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)/2pi 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 proper, 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., c0.

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 cn 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)nf(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 nm 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 zk ® 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 zk ® 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 zk®a so that f(zk) = 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 zk ® 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 2ze 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 zk =1/k of this theorem will be zk = -1/k, k = 1, 2, 3, ···, because, obviously,

for A = 0, one may take zk = -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 functions: 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(f1, f2, ···, fn) 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 z² 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 z²). The residue at z = 0 is, by (3),

(this follows also from the fact that the Laurent series of cotan z² with centre at z = 0 may only contain even powers of z. By (5), the residue at the point with j = cos z², Y = sin z² 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 a1, a2, ··· , an, 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 z0 of the boundary

where z ® z0 via points of the region D.

The proof follows from Cauchy's Theorem for multi-connected domains (13.). We surround every point an by a circle gk: |z - ak| = rk 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 gk, 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 gk 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 z0 of the boundary there exists

where z ® z0 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 cn 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 [lnf(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 b1, b2, ··· , bm of multiplicity p1, p2, ··· , pm , 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 a1, a2, ··· , al and their multiplicities by n1, n2, ··· , nl , 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 2p - 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 a1 a2, ···, an are a-points of the function f(z) in the region D of orders* n1, n2, ···, nl and poles b1, b2, ···, bl , of orders p1, p2, ···, pl, 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 |zn| converges to the point at infinity, if there is for any M>0 a number n0, beginning with which |zn| > M (we have done this above in similar cases). This terminology is justified in that, in fact, the stereographic projection Zn maps the point zn 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 z0 and w0 the definition in 5. of the limit of a function with the aid of neighbourhood. Then, the function tending to the limit w0 = ¥ 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)| < M1 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)| < M2 . However, then the function f(z) is bounded in the entire plane: For all z, one has |f(z)| < M = max(M1, M2).

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.

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 a1, a2, ··· , an, 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 D0 and D1 without common points have a commen boundary section g and in these regions be given (single-valued) analytic functions f0(z) and f1(z), respectively. We will say that the function f1(z) is a direct analytic continuation of the function f0(z) in the region D1, if there exists a function f(z) in D0 + g + D1 which equals f0(z) at all points of D0 and f1(z) at all points of D1:

By the uniqueness theorem of 20., there is given for the domains D0 and D1 and the section of the boundary g the single valued analytic continuation of the given function f0(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 D0 and D1 without common points such that their boundaries have one common section g and in these regions analytic functions f0(z) and f1(z), respectively. Moreover, if these functions are continuous in D0 + g and D1 + g and coincide at all points of the curve g, then the function f1(z) is the direct analytic continuation of the function f0(z) in the region D1.

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 = D0 + g + D1; 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 D0 or D1, this is a direct consequence of Cauchy's Theorem. However, if C belongs to D0 and D1, then, denoting by C0 and C1 the sections of the contour C lying in D0 and D1, 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

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 f0(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 f0(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 f0(x) and f1(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 D0 and D1 have in common beside the boundary segment g an internal point (Fig. 30) and that there are given in these domains the analytic functions f0(z) and f1(z). We will then say that f1(z) is the direct analytic continuation of f0(z) through the arc g if f0(z) and f1(z) are continuous in D0 + g and D1 + g and their values on g coincide.

If D0 and D1 do not have common internal points, this definition coincides with the initial one. However, if D0 and D1 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 f0(z) and f1(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 D0, D1, ··· , Dn such that each pair of adjoining domains Dk and Dk+1 have a common boundary segment gk,k+1 (Fig. 31) and let there be given in each domain Dk the single-valued analytic function fk(z). We will say that fn(z) is the analytic continuation* of the function f0(z) in the domain Dn through the chain of domains, if for any k = 0, 1, ··· , n -1 the functions fk(z) and fk+1(z), respectively, are continuous in Dk + gk,k+1 and Dk+1 + gk,k+1 and their values on gk,k+1 coincide. For n = 1, we obtain the preceding definition.

