10. The general power w = za , where a = a + ib is an arbitrary complex number, is defined by
Setting here z = reij, we obtain Ln z = ln r + i(j + 2kp) and, consequently,
where k is an arbitrary integer, whence for b ¹ 0 the function za has always infinitely many values which lie (for fixed z and a) on the circle |w| =rk with the radius
forming a geometric progression, infinite on both sides, with the denominator e-2pb. The arguments of these values
form likewise on both sides arithmetic progressions with the difference 2pa..
For b = 0, i.e., for real a, the values za lie on the circle |w| = ealn r = ra with the arguments
If a = p/q is rational (we assume that the fraction p/q has been reduced), then all the values qk will differ from q among these values (for example, q1 , q2, ···,qq-1) by a multiple of 2p. Consequently, in this case, the function w = za is finite-valued and coincides with the function ,
However, if a is an irrational real number, then there are among the values qk in (5) those which do not differ from integral multiples of 2p and, consequently, the function za = eaLnz is infinitely-valued.
The multi-valuedness of the general power, like those elementary functions considered above, is caused by the multi-valued argument. The methods of their subdivision into single-valued branches have been given above; the branch point is z = 0.
Side by side with the general power, the function (1) may be considered as the general exponential function
In contrast to (1), Formula (7) represents a set of individual separate, not interconnected functions which differ by the factor e2kp iz, where k is an integer.
1.4 Integration of functions of a complex variable We will consider the concept of the integral of a function of a complex variable and the most important properties of analytic functions, connected with the concept of integral or operating on it. In particular, for example, there will be established the equivalence of the concept of analytic functions as functions which are differentiable at every point of the region of their definition as well as of the functions the integral of which does not depend on the path (cf. Theorem 1 in 12. and Theorem 3 in 17.). This yields a new concept for the construction of a theory of analytic functions. Applications of the concept of integral and theorems, based on it, will be studied in the following chapters.
11. Integral of a function of a complex variable Let there be given some oriented curve C and on it the function f(z) of a complex variable. By definition, the integral of f(z) along C is
where z0 = a, z1, ··· , zn+1 = b are consecutive points which subdivide C into n segments, a and b denoting the ends of C, zk an arbitrary point on the segment |zk, zk+1| of C and the limit is taken under the assumption that max |zk+1- zk| ® 0.
If C is a sectionally smooth curve and f(z) is a sectionally continuous and bounded function, then the integral (1) exists. The proof reduces to the theorem, known from analysis, regarding the existence of the curvi-linear integral of a function of a real variable *.
* Cf. Fikhtengoltz, Vol. III, p. 27 or Smirnov, Vol. II, p. 206 et seq.
In fact, setting
The sums on the right hand side of (3) are integral sums for the corresponding curvi-linear integrals. Under our conditions, these integrals exist and, consequently, there exists
With the aid of (4), the calculation of the integral of a function of a complex variable is reduced to the calculation of real integrals.
Applying the introduced definitions, it is readily seen that the derivative and integral of a complex function of a real variable w(t) = j (t)+ iy(t) is represented by the linear combinations
Let z = z(t) = x(t) + iy(t) be a parametric representation of the curve C, where z(a) = a, z(b) = b; then, using (4), we reduce the calculation of the integral of f(z) along C to the calculation of the integral of the complex function of a real variable
Formula (4) likewise yields that the usual properties of curvi-linear integrals transfer to integrals of a function of a complex variable:
(a and b being complex constants, C1 + C2 denoting the curve comprising C1 and C2, C- the curve coincident with C, but running in the opposite direction).
We will yet prove one property of the integral:
Let M = max |f(z)| on the curve C and l be the length of C, then
The proof follows directly from the defined integral. In fact, we have
where is the length of the segmented line z0z1 ··· zn, inscribed in the curve C and in the limit, as max |Dzk|®0, we find (11).
12. Cauchy's theorem In the general case, depends on the integrand f(z) as well as on the curve C. However, if f(z) is analytic in some simply connected region, contained inside the curve C, then the integral is completely determined by the positions of the ends of C and does not depend on the shape of this line. In other words, we have
Theorem 1 (A.Cauchy, 1825) If the function f(z) is analytic in the simply connected region D, then for all curves C, lying in this region and having common ends, the integral has one and the same value.
We will prove this theorem under the additional assumption * concerning the continuity of the derivative f '(z) (in the definition of analyticity in 5. was only required the existence of this derivative).
* For a complete proof, cf. Markushevich, , pp. 137 - 143, 154 - 262.
As always, let f(z) = u(x, y) + iv(x, y); by the relation
(cf. (4) in 11.), the question of the independence of the integral on the path reduces to that regarding the independence on the path of the curvilinear integrals
However, as we know from analysis*, it is a necessary and sufficient condition for the independence in a simply connected region on the path of the curvi-linear integral where P and Q are functions with continuous partial derivatives, that the integrand of this integral is a total differential, i.e., that at every point of the region D applies the relation
* Cf. Fikhtengoltz, Vol. III, p. 68 or Smirnov, Vol. II, p. 216.
For the integrals (2), this relation becomes
however, the continuity of the partial derivatives follows from the assumption of the continuity of f '(z). Equation (3) coincides with the Laplace-Euler conditions and satisfies them, since f(z) is an analytic function, whence the theorem has been proved.
