Page images
PDF

n = co. The convergence is said to be “uniform” in an interval if, after specification of e, the same number n suffices at all points of the interval to make |f(x)-f.(x)|< e for all values of m which exceed n. The numbers n corresponding to any e, however small, are all finite, but, when e is less than some fixed finite number, they may have an infinite superior limit (§ 7); when this is the case there must be at least one point, a, of the interval which has the property that, whatever number N we take, e can be taken so small that, at some point in the neighbourhood of a, n must be taken > N to make|f(x)-f.(x)|<e when m > n; then the series does not converge uniformly in the neighbourhood of a. The distinction may be otherwise expressed thus: Choose a first and e afterwards, then the number n is finite; choose e first and allow a to vary, then the number n becomes a function of a, which may tend to become infinite, or may remain below a fixed number; if such a fixed number exists, however small e may be, the convergence is uniform. For example, the series sin x-? sin 2x+3 sin 3x-...is conver£ for all real values of x, and, when * > x > -r its sum is 3.x: ut, when x is but a little less than x, the number of terms which must be taken in order to bring the surn at all near to the value of *x is very large, and this number tends to increase indefinitely as x approaches r. This series does not converge uniformly in the neighbourhood of x = r. Another example is afforded by the series .#1-###". of which the remainder after n terms

is nx/(n'x'+1). If we put x =1/n, for any value of n, however great, the remainder is 3; and the number of terms required to be taken to make the remainder tend to zero depends upon the value of z when x is near to zero-it must, in fact, £ large compared with 1/x. The series does not converge uniformly in the neighbourhood of x=o. As regards series whose terms represent continuous functions we have the following theorems: (1) If the series converges uniformly in an interval it represents a function which is continuous throughout the interval. (2) If the series represents a function which is discontinuous in an interval it cannot converge uniformly in the interval. (3) A series which does not converge uniformly in an interval may nevertheless represent a function which is continuous throughout the interval. (4) A power series converges uniformly in any interval contained within its domain of convergence, the end-points being excluded.

[merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small]
[merged small][ocr errors][merged small][ocr errors][ocr errors]

Fourier's theorem is that, if the periodic interval can be divided into a finite number of partial intervals within each of which the function is ordinary (§ 14), the series represents the function within each of those partial intervals. In Fourier's time a function of this character was regarded as completely arbitrary. By a discussion of the integral (ii.) based on the Second Theorem of the Mean ($15) it can be shown that, if f(x) has restricted oscilla. tion in the interval (§ 11), the sum of the series is equal to #f(x+o)+ f(x-o)}, at any point x within the interval, and that it is equal to } {f(+o)+f(1-0)} at each end of the interval. (See the article FouriER's SERIEs.) It therefore represents the function at any point of the periodic interval at which the function is continuous (except p' ly the end-points), and has a definite value at each point of discontinuity. The condition of restricted oscillation includes all the functions contemplated in the statement of the theorem and some others. Further, it can be shown that, in any partial interval throughout which f(x) is continuous, the series converges uniformly, and that no series of the form (i), with coefficients other than those determined by Fourier's rule, can represent the function at all points, except points of discontinuity, in the same periodic interval. The result can be extended to a function f(x) which tends to become infinite at a finite number of points a of the interval, provided (1) f(x) tends to become determinately infinite at each of the points a, (2) the # definite integral of f(x) through the interval is convergent, (3)f(x) has not an infinite number of discontinuities or of maxima or minima in the interval. 24. Representation of Continuous Functions by Series.—If the series for f(x) formed by Fourier's rule converges at the point a of the periodic interval, and if f(x) is continuous at a, the sum of the series is f(a); but it has been proved by P. du Bois Reymond that the function may be continuous at a, and yet the series formed by Fourier's rule may be divergent at a. Thus some continuous functions do not admit of representation by Fourier's series. All continuous functions, however, admit of being represented with arbitrarily close approximation in either of two forms, which may be described as “terminated Fourier's series” and “terminated power series,” according to the two following theorems: (1) If f(x) is continuous throughout the intervai between o and 2r, and if any positive number e however small is specified, it is possible to find an integer n, so that the difference between the value of f(x) and the sum of the first n terms of the series for f(x), formed by Fourier's rule with periodic interval from o to 2r, shall be less than eat all points of the interval. This result can be extended to a function which is continuous in any given interval. (2) If f(x) is continuous throughout an interval, and any positive number e however small is specified, it is possible to find an integer n and a polynomial in x of the nth degree, so that the difference between the value of f(x) and the value of the polynomial shall be less than eat all points of the interval. Again it can be proved that, if f(x) is continuous throughout a given interval, polynomials in x of finite degrees can be found, so as to form an infinite series of polynomials whose sum is equal to f(x) at all points of the interval. Methods of representation of continuous functions by infinite series of rational fractional functions have also been devised. Particular interest attaches to continuous functions which are not differentiable. Weierstrass gave as an example the function represented by the series2a"cos (b"xx), where a is positive and less o

