Page images
PDF
EPUB
[ocr errors]
[ocr errors]
[ocr errors]

nr

n=00. The convergence is said to be “uniform” in an interval | the interval between-c and c, was given by Fourier, viz. we if, after specification of e, the same number n suffices at all have points of the interval to make 1/(x)-fm(x)\< e for all values of in which exceed n. The numbers n corresponding to any e,

---cos sin however small, are all finite, but, when e is less than some fixed The interval between - c and c may be called the “periodic finite number, they may have an infinite superior limit ($ 7); interval," and we may replace it by any other interval, e.g. that when this is the case there must be at least one point, a, of the between o and 1, without any restriction of generality. When interval which has the property that, whatever number -N we this is done the sum of the series takes the form take, e can be taken so small that, at some point in the neigh

LI bourhood of a, n must be taken > N to make \/(x) – Sm(x)]<

E f(?) cos(211 (8-x)}de, when m>n; then the series does not converge uniformly in the and this is neighbourhood of a. The distinction may be otherwise expressed

LI

sin {(2n+1)(2-x)+] dz. thus : Choose a first and e afterwards, then the number n is

sin (3-X)}

(ii.) finite; choose e first and allow a to vary, then the number n Fourier's theorem is that, if the periodic interval can be divided becomes a function of a, which may tend to become infinite, or into a finite number of partial intervals within each of which the may remain below a fixed number; if such a fixed number function is ordinary (8 14), the series represents the function exists, however small e may be, the convergence is uniform. within each of those partial intervals. In Fourier's time a

For example, the series sin x-sin 2x+} sin 3x = ...is conver- function of this character was regarded as completely arbitrary, gent for all 'real values of X, and, when 7 >x>- its sum'is .x; but, when x is but a little less than 7, the number of terms which

By a discussion of the integral (ii.) based on the Second Theorem must be taken in order to bring the sum at all near to the value of

of the Mean ($ 15) it can be shown that, if f(x) has restricted oscilla. fx is very large, and this number tends to increase indefinitely as f(x-0)} at any point x within the interval, and that it is equal to

tion in the interval (811), the sum of the series is equal to f(x+o) + * approaches T. This series does not converge uniformly in the ito) +5(1-0)| at each end of the interval. (See the article neighbourhood of *=*. Another example is afforded by the series Fourier's SERIES.) It therefore represents the function at any 3

(n+1)x conti (n+1Jeriti, of which the remainder after n terms point of the periodic interval at which the function is continuous is nx/(mox? +1). If we put x=11n, for any value of n, however point of discontinuity. The condition of restricted oscillation great, the remainder is ; and the number of terms required to be includes all the functions contemplated in the statement of the taken to make the remainder tend to zero depends upon the value of theorem and some others. Further, it can be shown that, in any x when x is near to zero-it must, in fact, be large compared with partial interval throughout which f(x) is continuous, the series 1/*. The series does not converge uniformly in the neighbourhood converges uniformly, and that no series of the form (i), with coof x=0.

efficients other than those determined by Fourier's rule, can represent

the function at all points, except points of discontinuity, in the same As regards series whose terms represent continuous functions periodic interval. The result can be extended to a function f(x) we have the following theorems:

which tends to become infinite at a finite number of points a of the (1) If the series converges uniformly in an interval it represents interval, provided (1) f(x) tends to become determinately infinite a function which is continuous throughout the interval.

at each of the points a, (2) the improper definite integral of f(x)

through the interval is convergent, (3) f(x) has not an infinite number (2) If the series represents a function which is discontinuous of discontinuities or of maxima or minima in the interval. in an interval it cannot converge uniformly in the interval.

24. Representation of Continuous Functions by Series.-If the (3) A series which does not converge uniformly in an interval series for f(x) formed by Fourier's rule converges at the point may nevertheless represent a function which is continuous

a of the periodic interval, and if f(x) is continuous at a, the throughout the interval.

sum of the series is (a); but it has been proved by P. du Bois (4) A power series converges uniformly in any interval con- Reymond that the function may be continuous at a, and yet the tained within its domain of convergence, the end-points being series formed by Fourier's rule may be divergent at a. Thus excluded.

some continuous functions do not admit of representation by (5) If fr(x) =f(x) converges uniformly in the interval Fourier's series. All continuous functions, however, admit of between a and b

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: or a series which converges unformly may be integrated term by (1) If [(x) is continuous throughout the intervai between o and term.

27, and if any positive number e however small is specified, (6) If $ (x) converges uniformly in an interval, then 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 fr(x) converges in the interval, and represents a continuous for f(x), formed by Fourier's rule with periodic interval from differentiable function, $(x); in fact we have

o to 2, shall be less than e at all points of the interval. This

result can be extended to a function which is continuous in any $'(x) =

given interval. or a series can be differentiated term by term if the series of

(2) If /(x) is continuous throughout an interval, and any derived functions converges uniformly.

positive number e however small is specified, it is possible to A series whose terms represent functions which are not con- find an integer n and a polynomial in x of the nth degree, so tinuous throughout an interval may converge uniformly in the that the difference between the value of f(x) and the value of the interval. If & fr(x),=f(x), is such a series, and if all the polynomial shall be less than e at all points of the interval.

