QA 1 "511 H3l UC-NRLF B M EMT 77fi Cambridge Tracts in Mathematics and Mathematical Physics kmmi General Editors J. G. LEATHEM, M.A. E. T. WHITTAKER, M.A., F.R.S. No. 2 HE INTEGRATION OF FUNCTIONS OF A SINGLE VARIABLE by G. H. HARDY, M.A. ^ V^^^ TnTbi M i.'i mi in t. i ^ Cambridge University Press Warehouse C. F. Clay, Manager London : P^etter Lane, E.G. Glasgow : 50, Wellington Street 1905 Price 2s. 6d. net Cambridge Tracts in Mathematics and Mathematical Physics General Editors \ J. G. LEATHEM, M.A. E. T. WHITTAKER, M.A., F.R.S. No. 2 The Integration of Functions of a Single Variable CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, C. F. CLAY, Manager. ILontiou: FETTER LANE, E.G. tPlasBoto: 50, WELLINGTON STREET. Ucip.ug: F. A. BROCKHAUS. l^cin gorfe: THE MACMILLAN COMPANY. Combao antj Calcutta: MACMILLAN AND CO., Ltd. [All rights reserved] THE INTEGRATION OF FUNCTIONS OF A SINGLE VARIABLE by G. H. HARDY, M.A. Fellow of Trinity College Cambridge: at the University Press 1905 (TambritigE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS. PREFACE. rriHIS pamphlet is intended to be read as a supplement to the -*- accounts of ' Indefinite Integration ' given in text-books on the Integral Calculus. The student who is only familiar with the latter is apt to be under the impression that the process of integration is essen- tially ' tentative ' in character, and that its performance depends on a large number of disconnected though ingenious devices. My object has been to do what I can to show that this impression is mistaken, by showing that the solution of any elementary problem of integration may be sought in a perfectly definite and systematic way. The reader who is familiar with the theory of algebraical functions and algebraical plane curves will no doubt find the treatment in Section V. of the integrals of algebraical functions sketchy and inadequate. I hope, however, that he will bear in mind the great difficulty of presenting even an outline of the elements of so vast a subject in a short space and without presupposing a wider range of mathematical knowledge than I am at liberty to assume. I have naturally not said much about particular devices which are only useful in special cases, but I have tried to show, where it is possible, how such devices find their place in the general theory. And I would strongly recommend any reader who is not already familiar with the general processes here explained to work through a number of examples (those for instance which have been set in the Mathe- matical Tripos in recent years) using in each case both the general method and any special method which he may find better adapted to the particular case. 1B9974 VI PREFACE 1 have bon-owed largely from the Cours d' Analyse of Hermite and Goursat, but my greatest debt is to Liouville, who published in the years 1830-40 a series of remarkable memoirs on the general problem of integration which appear to have fallen into an oblivion which they certainly do not deserve. It Avas Liouville who first gave rigid proofs of wliole series of theorems of the most fundamental importance in analysis — that the exponential function is not algebraical, that the logarithmic function cannot be expressed by means of algebraical and exponential functions, and that the standard elliptic integrals cannot be expressed by algebraical, exponential and logaritlmaic functions. That such theorems require proof is too often altogether forgotten. I have added a list of references for the benefit of more advanced readers. G. H. H. November, 1905. CONTENTS. I. Introduction II. Elementary functions and their classification III. The integration of elementary functions. Summary of results IV. The integration of rational functions ..... 1. The method of partial fractions 2. Integration by means of rational operations 3. Hermite's method of integration ..... 4. Particular problems of integration .... 5. The limitations of the methods of integration . associated with V. The integration of algebraical functions 1. Algebraical functions . 2. Integration by rationalisation. Integrals conies ..... 3. The integral jB(.i; '^ax^ + 2gx + c)dx 4. Unicursal plane curves 5. Particular cases .... 6. Unicursal curves in space . 7. Integrals of algebraical functions in general 8. The general form of the integral of an algebraical function Integrals which are themselves algebraical 9. Discussion of a particular case. The transcendence of e* and log a; 10. Laplace's principle 11. The general form of the integral of an algebraical function {continued). Integrals expressible by algebraical functions and logarithms 36 PAGE 1 3 7 10 10 12 13 14 16 18 18 19 21 25 27 29 29 30 33 35 VUl CONTENTS VI. 12. Elliptic and pseudo-elliptic integrals. Binomial Integrals 13. Curves of deficiency 1. The plane cubic 14. Abelian integrals in general 15. The classification of elliptic integrals Tran.scendental Functions Preliminary The integral J/e(e% e*% ..., e*^) c?.r The integral jP{.v, e"',^^, ...)dx. 