Note that also here for fixed D0, D1, ··· , Dn and g01, g12, ··· , gn-1,n the analytic continuation of the function f0(z) in the domain Dn (if it is possible) is determined single-valuedly. Already for a change of intermediate links of the chain Dk or even for a replacement of any arc gk,k+1 by another common arc of the boundary of Dk and Dk+1, the value of the analytic continuation may change.

We present a related simple example. Let D0 and D1 represent the upper and lower halves of a ring 1 < |z| < 2, for which, Im z > 0 and Im z < 0, respectively, and f0(z) is the branch of Öz, characterized by the condition 0 < arg z < p. If we take for g01 the segment (-2, -1), then the analytic continuation f0(z) in D1 is completely determined as the branch of Öz, for which p < arg z < 2p (on g01 the value of the argument must change continuously). Now, if retaining without change f0(z) and the domains D0 and D1, we replace the segment g01by the segment then the analytic continuation f0(z) in D1 is determined as the branch of Öz for which -p < arg z < p (now, the value of the argument must change continuously on the segment ). These continuations f1(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 f0(z), D0, D1 and g0 have the previous meaning; we select yet one link of the chain D2 = D0, linked to D1 by :(1, 2); then the analytic continuation f2(z) is determined as the branch of Öz for which 2p < arg z < 3p and at any point z of the upper half of the ring the values of f0(z) and f2(z) will differ (differ in sign). We see that when the domains D0 and Dk 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 D0 a single-valued analytic function f0(z). It may happen that f0(z) is not continuable either through one or another arc of its boundary C of the region D0. For example, let D0 be the circle |z| < 1 and

The function f0(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 2n-polygon, inscribed in the circle |z| = 1, the function f0(z) has singularities. Thus, the set of the singularities of the function f0(z) is everywhere dense on the circle |z| = 1 and f0(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 f0(z); we will call D0 an essential domain of this function and the function itself a complete analytic function.

Now, let f0(z) be continueable beyond the limit D0. 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 f0(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 Dk (k = 0, ±1, ±2, ···), represnting themselves the ring 0 < |z - a| < R cur along some radius , for example, arg |z - a| = 0. Let f0(z) be a branch of f(z), single-valued and analytic in any ring D0 with cut g0. If the values of f0(z) on both shores of the cut g0 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 f0(z) in the other rings Dk coincide with f0(z) ). We have cinsidered such special points in 22.

However, if the values of f0(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 , D1, ··· , Dn-1 of rings, joimed consecutively, for example, counter-clockwise (i.e., the lower shore of the cut on D0 , joins the upper shore of the cut at D1, the lower at D1 the upper shore at D2, etc.) such that on the remaining free shores of the cuts D 0 and Dn-1(the upper shore of D 0 and the lower one of Dn-1) the values of f0(z) and fn-1(z) coincide. The, by the uniqueness theorem of 20., fn-1(z) º f1(z), ··· , f2n-1(z) º fn-1(z) and, in general, the values fk(z) for k , changing from -¥ to +¥ repeat periodically the values f0(z), f1(z), ··· , fn-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 fk(z) tend for z ® a to one finite or infinite limit, one says that a is an algebraic branch point. For example, such are the points z = 0 and z = ¥ for the function

However, if the limit fk(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 Dk 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 Dn (n = 0, 1, ··· , n -1) in the z-plane will correspond to the chain of adjacent sectors D of the ring

.