Again it can be proved that, if f(x) is continuous throughout functions fr(x) have limits at a, then f(x) has a limit at a, which a given interval, polynomials in x of finite degrees can be found, is Ê LI [(x). A similar theorem holds for limits on the left 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 or on the right.

of continuous functions by infinite series of rational fractional 23. Fourier's Series.-An extensive class of functions admit functions have also been devised. of being represented by series of the form

Particular interest attaches to continuous functions which are

not differentiable. Weierstrass gave as an example the function cot .cosuri +ða sinoma). (i.)

represented by the series & a* cos (buxa), where a is positive and less and the rule for determining the coefficients an, bn of such a

than unity, and B is an odd integer exceeding (1 +17)la. It can be series, in order that it may represent a given function f(x) in shown that this series is uniformly convergent in every interval,

[ocr errors]

-0

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

and that the continuous function f(x) represented by it has the asymptotic expansions for the sum, difference, product, quotient, property that there is, in the neighbourhood of any point xs, an or integral, as the case may be. infinite aggregate of points x', having to as a limiting point, for 26. Interchange of the Order of Limiting Operations.-When which f(x')-f(x))/(x' - x) tends to become infinite with one sign when i'-* approaches zero through positive values, and

we require to perform any limiting operation upon a function infinite with the opposite sign when x-xc approaches zero through which is itself represented by the result of a limiting process, negative values. Accordingly the function is not differentiable at the question of the possibility of interchanging the order of the any point. The definite integral of such a function f(x) through the two processes always arises. In the more elementary problems interval between a fixed point and a variable point x, is a continuous differentiable function F(x), for which F'(x) = f(x); and, if f(x) is

of analysis it generally happens that such an interchange is one-signed throughout any interval F(x) is monotonous throughout possible; but in general it is not possible. In other words, the that interval, but yet F(t) cannot be represented by a curve. In performance of the two processes in different orders may lead any interval, however small, the tangent would have to take the to two different results; or the performance of them in one of the same direction for infinitely many points, and yet there is no interval

two orders may lead to no result. The fact that the interchange in which the tangent has everywhere the same direction. Further, it can be shown that all functions which are everywhere continuous is possible under suitable restrictions for a particular class of and nowhere differentiable are capable of representation by series of operations is a theorem to be proved. the form EO.O.(x), where Ean is an absolutely convergent series of Among examples of such interchanges we have the differentiation numbers, and on(s) is an analytic function whose absolute value and integration of an infinite series term by term ($ 22), and the never exceeds unity.

differentiation and integration of a definite integral with respect to 25. Calculations with Divergent Series.—When the series integration (8 19). As a last example we may take the limit of the

a parameter by performing the like, processes upon the subject of described in (1) and (2) of § 24 diverge, they may, nevertheless, sum of an infinite series of functions at a point in the domain of be used for the approximate numerical calculation of the values convergence. Suppose that the series 3 (x) represents a function of the function, provided the calculation is not carried beyond a certain number of terms. Expansions in series which have the (fr) in an interval containing a point a, and that each of the functions

If we first put r=a, and then sum the series, property of representing a function approximately when the (x) has a limit at a.

we have the value f(a); if we first sum the series for any x, and expansion is not carried 100 far are called “ asymptotic expan- afterwards take the limit of the sum at x=a, we have the limit of sions." Sometimes they are called “semi-convergent series "; f(x) at a; if we first replace each function fr(x) by its limit at a, and but this term is avoided in the best modern usage, because then sum the series, we may arrive at a value different from either it is often used to describe series whose convergence depends second results are equal; if the functions 1,(x) are all continuous at

of the foregoing. li the function f(x) is continuous at a, the first and upon the order of the terms, such as the series 1-1+1-... a, the first and third results are equal; if the series is uniformly

In [. neral, let f.(x)+fi(x) +... be a series of functions which convergent, the second and third results are equal. This last case does not converge in a certain domain. It may happen that, if any is an example of the interchange of the order of two limiting operanumber e, however small, is first specified, a number n can after tions, and a sufficient, though not always a necessary, condition, wards be found so that, at a point a of the domain, the value f(a) of for the validity of such an interchange will usually be found in some a certain function f(x) is connected with the sum of the first nti suitable extension of the notion of uniform convergence.