'i'iic integral jc^ iff (.r) o?.r. The logarith integrals Liou villa's general theorem . The integral j\og x R{.v)d.v . Conclusion Appendix. List of references m-, sine- and cosine PAGE 37 39 40 41 42 42 42 45 45 49 49 51 52 1 THE INTEGRATION OF FUNCTIONS OF A SINGLE VARIABLE. I. Introduction. The subject of the following pages is what may fairly be described as the fundamental problem of the Integral Calculus properly so called : '' to find a function whose difftrential coefficient is a given function,' or to solve the differential equation l=-^(-) w- It may seem at first sight that the Integral Calculus thus defined is merely a very small department of the theory of Differential Equations. Indeed Euler, the first systematic writer on the Calculus, defines the Integral Calculus in a way which includes tlie whole of that theory in its scope : ' calculus integralis est methodus, ex data differen- tialium relatione inveniendi relationem ipsarum quantitatum*.' Or again it may seem as if, according to our definition, the Integral Calculus is only a small part of the Theory of Definite Integrals. The latter theory, starting from the -definition of the definite integral \'^J\t)dt, J to as the limit of a certain sum, shows us that, under certain conditions on which we need not insist, the solution of the equation (1) is given by 2/ jy(t)dt. Every problem of what is usually (though not very happily) called ' indefinite integration ' may therefore be regarded as a problem in the * Institutiones Calculi Interjralis, p. 1. H. 2 INTRODUCTION [l theory of definite integrals, wliile tlie latter theory obviously includes many prol)leuis which fall outside the former. In spite of this ' indefinite integration ' in reality forms an inde- pendent theory, proceeding by its own methods and meeting with difficulties peculiar to itself When we say that we have solved a dilferential equation, for exami>le, we mean that we have succeeded either in expressing y oxjilicitly in terms of x by functional signs, one of which may be the sign of indefinite integration, or implicitly by means of some relation such as an algebraical ecpiation. We have in other words removed the difficulties of the problem from the field proper to the theory of differential equations to that of some other theory whose results are taken for granted. If our result involves the sign of indefinite integration the further question arises as to whether the process indicated can actually be carried out, and this is a question not in differential equations but in integral calculus. Much the same may be said of the relations of the theory of ' indefinite integration ' to the theory of definite integrals, or rather to the part of the latter theory which is concerned with the evaluation of particular integrals. To evaluate « Jo for instance, is to express it explicitly as a function of ?/, and in this expression the sign of indefinite integration may perfectly well occur. With the other side of the theory of definite integrals, the side which is really part of what is called the ' Theory of Functions of Real Variables,' and deals with questions concerning limits, continuity, and convergence, our present subject has really very little connection. We may indeed draw from it the one result, to which allusion has already been made, as to the existence of a theoretical solution of the equa- tion (1). But from our present point of view this result is entirely uninteresting and unimportant. What we are concerned with is the form of the solution, and the only proof of its existence which is of any value to us is that which consists in actually expressing it in terms of x. And we shall not be troubled in the least by any difficulties concerning continuity. The functions with which we shall be dealing will be always such that they and their differential coefficients are continuous except for certain special values of .r, and these values of ,r we shall simply Il] ELEMENTARY FUNCTIONS AND THEIR CLASSIFICATION 3 omit from coiisideratiou. It no way affects the meaning of the equations dhgw _ 1 [dx QjOC 00 log X that log A' and \\x become infinite for x = 0. After these preliminary remarks we may proceed to define our subject more precisely. II. Elementary functions and their classification. An elementary function is a member of the class of functions which comprises (i) rational functions, (ii) algebraical functions, explicit or implicit, (iii) the exponential function e^, (iv) the logarithmic function log x^ (v) all functions which can be defined by means of any finite combination of the symbols proper to the preceding four classes of functions. A few remarks and examples may help to elucidate this definition. 