By this theorem for functions, analytic in simply connected domains, we can replace
where z0 and z denote the ends of the curve C.
Based on Theorem 1, there may be proved several propositions, analogous to the usual ones of the integral calculus. First of all, there applies
Theorem 2 If the function f(z) is analytic in the simply connected region D, then the integral
considered to depend on its upper limit, likewise is a function analytic in D, where
In fact, by the definition of the derivative and the property of the integrals (9) and (10) of 11., we have
By the strength of the continuity*, f(z) may be written at the point z
where h(z) ® 0 as z ® z; substituting this into (6), we find
* The continuity of f(z) is a consequence of its analyticity.
Since f(z) is constant for the integration with respect to z, we have
because it follows continuously from the definition that Moreover, Inequality (11) of 11. yields
(by Theorem 1, the path of integration from z to z + h may be assumed to be straight, whence its length is |h|). Thus, the first limit in (7) is equal to f(z) and the second one equal to 0, i.e., F'(z) = f(z), as was to be proved.
The function, the derivative of which is equal to a given function f(z), is said to be its primitive. The theorem proved asserts that the integral of f(z), considered as a function of its upper limit, is one of the primitives of the function f(z).
Theorem 3. Any two primitives of one and the same function differ from each other only by a constant term.
Let F1(z) and F2(z) be these two versions and
It is sufficient for the proof of the theorem to show that F(z) is constant.
By the formula for the derivative (cf. (15), 5.), we have
because by our condition
so that u(x, y) and v(x, y) are constants and the theorem has been proved..
The following theorem allows to calculate integrals with the aid of primitives.
Theorem 4. If F(z) is an arbitrary primitive of the analytic function f(z), then
In fact, by Theorem 2, the function
is one of the primitives of f(z), by assumption, so is F(z), whence by Theorem 3
where C is some constant. Setting here z = z0, we find F(z0) + C = 0, whence C = - F(z0), i.e., (8).
Note still that Cauchy's theorem, proved at the start of this section, may be given the form:
Theorem If the function f(z) is analytic in the simply connected region D, then its integral along any closed curve C, lying in D, vanishes:
The proof is based on the fact that the closed contour C may be divided into two contours C1 and C2,with common starting and end points (Fig. 18), By the properties of integrals
whence the vanishing of the integral along C is equivalent to the equality between the integrals along C1 and C2.
Finally, we will prove one generalization of Cauchy's theorem which will be useful in the future. In fact, we spoke in this theorem in its last formulation about the integral along a contour, which lies entirely inside the region of analyticity of the function, while one has sometimes to consider integrals along curves on which the function, while remaining continuous, ceases to be analytic. It turns out that Cauchy's theorem remains valid also for this case:
Theorem 5 If the function f(z) is analytic in a simply connected domain D and continuous in the closed domain , then the integral of f(z), taken along the boundary C of this domain, vanishes:
We assume at first that C is a "starry" contour, i.e., there exists a point z0 such that any ray with vertex at this point intersects C in one and only one point. Without restricting the generality, it may be assumed that z0 = 0 (i.e., this means a displacement of the z-plane), then the curve C may be given the the equation z = r(j)eij, where r is a single-valued function. We denote by Cl the contour, determined by the equation z = lz = lr(j)eij, 0 < l < 1 (Fig. 19 below). Since Cl lies inside D, by Cauchy's theorem,
However, as the point z describes Cl, the point z = z /l describes C, whence (10) may be rewritten
Since the function f(z) is uniformly continuous in (cf. 5.), one may find for any e > 0 a d > 0, satisfying the inequality |z-z|<d for any pair of points z, z , whence
Let l be the length of the contour C and R = max r(j); with l > 1 - d/R; we will have for any pair of points z and z
whence (12) will be fulfilled and we obtain by (11)
Since here e is as small as we please and the integral does not depend on e, this integral is zero and the theorem has been proved for "starry" contours.
Now, let C be an arbitrary sectionally smooth curve. If C has sharp points, we remove from the region D circles of small radius e with centres at these points, so that the boundary of the region De obtained has no longer such points (Fig. 20). Producing inside De a finite number of lines gk (k = 1, 2, ··· , m), this region may obviously be broken up into parts Dk, bounded by "starry" lines Ck (k = 1, 2, ··· , n)*. By what has been proved above, the integral along any line Ck vanishes:
* It is readily seen that a segment of the sectionally smooth curve in a sufficiently small neighbourhood of its points, not being a point of return, is itself a "starry" curve. However, in the neighbourhoods of the starry points, the curve may or may not be starry (for example, the curve consisting of sections of the parabolas y = x² and y = 2x², for which x ³ 0, near the point z = 0).
We will assume that the lines Ck continue in one, for example, positive direction and add all the equations (13). Since for us every line is travelled twice and in opposite directions, then all integrals along gk cancel each other (cf. (9) and (10) in 11.).and the remaining parts of the boundary Ck form the boundary Ce of the region De and, consequently, the integral along this boundary vanishes:
There remains to show that the integral along the boundary C of the region D vanishes; however, this follows directly from the fact that C and Ce differ only by a finite number of small arcs and since the function f(z) is bounded, its integrals along these arcs likewise are small. Thus, the integral along C differs as little as one pleases from the integral along Ce which vanishes, whence it is also equal to zero. This proves the theorem completely.