and this is

* * than unity, and b is an odd integer exceeding (1+1+)/a. It can be shown that this series is uniformly convergent in every interval, and that the continuous function £ represented by it has the property that there is, in the neighbourhood of any point x, an infinite aggregate of points x', having x, as a limiting point, for which {f(x')-f(x))/(x'-x) tends to become infinite with one sign when x'-3', approaches zero through positive values, and infinite with the opposite sign when x'-xt approaches zero through negative values. rdingly the function is not differentiable at any point. The definite integral of such a function f(x) through the interval between a £ and a variable point x, is a continuous differentiable function F(x), for which F(x)=f(x); and, if f(x) is one-signed throughout any interval F(x) is monotonous throughout that interval, but yet F(x) cannot be represented by a curve. In any interval, however small, the tangent would have to take the same direction for infinitely many points, and yet there is no interval in which the tangent has everywhere the same direction. Further, it can be shown that all functions which are everywhere continuous and nowhere differentiable are capable of representation by series of the form Xant.(x), where Xa, is an absolutely convergent series of numbers, and din(x) is an analytic function whose absolute value never exceeds unity.

25. Calculations with Divergent Series.—When the series described in (1) and (2) of § 24 diverge, they may, nevertheless, be used for the approximate numerical calculation of the values of the function, provided the calculation is not carried beyond a certain number of terms. Expansions in series which have the property of representing a function approximately when the expansion is not carried too far are called “asymptotic expansions.” Sometimes they are called “semi-convergent series”; but this term is avoided in the best modern usage, because it is often used to describe series whose convergence depends upon the order of the terms, such as the series 1-3+}-. . .

In 1..neral, let fo(x)+f (x)+... be a series of functions which does not converge in a certain domain. ... It may happen that, if any number e, however small, is first specified, a number n can afterwards be found so that, at a point a of the domain, the value f(a) of a certain function f(x) is connected with the sum of the first n + 1

[ocr errors][merged small][ocr errors][ocr errors][ocr errors][merged small][ocr errors]

When the series is integrated term by term, the right-hand member of the equation for c(x) takes the form #3–?:#####–. 1.2 x 3.4 x* '5.6 x* * * *

This series is divergent; but, if it is stopped at any term, the difference between the sum of the series so terminated and the value of c(x) is less than the last of the retained terms. Stirling's formula is obtained by retaining the first termonly. Other well-known examples of asym totic expansions are afforded by the descending series for Bessel's functions. Methods of obtaining such expansions for the solutions of linear differential equations of the second order were investigated by G. G. Stokes (Math, and Phys. Papers, vol. ii. p. 329), and a general £ of asymptotic expansions has been developed by H. Poincaré. A still more general theory of divergent series, and of the conditions in which they can be used, as above, for the purposes of approximate calculation has been worked out by E. Borel. The great merit of asymptotic expansions is that they admit of addition, subtraction, multiplication and division, term by term, in the same way, as absolutely convergent, series, and they admit also of integration term: by term; that is to say, the results of such operations are