AUTHORITIES.-Among the more important treatises and memoirs terms of the series by the relation 1f(a) Efr(e)}<c. It must connected with the subject arc: R. Baire, Fonctions discontinues also happen that, if any number N, however great, is specified, a Ę. Borel, Théorie des fonctions (Paris, 1898) (containing

an introsumber :'(>n) can be found so that, for all values of m which exceed ductory account of the Theory of Aggregates), and Séries divergentes

(Paris, 1901), also Fonctions de variables réelles (Paris, 1905); T. J. m'. 12$.(a)|>N. The divergent series fo(x) +Sı(x) +... is then an I'A. Bromwich, Introduction to the Theory of Infinite Series (London,

1908); H. S. Carslaw, Introduction to the Theory of Fourier's Series asymptotic expansion for the function f(x) in the domain.

and Integrals (London, 1906); U. Dini, Functionen e. reellen Grosse The best known example of an asymptotic expansion is Stirling's (Leipzig. 1892), and Serie di Fourier (Pisa, 1880); A. Genocchi formula for a! when n is large, viz.

u. G. Peano, Diff- u. Int.-Rechnung (Leipzig, 1899); J. Harkness n! = V(27)in**le-ntonan,

and F. Morley, Introduction to the Theory of Analytic Functions where is some number lying between o and 1. This formula is (London, 1898); A. Harnack, Diff.and Int. Calculus (London, 1891); included in the asymptotic expansion for the Gamma function. Theory of Fourier's Series (Cambridge, 1907); C. Jordan, Cours

È. W. Hobson, The Theory of Functions of a real Variable and the We have in fact

d'analyse (Paris, 1893-1896); L. Kronecker, Theorie d. einfachen log ((x)= (x-4) log x-x+1 log 2a+(x),

u. vielfachen Integrale (Leipzig, 1894); H. Lebesgue, Leçons sur sbere o(s) is the function defined by the definite integral

l'intégration (Paris, 1904); M1. Pasch, Diff. u. Int.-Rechnung

(Leipzig, 1882); E. Picard, Traité d'analyse (Paris, 1891); 0. a(x) = $."141-647-61-f16ede.

Stolz, Allgemeine Arithmetik (Leipzig, 1885), and Diff. u. Ini.. The multiplier of ets under the sign of integration can be expanded (Paris, 1886); W. H. and G. C. Young, The Theory of Sets of Points

Rechnung (Leipzig, 1893-1899); J. Tannery, Théorie des fonctions in the power series

Cambridge, 1906); Brodén," Stetige Functionen e reellen Veränderlichen," Crelle, Bd. cxviii.; G. Cantor, A series of memoirs on the

* Theory of Aggregates" and on Trigonometric series " in Acta where B, Bz.... are “ Bernoulli's numbers " given by the formula Math. tt. ii., vii., and Math. Ann. Bde.iv.-xxiii.; Darboux, “Fonctions B. = 2.2m! (2x)-2- (nm).

discontinues," Ann. Sci. Ecole normale sup. (2), t. iv.; Dedekind, Was sind u. was sollen d. Zahlen? (Brunswick, 1887), and Steligkeit

u. irrationale Zahlen (Brunswick, 1872); Dirichlet, “ Convergence When the series is integrated term by term, the right-hand member des séries trigonométriques," Crelle. Bd. iv.; P. Du Bois Reymond, of the equation for a(x) takes the form

Allgemeine Functionentheorie (Tübingen, 1882), and many memoirs B. !_B2 1 B, 1

in Crelle and in Math. Ann.; Heine, Functionenlehre," Crelle, 1.2 X

Bd. Ixxiv.; J. Pierpont, The Theory of Functions of a real Variable

(Boston, 1905); F. Klein, “. Allgemeine Functionsbegriff,". Math. This series is divergent; but, if it is stopped at any term, the difference Ann. Bd. xxii.;' W. F. Osgood, “On Uniform Convergence," Amer. between the sum of the series so terminated and the value of a(x) is J. of Math. vol. xix.; Pincherle, “ Funzioni analitiche secondo less than the last of the retained terms. Stirling's formula is obtained Weierstrass," Giorn. di mat, t. xviii.; Pringsheim, “ Bedingungen by retaining the first term only. Other well-known examples of asymp. d. Taylorschen Lehrsatzes," Math. Ann.°Bd. xliv.; Riemann, totic expansions are afforded by the descending series for Bessel's Trigonometrische Reihe," Ges. Werke (Leipzig, 1876): Schoenflies, functions. Methods of obtaining such expansions for the solutions of "Entwickelung d. Lehre v. d. Punktmannigfaltigkeiten," Jahresber. linear differential equations of the second order were investigated by d. deutschen Math.- Vereinigung, Bd. viii.; Study, Memoir on G. G. Stokes (Moth. and Phys. Papers, vol. ii. P- 329), and a general Functions with Restricted Oscillation," Math. Ann. Bd. xlvii.; theory of asymptotic expansions has been developed by H. Poincaré. Wcierstrass, Memoir on " Continuous Functions that are not DifferA still more general theory of divergent series, and of the conditions entiable," Ges. math. Werke, Bd. ii. p. 71. (Berlin, 1895), and on the in which they can be used, as above, for the purposes of approximate Representation of Arbitrary Functions," ibid. Bd. iii. p. 1; W. H. akulation has been worked out by E. Borel. The great merit of Young, “ On Uniform and Non-uniform Convergence," Proc. London asymptotic expansions is that they admit of addition, subtraction, Math. Soc. (Ser. 2) t. 6. Further information and very full references en altiplication and division, term by term, in the same way as will be found in the articles by Pringsheim, Schoenflics and Voss in absolutely convergent series, and they admit also of integration the Encyclopädie der math. Wissenschaften, Bde. i., ji. (Leipzig, 1898, term by term; that is to say, the results of such operations are | 1899).