13 Extension to multiply connected domains Cauchy's theorem does not, generally speaking, apply to multi-connected regions. In fact, the function f(z) = 1/z is analytic everywhere in the ring 1/2 < |z| < 2, but the integral from -1 to 1 along the upper and lower halves of the circle |z| = 1 differ from each other. In fact, along the upper semi-circle C1, where z = eij, 0<j<p,
while along the lower semi-circle C2, where z = eij, -p<j<0,
Hence we will adopt sometimes for the notation of the integral from a to b along C in multi-connected regions the symbol
However, if in a multi-connected region curves C1 and C2 with common ends are such that they bound one simply connected domain, belonging to D, then, obviously, the integrals along such curves are equal, whence it follows that the value of an integral of an analytic function in a multi-connected domain D is not changed, if the contour of integration is deformed continuously so that its ends remain fixed and it remains all the time in D.
Let there be given in the multi-connected domain D
the points a and b, joined by the simple, i.e.,
without points of self-intersection, curve C0.
Let C be any other curve, joining these points
(Fig. 21a). According only to this statement, one may,
without changing the value of the integral, deform the curve C
into another curve , lying in D consisting of :
1) the curve which together with C0 bounds a simply connected region, belonging to D;
2) the union of the simple closed curves gk (k = 1, 2, ··· , m), each of which contains inside it one connected part of D (Fig. 21b). pass several times and in different directions (as in Fig. 21b, the curve g1 runs three times in a clockwise direction, while g2 once in an anti-clockwise direction).
For the sake of convenience, we agree to denote by gk (k = 1, 2, ··· , m) curves, running in the anti-clockwise direction and introduce yet curves gk (k = m + 1, ··· , n), surrounding the linked parts of D and not entering into the composition of (as g3 in Fig. 21b).
We introduce the notation
for continuous deformation gk , for which these curves remain inside D, integrals (2) do not change, whence the quantities Gk are only determined by the function f(z) and the domain D. Let Nk be the integer which states the number and the direction of the passage of gk while constituting ; these numbers may be positive, bounded or equal to zero (for example, in Fig. 21, N1 = -3, N2 = 1, N3 = 0.) We find from this and the properties of the integrals (9) and (10) in 11.
The quantities Gk are are called periods of the integral of the function f(z) in the multi-connected domain D or cyclic constants.
Example Let f(z) = 1/z and D be the "ring" 0 < |z| < R, where R is arbitrarily large. Any path C which joins the points 1 and z may be deformed as above into the path , consisting of several passages of the unit circle |z| = 1 and the simple line C0 joining the points 1 and z (Fig. 22).The integral along the circle, counter-clockwise, when z = eij and j grows from 0 to 2p, is
where k is an integer, showing how many times and in what direction the circle |z| = 1 has contributed to the composition of C (in Fig. 22, k = -2). By Theorem 4 of 12.,
where ln denotes the value of the logarithm, which is 0 at the point z = 1 and increases continuously along C0.
Assuming that C is an arbitrary path and denoting the value of the integral of 1/z along it by Ln z, Equation (5) yields
Thus, we have arrived again at the multi-valued function Ln z and explained its multi-valuedness from a new point of view.
Finally, we note that one may give the Cauchy theorem of 12. a somewhat new meaning so that it remains true also for multi-connected domains. Let the function f(z) be analytic in the multi-connected domain D, bounded by the curves C1, C2, ··· , Cn (Fig. 23) and continuous in . Introduce the cuts g1, ··· , gn , converting D into a simply connected region D*, and denote by C* the boundary of this region - the curve consisting of the partial curves Ck and the curves gk, where the last are travelled twice in the counter-clockwise directions (indicated in Fig. 23 by arrows).
The function f(z) is analytic in the simply connected domain D* and continuous in ,whence, by Theorem 5 of 12. and the properties of Integrals (9) and (10) of 11.
(the integrals along gk cancel each other and the remaining part along C* coincides with ). For this, we must assume that the curves C0 and C1, C2, ··· , Cn are covered so that the domain D remains all the time on one side (for example, on the left hand side as in Fig. 23). Thus, for regions of any connectivity, Cauchy's theorem holds in the form:
Theorem: If the function f(z) is analytic in D and continuous in , then its integral along the boundary of this region , proceeding such that the region D remains all the time on one side, is equal to zero.
14. Cauchy's formula and the theorem of the mean Let the function f(z) be analytic in the n-connected domain D and continuous in . We will show that there holds for any internal point of this region Cauchy's formula (1831)
where C - the boundary of D - is travelled in such a way that D remains all the time on the left hand side.
We note that there enters on the right hand side of Cauchy's formula only the value of f(z) on the boundary C of the region. Thus, under the given conditions, the values of the function inside the region are completely determined by its values on the boundary: Cauchy's formula allows to evaluate the function at any point of the region as soon as the boundary values of this function are known.
For the derivation of Cauchy's formula, we remove from the region D a small circle of radius r with centre at the point z and note that in the (n+1)-connected region D* the numerator and denominator of the integrand are analytic with respect to z, where the denominator does not vanish. Consequently, the integrand is analytic with respect to z in D*; since it is continuous in , one has by Cauchy's theorem (Formula (8) of 13.)
where the circle gr is travelled in the clockwise direction, whence
where gr is travelled in the counter-clockwise direction. We have on the circle gr that z - z = reij, whence, since the factor f(z) under the integral sign is constant with respect to z, we find
By (2) and (3), we have
we now estimate this difference. By (11) of 11.,
whence we see that our difference as r decreases may be as small as we please. On the other hand, as is seen from the left hand side of (4), this difference does not depend on r. Consequently, the difference under consideration vanishes and Cauchy's formula has been proved.