asymptotic expansions for the sum, difference, product, quotient, or integral, as the case may be. 26. Interchange of the Order of Limiting Operations.—When we require to perform any limiting operation upon a function which is itself represented by the result of a limiting process, the question of the possibility of interchanging the order of the two processes always arises. In the more elementary problems of analysis it generally happens that such an interchange is possible; but in general it is not possible. In other words, the performance of the two processes in different orders may lead to two different results; or the performance of them in one of the two orders may lead to no result. The fact that the interchange is possible under suitable restrictions for a particular class of operations is a theorem to be proved. Among examples of such interchanges we have the differentiation and integration of an infinite series term by term (§ 22), and the differentiation and integration of a definite integral with respect to a parameter by £ the like processes upon the subject of integration ($.19). As a last example.we may take the limit of the sum of an infinite series of functions at a point in the domain of

convergence. Suppose that the series $f;(x) represents a function

r (fr) in an interval containing a point a, and that each of the functions f(x) has a limit at a... If we first put x = a, and then sum the series, we have the value f(a); if we first sum the series for any x, and afterwards take the limit of the sum at x = a, we have the limit of f(x) at a ; if we first replace each function f.(x) by its limit at a, and then sum the series, we may arrive at a value different from either of the foregoing. If the function f(x) is continuous at a, the first and second results are equal; if the functions f.(x) are all continuous at a, the first and third results are equal; if the series is uniformly convergent, the second and third results are equal. This last case is an example of the interchange of the order of two limiting operations, and a sufficient, though not always a necessary, condition, for the validity of such an interchange will usually be found in some suitable extension of the notion of uniform convergence. AUThoRiTIES.-Among the more important treatises and memoirs connected with the subject are: R. Baire, Fonctions discontinues (Paris, 1905); O. Biermann, Analytische Functionen (Leipzig, 1887); E. Borcl, Théorie des fonctions { ris, 1898) (containing an introductory account of the Theory of Aggregates), and Séries divergentes (Paris, 1901), also Fonctions de variables réelles (Paris, 190 }: T. J. I’A. Bromwich, Introduction to the Theory of Infinite Series # ondon, 1908); H. S. Carslaw, Introduction to the Theory of Fourier's Series and Integrals (London, 1906); U. Dini, Functionen e. reellen Grosse

(Lei # 1892), and Serie di Fourier (Pisa, 1880); A. Genocchi u. #. eano, Diff- u. Int::Rechnung (Leipzig, 1899); J. Harkness and F. Morley, Introduction to the Theory of Analytic Functions

[ocr errors]
[ocr errors]
[ocr errors]
[graphic]