(A. E. H. L.)

it

3.4777

5.6

periods; (§ 23), Geometrical applications of Elliplic Functions, II.-FUNCTIONS OF COMPLEX VARIABLES

shows that any plane curve of deficiency unity can be expressed In the preceding section the doctrine of functionality is dis- by elliptic functions, and gives a geometrical proof of the addition cussed with respect to real quantities; in this section the theory theorem for the function P(x); (§ 24), Integrals of Algebraic when complex or imaginary quantities are involved receives Functions in connexion with the theory of plane curves, discusses treatment. The following abstract explains the arrangement the generalization to curves of any deficiency, ($25), Monogenic of the subject mattet: ( 1), Complex numbers, states what a functions of several independent variables, describes briefly the complex number is; (§ 2), Plotting of simple expressions involving beginnings of this theory, with a mention of some fundamental complex numbers, illustrates the meaning in some simple cases, theorems: ($ 26), Multiply-Periodic Functions and the Theory introducing the notion of conformal representation and proving of Surfaces, attempts to show the nature of some problems now that an algebraic equation has complex, if not real, roots; ($ 3), being actively pursued. Limiling operations, defines certain simple functions of a complex Beside the brevity necessarily attaching to the account here variable which are obtained by passing to a limit, in particular given of advanced parts of the subject, some of the more elethe exponential function, and the generalized logarithm, here mentary results are stated only, without proof, as, for instance: denoted by N(3); ($ 4), Functions of a complex variable in general, the monogeneity of an algebraic function, no reference being after explaining briefly what is to be understood by a region of made, moreover, to the cases of differential equations whose the complex plane and by a path, and expounding a logical integrals are monogenic, that a function possessing an algebraic principle of some importance, gives the accepted definition of a addition theorem is necessarily an elliptic function (or a particular function of a complex variable, establishes the existence of a case of such); that any area can be conformally represented on complex integral, and proves Cauchy's theorem relating thereto; a half plane, a theorem requiring further much more detailed (8 5), Applicalions, considers the differentiation and integration consideration of the meaning of arca than we have given; while of series of functions of a complex variable, proves Laurent's the character and properties, including the connectivity, of a thcorem, and establishes the expansion of a function of a complex Riemann surface have not been refcrred 10. The theta functions variable as a power series, leading, in (8 6), Singular points, to are referred to only once, and the principles of the theory of a definition of the region of existence and singular points of a Abelian Functions have been illustrated only by the developfunction of a complex variable, and thence, in ($ 7), Monogenic ments given for elliptic functions. Functions, to what the writer believes to be the simplest definition § 1. Complex Numbers.-Complex numbers are numbers of of a function of a complex variable, that of Weierstrass; (§ 8), the form x+iy, where x, y are ordinary real numbers, and i is a Some elementary properties of single valued functions, first discusses symbol imagined capable of combination with itself and the the meaning of a pole, proves that a single valued function with ordinary real numbers, by way of addition, subtraction, multionly poles is rational, gives Mittag-Leffler's thcorem, and Weier- plication and division, according to the ordinary commutative, strass's theorem for the primary factors of an integral function. associative and distributive laws; the symbol i is further such stating generalized forms for these, leading to the theorem of that i=-1. (9), The construction of a monogenic function with a given region of Taking in a plane two rectangular axes Ox, Oy, we assume that existence, with which is connected (§ 10), Expression of a monogenic 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 function by rational functions in a given region, of which the associated with a single complex number; in particular, for every method is applied in ($ 11), Expression of (1-2)" by polynomials, point of the axis Ox, for which y=0, the associated number is an to a definite example, used here to obtain (§ 12), An expansion ordinary real number; the complex numbers thus include the real of an arbitrary function by means of a series of polynomials, over

numbers. The axis Ox is often called the real axis, and the axis Oy

the imaginary axis. If P be the point associated with the complex a star region, also obtained in the original manner of Mittag- variable 2=x+iy, the distance OP be called ,, and the positive Leffler; (§ 13), Application of Cauchy's theorem to the delermination angle less than 2* between Ox and OP be called 0, we may write of definite integrals, gives iwo examples of this method; (8 14), = r(cos e +i sin o); then, is called the modulus or absolute value Doubly Periodic Functions, is introduced at this stage as furnish- of 2 and often denoted by Island o is called the phase or amplitude ing an excellent example of the preceding principles. The by additive multiples of 27 11 2 = x'+iy be represented by P. reader who wishes to approach the matter from the point of view the complex argument :'+e is represented by a point P' obtained of Integral Calculus should first consult the section (§ 20) below, by drawing from pa line equal to and parallel to OP; the geodealing with Elliptic Integrals; ($ 15), Potential Functions, 1 metrical representation involves for its validity certain properties Conformal representation in general, gives a sketch of the con

the possibility of constructing a parallelogram (with OP'asdiagonal). nexion of the theory of potential functions with the theory of It is important constantly to bear in mind, what is capable of easy conformal representation, enunciating the Schwarz-Christoffel algebraic proof (and geometrically is Euclid's proposition III.