In particular, if the curve C is itself the circle |z - z| = R, then, setting z - z = Reij, we obtain Cauchy's formula
This formula expresses the so-called Theorem of the Mean for analytic functions:
Theorem: If the function f(z) is continuous in a closed circle and analytic inside this circle, then its value at the centre of the circle equals the mean of the arithmetic values on the circle.
15. Maximum Principle and Schwarz's Lemma We will prove first the simple
Lemma: If in some domain D: 1) the real part of an analytic function f(z) or 2) its modulus is constant, then this function itself is constant.
For 1), the assertion follows directly from from the d'Alembert - Euler condition: We have whence, by these equations, also so that also We conclude therefore that v, but that means that also f(z) is constant in D.
We will now give the proof of the Lemma under Condition 20. Let |f(z)| º M, where M is constant; for M = 0, the statement is obvious. However, if M ¹ 0, we consider the function f(z) = ln |f(z)| + i arg f(z), which in this case is analytic. Its real part is constant ( = ln M), whence, by what has already been proved, the function ln f(z) is constant, which means that f(z) is constant, and the Lemma has been proved.
We will now prove the Maximum modulus principle for analytic functions.
Theorem If the function f(z), not equal to a constant, is analytic in D and continuous in , then its modulus cannot attain a maximum value at an internal point of D.
By the properties of continuous functions (cf. 5.), |f(z)| attains its maximum M inside or on the boundary of D (Fig. 24). In contradiction, we assume that |f(z)| attains its maximum M inside D and denote by the set of all points of D for which |f(z)| = M, i.e., |f(z)| is constant. It then follows from the Lemma that also f(z) is constant in D, and this contradicts the condition of the Theorem.
If does not coincide with D, then there exists a boundary point* z0 of this set, which is an internal point of D. By the continuity of f(z), we have |f(z0)| = M, because in any neighbourhood of z0 is the point . We construct the circle C: |z-z0|=r, belonging to D, so that on it there would be also a point z1, not belonging to (this may always be done, because z0 is a boundary point of ). Then |f(z1)| < M and for any sufficiently small e > 0, by the continuity of f(z), we may always say this of the point z1as part of C1of the circle C on which
* Boundary and internal points of a set are defined in the same way as for regions (cf. 3.)
We denote by C2 the remaining part of the circle; obviously, on it,
By the mean value theorem
where ds = rdj is the length element of the circle.
Going in (3) to absolute values and taking into consideration Inequalities (1) and (2), we find
where l1 and l2 are the leng())ths of C1 and C2 ( l1 + l2 = 2pr). However, the last inequality if impossible, whence the principle has been proved..
NOTE If the function f(z) is not constant, analytic in D and continuous in and, besides, does not vanish, then also the minimum of |f(z)| cannot be attained inside D.
It is sufficient for the proof to apply the maximum principle to teh function g(z) = 1/f(z).
The maximum principle of the modulus follows the useful for further applications
Lemma (G.Schwarz *). If the function f(z) is analytic in the circle |z| < 1 and continuous in the closed circle, where f(0) = 0 and if everywhere in the circle |f(z)| £ 1, then in the same circle
For this, if there should be at one internal point of the circle |f(z)| = |z|, then the last equality holds in the entire circle and
where a is a real constant.
* Hermann Schwarz 1843 - 1921.
We consider for the proof the function
By the conditions of the Lemma, j(z) is analytic in the ring 0 < z < 1 and continuous in the closed circle |z| £ 1 (the continuity of the point z = 0 follows from the fact that
We will prove in
14.), we have, g 0 < |z| < 1 and continuous in the closed circle |z| £ 1 (continuity at z = 0 follows from the fact that
We will prove in 22. that from this follows the analyticity of j(z) at the point z = 0; thus, we apply to j(z) the modulus maximum modulus principle. Since on the circle |z| = 1, we have |j(z)| = |f(z)/z| £ 1; then also, by this principle, everywhere in the circle |j(z)| = £ 1, i.e., |f(z)| £ |z|, and the first part of the Lemma is proved.
If now at every internal point |f(z0)| = |z0|, then at each point |j(z0)|=1; however, then, by the maximum principle, |j(z)|º1 at all points of the circle and, by the Lemma above, j(z) is constant. Since |j(z)| º 1, one may enter this constant in the form eia ,where a is a real number; consequently, f(z)=eiaz, and Schwarz's Lemma is proved.
Geometrically, Schwarz' Lemma states that, for any mapping of the unit circle onto a domain D, lying inside the unit circle, with the aid of the analytic function w = f(z), f(0) = 0, the image of an arbitrary point z lies closer to the origin than any point z (Fig. 25); however, if the image of anyone point z were to lie at the same distance as the very point, then D coincides with the unit circle and the mapping is reduced to a rotation.
16. Uniform convergence The following work has an auxiliary character. We will consider the questions, important for the sequel, of the uniform convergence of sequences and series of analytic functions.