II.-FUNCtions of CoMPLEx VARIABLES

In the preceding section the doctrine of functionality is discussed with respect to real quantities; in this section the theory when complex or imaginary quantities are involved receives treatment. The following abstract explains the arrangement of the subject matter: (§ 1), Complex numbers, states what a complex number is; (§ 2), Plotting of simple expressions involving complex numbers, illustrates the meaning in some simple cases, introducing the notion of conformal representation and proving that an algebraic equation has complex, if not real, roots; ($ 3), Limiting operations, defines certain simple functions of a complex variable which are obtained by passing to a limit, in particular the exponential function, and the generalized logarithm, here denoted by A(2); (§ 4), Functions of a complex variable in general, after explaining briefly what is to be understood by a region of the complex plane and by a path, and expounding a logical principle of some importance, gives the accepted definition of a function of a complex variable, establishes the existence of a complex integral, and proves Cauchy's theorem relating thereto, (§ 5), Applications, considers the differentiation and integration of series of functions of a complex variable, proves Laurent's theorem, and establishes the expansion of a function of a complex variable as a power series, leading, in (§6), Singular points, to a definition of the region of existence and singular points of a function of a complex variable, and thence, in (§ 7), Monogenic Functions, to what the writer believes to be the simplest definition of a function of a complex variable, that of Weierstrass; ($ 8), Some elementary properties of single valued functions, first discusses the meaning of a pole, proves that a single valued function with only poles is rational, gives Mittag-Leffler's theorem, and Wcierstrass's theorem for the primary factors of an integral function. stating generalized forms for these, leading to the theorem of ($ 9), The construction of a monogenic function with a given region of existence, with which is connected (§ 10), Expression of a monogenic function by rational functions in a given region, of which the method is applied in (§ 11), Expression of (1-z)" by polynomials, to a definite example, used here to obtain (§ 12), An expansion of an arbitrary function by means of a series of polynomials, over a star region, also obtained in the original manner of MittagLeffler; (§ 13), Application of Cauchy's theorem to the delermination of definite integrals, gives two examples of this method; (§ 14), Doubly Periodic Functions, is introduced at this stage as furnishing an excellent example of the preceding principles. The reader who wishes to approach the matter from the point of view of Integral Calculus should first consult the section ($20) below, dealing with Elliptic Integrals; (§ 15), Potential Functions, Conformal representation in general, gives a sketch of the connexion of the theory of potential functions with the theory of conformal representation, enunciating the Schwarz-Christoffel theorem for the representation of a polygon, with the application to the case of an equilateral triangle; (§ 16), Multiple-valued Functions, Algebraic Functions, deals for the most part with algebraic functions, proving the residue theorem, and establishing that an algebraic function has a definite Order; (§ 17), Integrals of Algebraic Fivictions, enunciating Abel's theorem; (§ 18), Indeterminateness of Algebraic Integrals, deals with the periods associated with an algebraic integral, establishing that for an elliptic integral the number of these is two; (§ 19), Reversion of an algebraic integral, mentions a problem considered below in detail for an elliptic integral; ($20), Elliptic Integrals, considers the algebraic reduction of any elliptic integral to one of three standard forms, and proves that thé function obtained by reversion is single-valued; ($ 21), Modular Functions, gives a statement of some of the more/elementary properties of some functions of great importance, with a definition of Automorphic Functions, and a hint of the connexion with the theory of linear differential equations; (§ 22), A property of integral functions, deduced from the theory of modalar functions, proves that there cannot be more than one value not assumed by an integral function, and gives the basis of the well-known expression of the modulus of the elliptic functions in terms of the ratio of the

periods; (§ 23), Geometrical applications of Elliptic Functions, shows that any plane curve of deficiency unity can be expressed by elliptic functions, and gives a geometrical proof of the addition theorem for the function {}(u); (§ 24), Integrals of Algebraic Functions in connexion with the theory of plane curves, discusses the generalization to curves of any deficiency, ($25), Monogenic Functions of several independent variables, describes briefly the beginnings of this theory, with a mention of some fundamental theorems: (§ 26), Multiply-Periodic Functions and the Theory of Surfaces, attempts to show the nature of some problems now being actively pursued.

Beside the brevity necessarily attaching to the account here given of advanced parts of the subject, some of the more elementary results are stated only, without proof, as, for instance: the menogeneity of an algebraic function, no reference being made, moreover, to the cases of differential equations whose integrals are monogenic, that a function possessing an algebraic addition theorem is necessarily an elliptic function (or a particular case of such); that any area can be conformally represented on a half plane, a theorem requiring further much more detailed consideration of the meaning of arca than we have given; while the character and properties, including the connectivity, of a Riemann surface have not been referred to. The theta functions are referred to only once, and the principles of the theory of Abelian Functions have been illustrated only by the developments given for elliptic functions.

§ 1. Complex Numbers.—Complex numbers are numbers of the form x+ iy, where x, y are ordinary real numbers, and i is a symbol imagined capable of combination with itself and the ordinary real numbers, by way of addition, subtraction, multiplication and division, according to the ordinary commutative, associative and distributive laws; the symbol i is further such that i =-1.