7). theorem for the representation of a polygon, with the application that the modulus of a sum or difference of two complex numbers is to the case of an equilateral triangle; (8 16), Multiple-valued moduli, and is greater than (or equal to) the difference of their Functions, Algebraic Functions, deals for the most part with moduli: the former statement thus holds for the sum of any number algebraic functions, proving the residue theorem, and establishing of complex numbers. We shall write E(10) for cos 8 + i sin e; it is that an algebraic function has a definite Order; ($ 17), Integrals at once verified that Elia). E(IB) = Eli(a + b)], so that the phase of a of Algebraic Functions, enunciating Abel's theorem; (8 18), product of complex quantities is obtained by addition of their

respective phases. Indeterminateness of Algebraic Integrals, deals with the periods

§ 2. Plotting and Properties of Simple Expressions involving associated with an algebraic integral, establishing that for an a Complex Number.-If we put $ = (2-1)/(2+i), and, putting elliptic integral the number of these is two; (8 19), Reversion of $=$+in, take a new plane upon which $, are rectanguan algebraic integral, mentions a problem considered below in iar co-ordinates, the cquations $=(r*+y2-1)/(ro+(y+1)°), detail for an elliptic integral; ($ 20), Elliptic Integrals, considers n=-2xy/(r?+(y+1)?) will determine, corresponding to any the algebraic reduction of any elliptic integral to one of three point of the first plane, a point of the second plane. There is standard forms, and proves that the function obtained by ihe one exception of 2=-;, that is, x=0, y=-1, of which the reversion is single-valued; (§ 21), Modular Functions, gives a corresponding point is at infinity. It can now be easily proved statement of some of the more elementary properties of some that as a describes the real axis in its plane the point $ describes functions of great importance, with a definition of Automorphic once a circle of radius unity, with centre at $ =0, and that there Functions, and a hint of the connexion with the theory of linear is a definite correspondence of point to point between points differential equations; (§ 22), A property of integral functions, in the z-plane which are above the real axis and points of the deduced from the theory of modular functions, proves that there s-plane which are interior to this circle; in particular s=; cannot be more than one value not assumed by an integral corresponds to $ =0. function, and gives the basis of the well-known expression of Moreover, 5 being a rational function of 2, both & and ? are conthe modulus of the elliptic functions in terms of the ratio of the Itinuous differentiable functions of x and y, save when s is infinite:

[ocr errors]

writing} = (x, y) =f(:-iy, y), the fact that this is really independent, be in absolute value less than a real positive quantity M, it can be of y leads at once to ajlax tidslay=0, and hence to

shown that for lel=n every term is also less than M in absolute value, $ _მო . On 2229

namely, lan <Mr. ll in every arbitrarily small ncighbourhood of dx dy oy or oxi töyk=0;

2=0 there be a point for which two converging power series £9.2", so that is not any arbitrary function of x, y, and when & is known il 2anza vanish at 2=0 there is a circle of finite radius about :=o as

Ebuz agree in value, then the series are identical, or an = ba; thus also is determinate save for an additive constant. Also, in virtue of

centre within which no other points are found for which the sum of these equations, if $. Ś be the values of $ corresponding to two

the series is zero. Considering a power series |(z) = ana" of radius of near values of 3, say : and s', the ratio (5-5)/(:'-2) has a definite

convergence R, if 1201 <R and we put 2=2+1 with !!! <R-Izol. limit when :' =:, independent of the ultimate phase of z'-2, this

the resulting series Ear(20+1)* may be regarded as a double scries limit being therefore equal to aslax, that is, aflax+idn/ox. Geo

in eo and 1, which, since 1:01+i<R, is absolutely convergent; metrically this fact is interpreted by saying that if two curves in the

it may then be arranged according to powers of l.

Thus we may :-plane intersect at a point P, at which both the differential co

write f()= EA!"; hence An= (20), and we have (30+1)-|(20)]/l = efficients aglar, onlax are not zero, and P', P' be two points near

Ann-, wherein the continuous series on the right reduces to A. to Pon these curves respectively, and the corresponding points of the s-plane be Q, Q', Q', then (1) the ratios PP/PP'. 90/QQ' are for i=0; thus the ratio on the left has a definite limit when 1=0, ultimately equal, (2) the angle P'PP" is equal to l'og; (3) the equal namely to A, or Ena,20*-4. In other words, the original series retation from PP to PP' is in the same sense as from QO'to 2Q"; may legitimately be differentiated at any interior point za of its circle it being understood that the axes of t. , in the one plane are related of convergence. Repeating this process wc find szo+1) = x/"fin) (zo)/n!, as are the axes of x, y. Thus any diagram of the ?-plane becomes a where f{) (20) is the nth differential coefficient. Repeating for this diagram of the s-plane with the same angles; the magnification, power series, in l, the argument applied about 2=0 for Zanz", we however, which is equal to

varies from point to

inser that for the series /(:) every point which reduces it to zero is

an isolated point, and of such points only a finite number lie within point. Conversely, it appears subsequently that the expression a circle which is within the circle of convergence of f(z). of any copy of a diagram (say, a map) which preserves angles requires --397... of which the radius of convergence is infinite. By the intervention of the complex variable.