The sequence of functions f1(z), f2(z), ··· is said to be uniformly convergent to the function f(z) in the domain D ( or on the curve C) if for any e > 0 there exists a number n0, which depends only on e, such that there holds for n ³ n0, for all z of D (or on C) the inequality
We will prove two theorems, analogous to the corresponding theorems of analysis.
Theorem 1 The limit f(z) of the sequence of continuous functions f1(z), f2(z), ··· fn(z) ··· converges uniformly in some domain D (or on a curve C) and is also a continuous function.
We give a number e > 0 and denote by z0 an arbitrary point of the region D (or the curve C). Due to the uniform convergence, one can find a number n such that for all z of D (on C)
By the continuity of fn(z) at the point z0, one can find a number d > 0 such that for all z of D (on C)
For such z and the n chosen above, we have by the inequalities (2) and (3):
and this means the continuity of (z).
Theorem 2 If the sequence of continuous functions f1(z), f2(z), ··· fn(z) ··· converges uniformly on the curve C to f(z), then there holds the limit relation
We give a number e > 0. By strength of the uniform convergence, we can find n0 such that for all n ³ n0 and all z on C
where l is the length of C. For such n,
which verifies the truth of (4).
The proved theorem allows to go to the limit under the integral sign in the case of uniform convergence of sequences of functions.
The concept of uniformly convergent sequences is closely linked to that of uniformly convergent series. Functional series are said to be uniformly convergent in the region D (or on the curve C) if the sequence of its partial sums
converges uniformly in this domain (on this curve).
As in analysis, it proves to be convenient for applications for sufficient indications of uniform convergence of functional series.
Theorem 3 If the functional series is exceeded by some uniformly convergent numerical series i.e., if for any point z of D
then the given functional series converges uniformly in D.
In fact, by a known comparison theorem, the given series converges at any point z of D. Denote this sum by s(z). In fact, any n remainder rn(z) = s(z) - sn(z) of this series satisfies, by (5), the inequality
We have here on the right hand side the remainder rn of the converging numerical series, which tends to zero as n ® ¥, whence one may find for any e > 0 a number n0, depending only on e, beginning with which there will be rn < e, and then, by (6), for any z in D and n ³ n0, one has the inequality
This as well is referred to as uniform convergence of the given series.
By Theorems 1 and 2, the sum of a uniformly convergent series consisting of continuous functions is continuous and such a series may be integrated term by term, i.e.,
The question of the possibility of term by term differentiation of a functional series will be studied in 19. (Weierstrass' theorem).
Consider now the family of functions f(z, a), depending on a (real or complex) parameter a. One says that f(z, a) tends for a®a0 to the function f(z) with respect to z in the domain D (or on the curve C) if for any e > 0 can be found d = d(e) such that for |a - a0| > d for all z of D (or on C) one has the inequality
In the same way, one can prove for sequences that the limit of a uniformly convergent family of of continuous functions is a continuous function and that for such a family there holds the limit relation
In the sequel, we will have to deal with integrals along unbounded curves - improper integrals. For this, we will always consider only such curves C, segments of which belonging to an arbitrary circle are sectionally smooth. The function f(z), given on C, will be assumed to be sectionally continuous and bounded.
We will now determine the integral of f(z) along an unbounded curve C. To start with, let C be unbounded only on one side and a be its end. We will then denote by C the part of C with end a and of length l and set by definition
where, if this limit exists, we will say that the (improper) integral converges. If C is unbounded in both directions, we define the integral as the sum of integrals of two parts, on which C is subdivided by an arbitrary point a.
Let f(x, z) be defined for all z in D and for all z on the line C. We will say that the integral
converges uniformly in D, if there exists for any e > 0 a number l0 such that for all z of D for any l > l0
(we assume that C is unbounded on one side; the extension to the general case is done as above).
Theorem 4 If the function f(z, z) is analytic with respect to z and sectionally continuous with respect to z for all z in the simply connected region D and for all z on the line C and the integral
converges uniformly in D, then it is analytic in this region (cf. Theorem 1 in 19.)
In order to prove this theorem, we employ the theorem, inverse to Cauchy's theorem, according to which the function F(z) is analytic in the simply connected domain D, if it is continuous in this domain and its integral along any closed curve, belonging to the domain, vanishes (for the proof of this theorem refer to 16.).
Under the conditions of the proved theorem, the continuity of the function F(z) is established as usual (as Theorem 2 of Relation (9)). There remains to prove that the integral of F(z) along the arbitrary closed contour G , belonging to D , vanishes. We have
By the uniform convergence of (12), according to a theorem, known from analysis*, one may interchange the order of integration and obtain
since the inner integral vanishes, by Cauchy's theorem. This proves Theorem 4.
** Cf. Fichtengoltz, Vol. II, p. 733: what is stated there relates to real definite integrals, but after introduction of a parameter and separation of real and imginary parts Integral (13) is reduced to such integrals.
Note that in the case of an unbounded curve C, it is unnecessary to make for the analyticity of the function F(z) any additional assumptions regarding the convergence of the integral (12); this follows from the possibility of interchanging without further assumptions the order of integration in (13).
17. Higher order derivatives By definition, an analytic function - a function of a complex variable - has a derivative at every point of a certain domain D (cf. 5.) We will show that there follows automatically from the analyticity of a function the existence and analyticity of all its consecutive derivatives.