Taking in a £ two rectangular axes Ox, # we assume that every point of the plane is definitely associated with two real numbers x, y (its co-ordinates) and conversely; thus any point of the plane is associated with a single complex number; in particular, for every point of the axis Ox, for which y =O, the associated number is an ordinary real number; the complex numbers thus include the real numbers. e axis Ox is often called the real axis, and the axis Oy the imaginary axis... If P be the point associated with the complex variable z = x + iy, the distance OP be called, r, and the positive angle less than 2x between Ox and OP be called 6, we may write 2 = r(cos. 6+ i sin 6); then f is called the modulus or absolute value of z and often denoted by |z|and 6 is called the phase or amplitude of z, and often denoted by ph (2); strictly the phase is ambiguous by additive multiples of 2r. =x'+iy’ be represented by P", the complex argument £4-2 is represented by a point P obtained by drawing from P' a line equal to and parallel to QP; the geometrical representation involves for its validity certain properties of the plane; as, for instance, the equation z'+z=z-i-z' involves the possibility of constructing a parallelogram (with OP'asdiagonal). It is important constantly to bear in mind; what is capable of easy algebraic proof (and geometrically is Euclid's proposition lii. 7), that the modulus of a sum or difference of two complex numbers is generally less than (and is never greater than) the sum of their moduli, and is greater than (or equal to) the difference of their moduli; the former statement thus holds for the sum of any number of complex numbers... We shall write E(0) for cos 0+i sin 6; it is at once verified that E(ia). E(i.6) = E[i(a +3)], so that the phase of a product of complex quantities is obtained by addition of their respective phases.

[ocr errors][merged small]
[ocr errors][ocr errors][merged small][merged small][merged small]
[ocr errors]

number. This proposition alone suffices to suggest the importance
of complex numbers.
§ 3. Limiting Operations.-In order that a complex number
# = #--in may have a limit it is necessary and sufficient that each
of $ and m has a limit. Thus an infinite series wo-Hwi-Hw:-H ...,
whose terms are complex numbers, is convergent if the real
series formed by taking the real parts of its terms and that
formed by the imaginary terms are both convergent. The
series is also convergent if the real series formed by the moduli
of its terms is convergent; in that case the series is said to be
absolutely convergent, and it can be shown that its sum is
unaltered by taking the terms in any other order. Generally
the necessary and sufficient condition of convergence is that,
for a given real positive e, a number m exists such that for every.
n > m, and every positive p, the batch of terms w.--w,41+
... +w, t, is less than € in absolute value. If the terms depend
upon a complex variable 2, the convergence is called uniform
for a range of values of z, when the inequality holds, for the
same e and m, for all the points z of this range.
The infinite series of most importance are those of which the
general term is a.s", wherein as is a constant, and s is regarded as
variable, n =o, 1, 2, 3,.... Such a series is called a power series.
If a real and positive number M exists such that for a = 2, and every
n, lanzo" | < M, a condition which is satisfied, for instance, if the
series converges for z = zo, then it is at once proved that the series
converges absolutely for every z for which |z| < # and con-
verges uniformly over every range | 3 | <r' for which r"< | 2 |.
To every power series there £ then a circle of convergence
within which it converges absolutely and uniformly; the function
of z represented by it is thus continuous within the circle (this being
the result of a general £ of uniformly convergent series of
continuous functions); the sum for an interior point z is, however,
continuous with the sum for a point zo on the circumference, as z
approaches to z, provided the series converges for z=z, as can be
shown without much difficulty. Within a common circle of con-
vergence two power series Zanz”, Xbus" can be multiplied together
according to the ordinary rule, this being a consequence of a theorem
for absolutely convergent series. . If r be less than the radius of
convergence of a series Sa..." and for |z|=ri, the sum of the series

[ocr errors][ocr errors][ocr errors][ocr errors]
[ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors]