As another illustration consider the case when s is a polynomial multiplication we have exp ().exp (2') =exp (2+2'). In particular in ,

when x, y are real, and s=x+iy, exp (s) = exp (x) exp (iy). "Now the $ = Poza + poznat... +Pmi

functions H being an arbitrary real positive number, it can be shown that a

Uo=sin y, Vo = 1 -cos y, U, = y-sin y, radius Ř can be found such for every 1:1 > R we have | 51 > H; consider the lower limit of 13 | for 1:1'<r; as this a real vi = 4y?--?+cos y, U, = 1y8=y+sin y: Vo = vye - $y2+1-cos y.... coatinuous function of x, y for 121 <R, there is a point (x, y),

all vanish for y=0, and the differential coefficient of any one after say (te, yo), at which is l'is least, say equal top: and therefore increasing when its differential cocfficient is positive, wc inser, for

the first is the preceding one; as a function (of a real variable) is within a circle in the s-plane whose centre is the origin, of radius a there are no points 5 representing values corresponding to 121 <R.

y positive, that each of these functions is positive; proceeding to a

limit we hence infer that But if so be the value of 5 corresponding to (xo. yo), and the expression of s-so near so = xo +1 yo, in terms of 2-20. be A(:-20)" + cos y=1-3y2 + y -..., sin y=y-]y3+iloys -..., B(:-%)*+1..., where A is not zero, to two points near to (xo, yo), for positive, and hence, for all values of y. We thus have exp (iy) =

cos y+isin y, and exp (2)=cxp (x). (cos y+isin y). In other words, say (51, 9u) or s4 and =2+(21-wo) (cosm ti sinh). will corre- the modulus of exp (2) is exp (x) and the phase is y. Hence also spond two points near to so, say Si, and 280-51, situated so that so exp (3+271) = exp (x) [cos (y+27) +i sin (y+27)], is between them. One of these must be within the circle (p). We which we express by saying that exp. (z) has the period 2ri, infer then that p=0, and have proved that every, polynomial in and hence also the period akni, where k is an arbitrary integer. 1 vanishes for some value of 2, and can therefore be written as a From the fact that the constantly increasing function exp (x) can product of factors of the form 3-a, where a denotes a complex vanish only for x=0, we at once prove that exp (2) has no other number. This proposition alone suffices to suggest the importance periods. of complex numbers.

Taking in the plane of z an infinite strip lying between the lines * 3. Limiting Operations.-In order that a complex number; y=25 and plotting the function = cxp (3) upon a new plane, s={tin may have a limit it is necessary and sufficient that each of 5 arises when 2 takes in turn all positions in this strip, and that of & and n has a limit. Thus an infinite series uitwitwat. no value arises twice over. The equation $=exp (2) thus definess, whose terms are complex numbers, is convergent if the real regarded as depending upon f; with only an additive ambiguity series formed by taking the real parts of its terms and that 2kri, where k is an integer. We write s=1(5):

when $ is real this formed by the imaginary terms are both convergent. The

becomes the logarithm of $; in general 1/3) = log 151 +i ph (5)+

2kri, where k is an integer; and when $ describes a closed circuit series is also convergent if the real series formed by the moduli surrounding the origin the phase of $ increases by 27, or k increases of its terms is convergent; in that case the series is said to be by unity. Differentiating the series for $ we have ds/ds = 5. so absolutely convergent, and it can be shown that its sum is that 2, regarded as depending upon $, is also differentiable, with unaltered by taking the terms in any other order.

Generally 3 (5-1)-:
dz/ds=5! On the other hand, consider the series 5-1-}(5-1) +

.; it converges when $ = 2 and hence converges for the necessary and sufficient condition of convergence is that, is-115 1; its differential coefficient is, however, 1-(-1)+ for a given real positive e, a number m exists such that for every. (S-1)-..., that is, (1+5-1)-2 Wherefore if (s) denote this >m, and every positive p, the batch of terms wn+watif series, for 's-11 <1, the difference (s)-0(5), regarded as a +++, is less than e in absolute value. If the terms depend take the value of 4(5) which vanishes when $ = 1 we infer thence

function of and m. has vanishing differential coefficients; if we upon a complex variable 2, the convergence is called uniform for a range of values of 2, when the inequality holds, for the that for 15-11<1, 1(3)= =(-)=

-(3-1)*. It is to be remarked same e and m, for all the points of this range.

that it is impossible for $ while subject to 15-11.<i to make a The infinite series of most importance are those of which the circuit about the origin. For values of $ for which 15-11*1, we general term is 0.3", wherein an is a constant, and z is regarded as

can also calculate 4() with the help of infinite series, utilizing the variable, *= 0, 1, 2, 3,... Such a series is called a power series.

fact that 1(53') = (5)+1(5'). Il a real and positive number M exists such that for 2 = 2 and every