Theorem 1 (A. Cauchy 1842) If the function f(z) is analytic in D and continuous in , then it has at every point of D derivatives of all orders, where the n-th derivative is given by
where C is the boundary of D.
Let z be an arbitrary internal point of D. By the definition of the derivative and Cauchy's formula in 14., which we apply for the points z and z + h, we have
However, obviously, as h ® 0, the function 1/(z - z - h) tends for all z on C to 1/(z - z) and, consequently, by Theorem 2 of 16. (for the case of a set of functions, depending on the parameter h), the limit exists, where
The theorem has been proved for n = 1. Assuming that it holds for any n - 1, one can prove in exactly the same way its truth for n and thereby prove the theorem completely.
Note 1. As is seen from the proof, the theorem may yet be formulated as follows: If the function f(z) is continuous on the boundary C of the domain D, then the function
representing Cauchy's formula, is analytic in this region.
Note 2. Equation (1) for the derivatives is obtained formally by differentiation of Cauchy's formula with respect to z; the theorem proved confirms the correctness of this differentiation.
Equation (1) yields the important Cauchy inequality. Denote by M the maximum modulus of f(z) in the domain D, by R the distance of the point z from the boundary D and by l the length of this boundary. Equation (1) yields
In particular, if f(z) is analytic in the circle |z - z0| < R, then, using this circle in the capacity of D, we obtain
which is Cauchy's inequality.
We will employ these results for the proof of two important theorems of the theory of analytic functions.
Theorem 2 (A.Cauchy, J. Liouville)* If the function f(z) is analytic throughout the plane and bounded, then it is constant.
* Th theorem was first proved by Cauhy in 1844, but used in an essential way in the work of J. Liouville (1809 - 1882).
Let everywhere |f(z)| £ M. For arbitrary points z of the plane and for any r, Inequality (5) yields for n = 1
Since here the left hand side does not depend on r and the right hand side may for increasing R be made as small as desired, one has |f '(z)| = 0. Thus, f '(z) º 0, whence, by Theorem 4 of 12.,
i.e., the function f(z) is constant and the theorem is proved.
Note Theorem 2 admits the generalization:
If the function f(z) is analytic throughout the plane and its modulus does not increase faster than M|zn|, where n is constant, then this function is a polynomial of degree not higher than n**.
** In particular, we obtain for n = 0 Theorem 2.
The proof is analogous to the preceding one: Let z0 be an arbitrary point of the plane; we have by Inequality (5)
and, noting that here |z| £ |z0| + R, we find after going to the limit R ® ¥ that f (n + 1)(z0) º 0. Since z0 is an arbitrary point of the plane, then f (n + 1)(z) º 0, whence, by the same method as above, one can arrive at the result required.
The following theorem is the inverse of Cauchy's theorem in 12.
Theorem 3 (G. Morera (1856 - 1909) 1886) If the function f(z) is continuous in the simply connected domain D and the integral along any closed contour in D vanishes, then f(z) is analytic in this domain.
The theorem states that in D the integral does not depend on the path of integration, i.e., it determines for fixed z0 a certain function of z:
Reverting literally to the proof of Theorem 2 in 12., we see that this function has the derivative F '(z) = f(z), i.e., it is analytic (in this theorem, we have used only the continuity of f(z) and the independence of the integral on the path). However, by Theorem 1 above, f(z) as the derivative of an analytic function is on its own merit an analytic function. Thus, Morera's Theorem has been proved.
1.5. Representation of analytic functions by series We will consider the problem of the representation of analytic functions by power series and their generalization - series of positive and negative powers of (z - a). The expansion of functions in series is not only of theoretical, but also of practical interest. For example, we say that with the aid of series we may compute approximate values of a function; in many problems of an applied character (solution of differential equations, etc.), solutions are obtained directly in the form of series.
We will restrict consideration here to the basic theoretical aspects, linked to expansion of functions in series; the majority of them will have a very important role in the following exposition of the theory of functions of a complex variable and its applications (for example, especially in Chapter V et seq.). In particular, there will be established the equivalence of the concepts of analytic functions (in the sense of 2.) as well as of functions , differentiable at every point of the region of definition and also of functions, differentiable at every point of their region of definition and also of functions, representable in the neighbourhood of every point as sums of power series (cf. Taylor's Theorem in 18. and Theorem 3 in 19.); this presents yet another concept of the theory of analytic functions. In 25, we generalize the concept of analyticity by extending it to multi-valued functions.
18. Taylor Series We begin with a generalization of formulae, known from analysis as Taylor's formulae, and prove on its basis that every function, analytic at a point, is represented in the neighbourhood of this point by the sum of a power series.
We will employ the formula for the sum of the terms of a geometric progression
by rewriting it in the form
(this formula is also true for complex q). Fixing some point a in the domain D of analyticity of f(z) and using (1), we write
Multiply now both sides of the equation by (1/2pi)f(z) and integrate it with respect toz along some closed contour C, lying in F and containing z and a.Using Cauchy's formula in 14. and the formulae for their derivatives in 17., we obtain Taylor's classical formula*
where the remainder term has the form
There arises the question under what conditions R ® 0 as n ® ¥ or, what is the same thing, under what conditions the function f (z) is represented by its Taylor expansion with centre at the point a, i.e.,
The answer to this question is given by the
Theorem (A. Cauchy, 1831) The function f(z) is represented by its Taylor series (4) in any open circle with centre at the point a, in which it is analytic. In every closed region, belonging to this circle, Taylor's expansion converges uniformly.
