: Theory of analytic and elliptic functions,
translated from German, ONTI, 1933

[7] E.T.Whittacker and G.N.Watson: Course of contemporry
analysis, translated from English, Fizmatgiz 1963

[8] V.I.Smirnov: Course of higher mathematics, Vol. III, Part 2,
GOSTECHDAT, 1957

[9] B.A.Fuks and B.V.Shabat: Functions of a complex variable and
some of their applications, "Nauka", 1964

[10] J.Springer: Introduction to the theory fo Riemann surfaces,
translatde from German, IL, 1960

[11] A.I.Markushevich: Essays on the history of the theory of
analytic functions, Gostekhizdat. 1951.