with central angles 2p/n. Consider the function j(z) = f(a +zn) and select in each sector Dn a corresponding branch fn of the function f. Obviously, the function j(z) is continuously continued from D0 to D1, D2, ··· , Dn-1 and its values on the free shores of the cuts D0 and Dn-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 Dk, 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 D0, D1, ··· , Dn with common boundary segments g01, g12, ··· , gn-1,n. Let the domains D0 and D1 have common sections, where in one of them the values f0(z) and f1(z) coincide and in others differ. We will choose sheets, corresponding to D0 and D1 and join them along the lines, corresponding to g01. We place these sheets on D0 + g01 + D1 so that each sheet lies on the corresponding domain and join their sections, spread over those common parts D0 and D1 in which f0(z) and f1(z) coincide; we will consider the joined parts as a single layer. However, over the common sections of the domains D0 and D1, in which the values of f0(z) and f1(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 f0(z) to the point of the first sheet, located above z, and the value of f1(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 D2 *, 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 D0 and D2). As a result, we obtain a pile of, generally speaking, multi-sheeted surfaces distributed over the domain D0 + g01 + D1 + ··· + gn-1,1 + Dn. 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 D2 overlays those common parts of D0 and D1in which the values f0(z) and f1(z) differ and if those values of f2(z) also differ from f0(z) and f1(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 z0 in the neighbourhood of z0 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 Dk the plane with the cut positive semi-axis: Dk is characterized by the inequalities 2kp < arg z < 2(k + 1)p (k = 0, ±1, ±2, ···). In the initial domain D0, we choose the branch f0(z), determined by the conditions 0 < arg z < 2p, continue it into D1, D2, ··· , Dn-1.Correspondingly, we store n copies of the sheets, having the same form as Dk, and join the lower shore of the cut of the domain D0 to the upper shore of the cut of the domain D1, the lower shore of the cut of the domain D1 to the upper shore of the cut of the region D2, etc. The values of f0(z) and fn(z) on the positive semi-axis (and in the entire domain Dn = D0) coincide. Consequently, we must join (without taking into consideration intersections which then arise) the remaining free upper shores of the cut on sheet D0 with the lower shore of the cut on Dn-1. The values of in the remaining domains Dk only repeat the selected values f0, f1, ··· , fn-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 Dk is the same as in the preceding example. We select in D0 the branch w = ln |z| + iarg z, where 0 < arg z < 2p, and this branch is continued without bound in the domains Dk for k = 0, ±1, ±2, ···. This corresponds to the fact that an innumerable multitude of samples of sheets of the same form as Dk are joined following the law: The lower shore of the cut of each sheet Dk is joined to the upper shore of the sheet Dk+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 Dk the plane with the removed cut [1, -1], denote by f0(z) and f1(z) those branches of the function which map D0 and D1 , respectively, onto the inside and the outside of the unit circle (cf. 7.). Since f0(z) maps the lower edge of the segment [-1, 1] onto the upper semi-circle, f1(z) onto the upper edge, we must join the lower edge of the cut in sheet D0 and the upper edge of the cut in sheet D1. The same must be done with the upper shore of the cut in D0 and the lower one in D1 , 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 Öz 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 = Öz (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 f0(z) ) maps the plane z with cuts along (- ¥ - 1) and (1, ¥ ) (we denote it by D0 ) onto the strip D0 : -p/2 < Re w < p/2 with corresponding boundaries. listed in Fig. 36. Since sin (w + p) = - sin w, then the strip D1 : p/2 < Re w < 3p/2 becomes through mapping z = sin w the same domain of the z-plane; we denote this domain by D1 and by f1(z) the function, realizing the inverse mapping. The corresponding boundaries D1and D1are shown in Fig. 36.

Obviously, the branch f1(z) is the analytic continuation of f0(z) in D1 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 D0 and D1 : (4) to (11) and (3) to (21). 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), (31), (41). Due to the periodicity, of the function z = sin w, the set of the strips D2and D3 maps onto such a two-sheet surface, consisting of the sheets D2and D3 and having the four free shores of the cuts (12), (22), (33) and (43). We must join the two constructed surfaces, across the free shores of the cuts to the sheets D1 and D2: (41) to (12) and (31) to (22) - this corresponds to a continuous continuation of the function z = sin w through the straight line Re w = 3p/2 (Fig. 36). Then, there appears under the point z = -1 the branch point, joining the sheets D1 and D2. Continuing such a design without a break to the right and the left from the basic strip D0, 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

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[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
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