* An expansion of sucha form (for real z) is first encountered in a paper in 1715 by Brook Taylor (1685 - 1731), but its systematic application began only in 1742 in the work of Colim Marlaurin (1698 - 1746).
Let R denote the radius of the circle of analyticity of the function f(z) (with centre at the point a) and consider an arbitrary number R', 0 < R' < R, and the circle |z - a| £ kR', where k < 1 is an arbitrary positive number. Let z be any point of this circle and C the circle |z - a| £ R'. We have |z - a| £ kR', |z - a| £ R', whence
and (3) yields
where M is the maximum modulus of f(z) in the circle |z - a| £ R' (the function f(z) is analytic in this circle, whence it is bounded). Since k < 1, this shows that Rn ® 0 as n ® ¥, where the estimate of Rn does not depend on z; thus, in any circle |z - a| < kR', where 0 < k < 1, the Taylor series converges uniformly.
We may embed an arbitrary closed region, lying in the circle of analyticity of the function f(z) in some circle |z - a| < kR', where 0 < k < 1, whence also in such a region the series converges uniformly, and the theorem has been proved.
Thus, every function, analytic in a circle, may be expanded in a power series. There arises the question whether conversely an analytic function is the sum of an arbitrarily converging power series? In order to answer this question, we must consider certain properties of power series. This will be done in 19. We now present the Taylor expansions of certain elementary functions:
they have been written down for those single-valued branches which are equal to 0 and 1, respectively, for z = 0. The methods for obtaining these expansions are the same as in ordinary analysis.
* In this formula, a is an arbitrary complex number; in the particular case of natural a = n, the series ends at the nth term and, consequently, converges in the entire plane.
19. Power series
We start with two general theorems relating to uniformly converging series, comprising analytic functions ; these theorems were first proved by K. Weierstrass in 1859.
The first of them shows that uniform passage to the limit preserves the property of analyticity.
Theorem 1 If the series
consisting of functions which are analytic in a simply connected domain D, is uniformly convergent in this domain, then its sum likewise is a function which is analytic in D.
In fact, according to 16. , the sum s(z) of Series (1) is continuous in D. Let C be an arbitrary closed contour, lying in D; by the uniform convergence of Series (1), this may be integrated term by term along C to yield
because, by Cauchy's Theorem in 12., the integral of analytic functions fn(z) along a closed contour in a simply connected domain vanishes. Now, by Morera's Theorem of 17., we may assert that the function s(z) is analytic in the region D, whence the theorem has been proved.
The second theorem shows that for analytic functions the question of the possibility of term by term differentiation of the series is resolved more simply than in ordinary analysis.
Theorem 2 The arbitrary series (1), consisting of functions analytic in the domain D and continuous in , uniformly converging in , may be obtained by differentiation any number of times in D.
Let z be an arbitrary point on the boundary C of the domain D, z an arbitrary internal point of D. Since the difference z - z is bounded for fixed z below by the modulus of a positive number, the series where k is an arbitrary natural number, converges uniformly with respect to z on C. Consequently, it may be integrated term by term along C, whence converges the series
(for every term of the series, we may use Cauchy's formula for the derivatives in 17. . It remains to prove that the sum of Series (2) is the k-th derivative of the sum s(z) of Series (1). However, by the uniform convergence of the left hand side, one may write (2) in the form
(we employ again the same Cauchy formula), as was required.
Note 1. In order to assert the uniform convergence of a series of analytic functions in a closed domain , it is sufficient to require its uniform convergence on the boundary of this region. This follows directly from the maximum principle of 15., according to which
Note 2. A simple example shows that one may assert in Theorem 2 the convergence of the series from the derivatives only in the domain D, and not in . In fact, obviously, the series converges uniformly in the closed circle |z| £ 1, because it becomes stronger than the convergent numerical series However, the derivative of the series (convergent by Theorem 2 for |z| < 1) diverges at the point z = 1 of the boundary of the circle.
In the sequel, the dominant role belongs to power series. The character of their convergence is explained by
Theorem 3 (N. Abel 1826 (1802 - 1829)) If the power series converges at the point z0, then it converges also at any point z, located closer to the centre a than z0 , where in any circle |z - a| £ k|z0 - a|, where 0 < k < 1 the series converges uniformly.
We assume that z is an arbitrary point of the last circle and represent the n-th term of the series in the form
By the convergence of the series at the point z0, its general term tends to zero and, consequently, is rounded at this point, i.e., |cn(z0 - a)n| £ M for all n. Besides, by assumption, |(z - a)/(z0 - a)| £ k, whence for all n
This yields the uniform convergence of the series in the circle |z - a| £ k |z0 - a| and finally the proof of Abel's theorem.
By Abel's Theorem, the region of convergence of the power series is the open circle with centre at the point a (which might likewise degenerate into a point or occupy the entire plane) and yet, may be, be several points on the boundary of the circle. The radius of this circle is referred to as the radius of convergence of the power series.
The formula for the determination of the radius of convergence is
where denotes the upper limit (Fichtengoltz, Vol. I, p. 107). The formula was obtained by A. Cauchy in 1821 and found essential use by J. Hadamard (in the 20-th Century). It is referred to as the formula of Cauchy-Hadamard.