The function exp (z) is used also to define a generalized form of the cosine and sine functions when z is complex; we write, namely, gos 2 - Mexp (i:) + exp (-i:)) and sin z = -41 exp (iv)-exp (-12)]. It will be found that these obey the ordinary relations holding when z is real, except that their moduli are not inferior to unity. For example, cos i = 1 + 1/21+1/4!-H...is obviously greater than unity. $4. Of Functions of a Complex Variable in General.—We have in what precedes shown how to generalize the ordinary rational, algebraic and logarithmic functions, and considered more general cases, of functions expressible by power series in z. With the suggestions furnished by these cases we can frame a general definition. So far our use of the plane upon which z is represented has been only illustrative, the results being capable of analytical statement. In what follows this representation is vital to the mode of expression we adopt; as then the properties of numbers-cannot be ultimately based upon spatial intuitions, it is necessary to indicate what are the geometrical ideas requiring elucidation. Consider a square of side a, to whose perimeter is attached a definite direction of description, which we take to be counterclockwise; another square, also of side a, may be added to this, so that there is a side common; this common side being erased we have a composite region with a definite direction of perimeter; to this a third square of the same size may be attached, so that there is a side common to it and one of the former squares, and this common side may be erased. If this £ be continued any number of times we obtain a region of the plane bounded by one or more polygonal closed lines, no two of which intersect; and at each portion of the perimeter there is a definite direction of description, which is such that the region is on the left of the describing point. Similarly we may construct a region by piecing together triangles, so that every consecutive two have a side in common, it being understood that there is assigned an upper limit for the reatest side of a triangle, and a lower limit for the smallest angle. # the former method, each square may be divided into four others by lines through its centre parallel to its sides; in the latter method each triangle may be divided into four others by lines joining the middle points of its sides;, this halves the sides and preserves the angles. When we speak of a region of the plane in general, unless the contrary is stated, we shall £ it capable of being gencrated in this latter way by means of a finite number of triangles, there being an upper limit to the length of a side of the triangle and a lower limit to the size of an angle of the triangle. We shall also require to speak of a path in the plane; this is to be understood as capable of arising as a limit of a polygonal path of finite length, there being a definite direction or sense of description at every point of the path, which therefore never meets itself. From this the meaning of a closed path is clear. The boundary points of a region form one or more closed paths, but, in general, it is only in a limiting sense that the interior points of a closed path are a region, There is a logical principle also which must be referred to. We frequently have cases where, about every, interior or boundary, point 20 of a certain # a circle can be put, sav of radius ro, such that for all points z of the region which are interior to this circle, for which, that is, Iz-zol<ro, a certain property holds, Assuming that to ro is given the value which is the upper limit for zo, of the ssible values, we may call the points 12-zol-ro, the neighbourood belonging to or proper to zo, and may speak of the property as the property (z,zo). he value of ro, will in general vary with zo; what is in most cases of importance is the question whether the lower limit of ro for all positions is zero or greater than zero. (A) This lower limit is certainly greater than zero provided the property (z,zo) is of a kind which we may call extensive; such, namely, that if it holds, for some position of zo, and all positions of z, within a certain region, then the property (2,2i) holds within a circle of radius R about any interior point 21 of this region for all points a for which the circle ## is within the region. Also in this case ro varies continuously with zo. (B) Whether the property is of this extensive character or not we can prove that the region can be divided into a finite number of sub-regions such that, for every one of these, the property holds, (1) for some point z within or upon the boundary of the sub-region, (2) for every point z within or upon the boundary of the sub-region. We prove these statements (A), (B) in reverse order. To prove (B) let a region for which the property (z,z) holds for all points z and sonne point zoof the region, be called suitable; if each of the triangles of which the region is built up be suitable, what is desired is proved; if not let an unsuitable triangle be subdivided into four, as before explained; if one of these subdivisions is unsuitable let it be again subdivided; and so on. Either the process terminates and then what is required is proved; or else we obtain an indefinitely continued sequence of unsuitable trianglos, each contained in the preceding, which converge to a point, say t, after a certain stage all these will be interior to the proper region of f; this, howcvcr, is contrary to the supposition that they are all unsuitable. We now make some applications of this result (B). Suppose a

[ocr errors][ocr errors][ocr errors][ocr errors][merged small][merged small][ocr errors][merged small]
« ՆախորդըՇարունակել »