The function (5) is required to define so when $ and a are complex #161 <M, a condition which is satisfied, for instance, if the numbers; this is defined as exp (a1(5)]. that is as E 2"[^(3) } */!. weries converges for 2 = 2o, then it is at once proved that the series

When a is a real integer the ambiguity of 1(5) is immaterial here, Converges absolutely for every z for which 121<!201. and con. verges uniformly over every range Is <"' for which r'< 150 1. where o 'is a positive integer, there are g values possible for jole, of

since exp (a (5) +2 kani]=exp la1(s)when a is of the form 119. To every power series there belongs then a circle of convergence within which it converges absolutely and uniformly; the function

the form exp

k=0, , of : represented by it is ihus continuous within the circle (this being the result of a general property of uniformly convergent series of values of k leading to one of these; the qth power of any one of continuous functions); the sum for an interior point : is, however, these values is $; when a = pl9, where p q are integers without continuous with the sum for a point zo on the circumference, as z common factor, 9 being positive, we have spia= (file). .. The approaches to so provided the series converges for 2 =20, as can be definition of the symbol s is thus a generalization of ihe ordinary Shawn without much difficulty. Within a common circle of con- definition of a power, when the numbers are real. As an exainple, vergere two power series Lanz", £b?" can be multiplied together let it be required to find the meaning of i': the number i is of according to the ordinary rule, this being a consequence of a theorem modulus unity and phase f*; thus 1 (i) = (+2ka); thus for absolutely convergent series. Il ri be less than the radius of

= exp 1-fr-2k=) =exp (-4*) exp(-2kw), ccavergence of a series East and for 1:1 = r, the sum of the series is always real, but has an infinite number of valucs.

The function exp (2) is used also to define a generalized form of definite finite rcal value attached to every interior or boundary the cosine and sine functions when s is complex; we write, namely point of the region, say f(x,y). It may have a finite upper limit H COS : = }(exp (is) + exp(-iz)) and sin 2=-filexp (iz) - exp(-2)]. for the region, so that no point (x,y) exists for which /(x,y) > H, It will be found that these obey the ordinary relations holding when but points (x,y) exist for which f(x,y) > H-6, however small e may s is rcal, except that their moduli are not inferior to unity. For be: if not we say that its upper limit is infinite. There then at example, cos i =1+1/2!+1/4!+. ..is obviously greater than unity. Icast one point of the region such that, for points of the region within $4. Of Functions of a Complex Varieble in Gencral.-We have a circle about this point, the upper limit of f(x,y) is Ħ, however

small the radius of the circle be taken; for is not we can put about in what precedes shown how to generalize the ordinary rational, every point of the region a circle within which the upper limit of algebraic and logarithmic functions, and considered more f(x,y) is less than H; then by the result (B) above the region general cases, of functions expressible by power series in z. consists of a finite number of sub-regions within each of which the With the suggestions furnished by these cases we can frame a

upper limit is less than H; this is inconsistent with the hypothesis

that the upper limit for the whole region is H. A similar statement general definition. So far our use of the plane upon which : is holds for the lower limit. A case of such a function f(x,y) is the represented has been only illustrative, the results being capable radius ro of the neighbourhood proper to any point so, spoken of of analytical statement. In what follows this representation is above. We can hence prove the statement (A) above. vital to the mode of expression we adopt; as then the properties lower' limit of rois acro? Let thens be a point such that the lower of numbers cannot be ultimately based upon spatial intuitions, limit of ro is zero for points

Zo within a circle about s however small; it is necessary to indicate what are the geometrical ideas requiring let • be the radius of the neighbourhood proper to s: take so elucidation.

that. 120-$1<dr; the property (2,8), being extensive, holds Consider a square of side a, to whose perimeter is attached a

within a circle, centre 20, of radius 9-120-$1; which is greater definite direction of description, which we take to be counter

than 20-51, and increases to as 120-$i diminishes; this being clockwise; another square, also of side a, may be added to this, so

true for all points zo near $, the lower limit of ro is not zero for the that there is a side common; this common side being erased we neighbourhood of $. contrary to what was supposed. This proves have a composite region with a definite direction of perimeter;1(A). Also, as is here shown that 4051 - 20-51, may similarly be to this a third square of the same size may be attached, so

shown that r10-120-5). Thus to differs arbitrarily little from that there is a side common to it and one of the foriner squares,

" when ! 30-$1 is sufficiently small; that is, to varies continuand this common side may be crased. If this process be continued definite finite" value at every point of the region considered, to be