For its derivation one must prove that for any z for which |z - a| £ kR, the power series converges and for any z, for which |z-a| > R, this series diverges. By definition of the upper limit, for any e > 0, one finds n0 beginning with which
We choose e such that
when for n ³ n0 and |z - a| £ kR
Since 2k/(k + 1) < 1, by a known theorem of comparison of series, formed by the terms on the left hand side, it converges.
Moreover, the determination of the upper limit yields that for any e > 0 an infinite sequence n = nk for which
However, for |z - a| > R, one may always choose e so that (1/R - e)|z - a| > 1, when for our sequence n = nk, corresponding to this e , the term will increase unboundedly and, consequently, the power series will diverge (its general term not tend to zero).
The theorems of Weierstrass and Abel yield an affirmative answer to the question in 18.
Theorem 4 The sum of any power series in its circle of convergence is an analytic function.
In fact, let |z - a| < R be the circle of convergence of our power series. In any circle |z - a| £ kR, where 0 < k < 1, the convergence is uniform by Abel's Theorem and, since the terms of the series cn(z - a)n are analytic functions, then, by Weierstrass' Theorem, this sum is analytic in this circle. However, since any internal point z of the circle of convergence may be enclosed in some circle |z - a| < kR, where 0 < k < 1, then has been proved thereby the analyticity of the sum of the series in the entire circle of its convergence.
Finally, we prove the truth of the
Theorem 5 Any power series is the Taylor series of its sum.
In fact, let in some circle
Setting here z = a, we find f(a) = c0. Differentiating (5) term by term and then setting z = a, we find f '(a) = c1. By consecutive differentiations and then setting z = a we find
and Series (5) actually is the Taylor series of f(z).
Theorem 5, referred to as the Theorem of the uniqueness of the Taylor series expansion, because it follows from it that having found by any method an expansion of an analytic function f(z) in a power series, it is the Taylor expansion of this function.
Moreover, one may conclude from this theorem and the theorems of 18. that the radius of convergence of the power series (5) coincides with the distance from the centre a to the closest points at which the analyticity of the sum f(z) of this series is violated. For example, the radius of convergence of Series (6) in 18. equals 1. because for z = -1 their sums lose their analyticity (naturally, we consider the second series when a is not an integer).
20. Uniqueness Theorem We have seen in 14. that an analytic function is determined completely by its values on the boundary of its domain of analyicity. We will show here, in agreement with this, that an analytic function is determined completely by its values by its values on an arbitrary sequence of points, converging to some internal point of the region of analyticity.
We will start with a theorem relating to the zeroes of an analytic function. Any point z = a at which a function f(z) vanishes, i.e., at which f(a) = 0, is called a zero of f(z). If an analytic function is not identically 0 in the neighbourhood of its zero a, then in its Taylor series with centre at a not all coefficients may be zero (otherwise, the sum of the series would be identically equal to zero). The number of the least different from zero coefficient of this expansion is called the order of the zero of a. Thus, in the neighbourhood of a zero of order n, the function's Taylor expansion has the form
where cn ¹ 0 and n ³ 1.
Obviously, one may determine the order of a zero a from the order of the least non-zero derivative f (n)(a).
Likewise it is obvious that in the neighbourhood of a zero of order n an analytic function may be written in the form
where the function
is likewise analytic in the neighbourhood of the point a (because it is represented by a convergent power series).
By the continuity of j(z), this function differs from 0 also everywhere in some neighbourhood of the point a, whence follows
Theorem 1 Let the function f(z) be analytic in the neighbourhood of its zero a and neither equal identically zero nor sero anywhere in its neighbourhood. Then, there exists a neighbourhood of the point a in which f(z) does not have zeroes other than a.
From the proved theorem follows the uniqueness theorem of the theory of analytic functions which we will discuss in 21.
Theorem 2 If the functions f1(z) and f2(z) are analytic in the domain D and their values coincide on some sequence of points an, converging to an internal point a of D, then everywhere in D
For the proof, we consider the function
It is analytic in D, has its zero points an, and by its continuity also the point a, because
whence f(z) is identically zero in some neighbourhood of a, because otherwise it would violate only the proved Theorem 1. Thus, the set of all zeroes of f(z) would be a sole internal point.
Denote by the set of all internal points of the set of zeroes of f(z). If coincides with D, our theorem has been proved. However, if comprises only a part of the domain D, then there exists a boundary point b of the set , which is an internal point of D. There exists a sequence of points bn of the set which converges to b; by the continuity of f(z), the point b is a zero of f(z). On the other hand, f(z) is not identically equal to zero in some neighbourhood of the point b, because then b would be an internal and not a boundary point of . Hence, it follows by Theorem 1 from this that in some neighbourhood of the point b is not a zero of f(z), which contradicts the fact that b is a boundary point of . This contradiction also proves the uniqueness theorem.
It follows from the uniqueness theorem that a function f(z), which is analytic in some domain and does not vanish identically, cannot vanish in some sub-region of D nor on some arc, lying in D, not even on a sequence of points of D, converging to its internal point.
However, it is easy to produce an example when an infinite sequence of zeroes of a function converges to a boundary point of its domain of analyticity: The function f(z) = sin 1/z becomes zero on the sequence of points z = (1/np) (n = ±1, ±2, ···), which converges to the point z = 0.