Next suppose the function (2,3), which has a 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

continuous but not necessarily real, so that about avery point ? tion, which is such that the region is on the left of the describing for all points of the region interior to this circle, we have each portion of the perimeter there is a definite direction of descrip- within or upon the boundary of the region, a being an arbitrary rcal triangles, so that every consecutive two have a side in common, ! f(x,y) = f(xo.yo!! <fn, and therefore (z'y") being any other point it being understood that there is assigned an upper limit for the

interior to this circle, 1/(x+%')-f(x,y)<n. We can then apply greatest side of a triangle, and a lower limit for the smallest angle the result (A) obtained above, taking for the neighbourhood proper in the former method, each square may be divided into four others to any point zo the circular area within which, for any two points by lines through its centre parallel to its sides; in the latter method (x.y). . (x,y), we have | S(x'.y') -f(x,y)|<n. This is clearly an each triangle may be divided into four others by lines joining the extensive property. Thus, a number is assignable, greater than middle points of its sides; this halves the sides and preserves the zero, such that, for any two points (x,y), (x,y')within a circle angles. When we speak of a region of the plane in general, unless 12-20] => about any point 20, we have f(x,y')-1(x,y) /<. the contrary is stated, we shall suppose it capable of being generated and, in particular, 1/(x,y) =f(x,y) 15 m, where in is an arbitrary in this latter way by means of a finite number of triangles, there

real positive quantity agreed upon beforehand. being an upper limit to the length of a side of the triangle and a

Take now any path in the region, whose extreme points are 1, 2, lower limit to the size of an angle of the triangle. We shall also

and let si, ... mi be intermediate points of the path, in order: require to speak of a path in the plane; this is to be understood as

denote the continuous function f(x,y) by f(x), and let f, denote any capable of arising as a limit of a polygonal path of finite length, quantity such that Ifo-(3)] = 15(4+1)-1(2) 1: consider the sum there being a definite direction or sense of description at every point

(31-2)fo+(2-2)fi+...+(3-3-1)f--. of the path, which therefore never meets itself. From this the By the definition of a path we can suppose, n being large enough, meaning of a closed path is clear. The boundary points of a region that the intermediate points 21....are so taken that if : form one or more closed paths, but, în general, it is only in a limiting its be any two points intermediate, in order, to % and Erzs, we have sense that the interior points of a closed path are a region. 120+1-31 1 <12+2-er li we can thus supposel - 1.12-21.... There is a logical principle also which must be referred to. We

2-%-all to converge constantly to zero. This being so, we can frequently have cases where, about every, interior or boundary, show that the sum above has a definite limit. For this it is sufficient, point so of a certain region a circle can be put, say of radius ro, such as in the case of an integral of a function of one real variable, to that for all points : of the region which are interior to this circle, prove this to be so when the convergence is obtained by taking new for which, that is, 12-01<7o, a certain property holds. Assuming points of division intermediate to the former ones. If, however, that to ro is given the value which is the upper limit for 2o. of the 21, 20,30 ... 2r,m- be intermediate in order to & and St. and possible values, we may call the points 13–201<to the neighbour. 1. -f(2.1) 1 <1 16.1+1)-1(2500) I, the difference between 2(+1-8}, hood belonging to or proper to zo, and may speak of the property and as the property (3,6).' The value of ro will in general vary with so; E|(20,1—-)$6,0+(2,2-2,1) fr.ct...+(2+1-2,-1) fr.m-1). 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)

which is equal to This lower limit is certainly greater than zero provided the property

(2,1+2,5)(1.6-fo), (2,2a) is of a kind which we may call extensive; such, namely, that if it holds, for some position of zo and all positions of 2, within a certain is, when 1-2 is small enough, to ensure 19(2+1) -S() [<7.

absolute value than region, then the property (2,21) holds within a circle of radius R about any interior point 2, of this region for all points for which

22n2120,1+1,il. the circle 12-5)=R is within the region. Also in this case to which, if S be the upper limit of the perimeter of the polygon from varies continuously with 2. (B) Whether the property is of this which the path is generated, is <2n5, and is therefore arbitrarily extensive character or not we can prove that the region can be divided small. into a finite number of sub-regions such that, for every oncof these, the property holds, (1) for some point 20 within or upon the boundary

The limit in question is called fas(-)ds

. In particular when of the sub-region, (2) for every point s within or upon the boundary (2) = 1, it is obvious from the definition that its value is :-*: of the sub-region. We prove these statements (A), (B) in reverse order. To prove value is 1(22-%); these results will be applied immediately.

when {(s) = 2, by taking fr = (+1-24), it is equally clear that its (B) let a region for which the property (32) holds for all points 2 and some point zoof the region, be called suitable: if cach of the triangles certain region there belong two definite finite numbers f(x). F(2),

Suppose now that to every interior and boundary point of a of which the region is built up be suitable, what is desired is proved; such that, whatever real positive quantity o may be, a real positive if not let an unsuitable triangle be subdivided into four, as before explained; if one of these subdivisions is unsuitable let it be again number e exists for which the condition subdivided; and so on. Either the process terminates and then

f(s)-f(20)

-F(-o) | <7, what is required is proved; or else we obtain an indefinitely continued sequence of unsuitable triangles, each contained in the which we describe as the condition (3,20), is satisfied for every points, preceding, which converge to a point, say, $; after a certain stage within or upon the boundary of the region, satisfying the limitation all these will be interior to the proper region of $; this, however, is 15--20<e. Then (20) is called a differentiable function of the contrary to the supposition that they are all unsuitable.

complex variable zo over this region, its differential cocfficient being We now make some applications of this result (B). Suppose a | F(s). The function () is thus a continuous function of the real

« PreviousContinue »