1. A rational function is a function defined by means of any finite combination of the elementary operations of addition, multiplication, and division, operating on the variable x. It is shown in elementary algebra that any rational function of X may be expressed in the form box"" + b^x''-' + ... +b„\ where m and u are positive integers and the «'s and ^'s constants. It is hardly necessary to remark that it is in no way involved in the definition of a rational function that these constants should be rational or algebraical* or real numbers. Thus x^ + X + i J2 X sj2 -e is a rational function. * An algebraical number is a number which is the root of an algebraical equation whose coefficients are integral. It is linown that many numbers (such as e and w) are not roots of any such equation, 1—2 4 ELEMENTARY FUNCTIONS AND THEIR CLASSIFICATION [ll 2. All e.rpUitt ((Igehm'icdl function is a function defined by means of any finite combination of the four elementary operations and any finite number of operations of root extraction. Thus m I are explicit algebraical functions. And so is .r» {i.e. ^.r"") for any in- tegral values of m and n. But x'^\ x'+' are not algebraical functions at all, but transcendental functions, as irrational or complex powers can only be defined by the aid of exponentials and logarithms. If y is an explicit algebraical function of x we can always find an equation whose coefficients are rational functions of x. Thus, for example, the function ■ y= Jx+ J{x + Jx) satisfies the eiiuatinn y-(4/ + 47/+l).r = 0. The converse is not true, since it has been proved that in general equations of degree higher than the fourth have no roots which are explicit algebraical functions of their coeflScients. A simple example is given by the equation 7/ - ?/ — ,r = 0. ■ We are thus led to consider a more general class of functions, implicit algebraical functions, wliicli includes the class of explicit algebraical functions. 3. An algebraical function of .r is a function wliich satisfies an equation whose coefficients are rational functions of x. We shall always suppose this equation to be irreducible, i.e. incapable of resolution into factors whose coefficients are also rational functions of .r. If it could be so resolved we could regard y as the root of an equation of lower degree than ni. Thus if y* — af = we must have either i/- + x = ox if-x-0. Each of these latter equations is irreducible. 2-4] ELEMENTARY FUNCTIONS AND THEIR CLASSIFICATION 5 The equation which y satisfies will have m - 1 roots other than y. No two roots can be equal, for if two roots were equal the equation would have a factor in common with the derived equation imf'-^ + {ni - 1) i?iy"-- + . . . = 0, and this common factor could be determined by the elementary theory of the greatest common measure of two polynomials, and would be rational in X. The original equation would therefore not be irreducible. Of the m roots of the equation we confine our attention 'to one, namely y. The relations which hold between y and the other roots are of the greatest importance in the theory of functions, but we are in no Avay concerned with them at present. 4. Elementary functions which are not rational or algebraical are called elementary transcendental functions, or elementary transcendents. They include all the remaining functions which are of ordinary occur- rence in elementary analysis. The trigonometrical (or circular) and hyperbolic functions, direct and inverse, may all be expressed in terms of exponential or logarithmic functions b3nneans of the ordinary formulae of elementary trigonometry. Thus, for example, sin x= —. (/^ - e'^"^), sinh x ^-( = S, where p, A, B, ..., JR, S are any algebraical functions of a; and a, (S, ..., p any constants. It is not a little surprising that the necessity of giving some proof of the theorems (a) — (c?) should be so generally overlooked by writers on elementary analysis. III. The integration of elementary functions. Summary of results. In the following pages we shall be exclusively concerned with the question of the integration of elementary functions. We shall endeavour to give as complete an account as the space at our disposal permits of the progress which has been made by mathematicians towards the solution of the following two problems : — (i) (/" /(•?') is ^'^ elementary function, how can we determine whether its integral is also an elementary function ? (ii) ifths integral is an elementary function, how can ice find it ? Complete answers to these questions have not and probably never will be given. But sufficient has been done to give us a tolerably complete insight into the nature of the answers, and to ensure that it shall not be difficult to find the complete answers in any particular case which is at all likely to occur in elementary analysis or in its applications. 8 THE INTEGRATION OF ELEMENTARY FUNCTIONS [ill It will probably be well for us at this point to summarise the principal results which have been obtained. 1. The integral of a rational function (rv.) is always an elementary function. It is either itself rational or is the sum of a rational function and of a finite number of constant multiples of logarithms of rational functions (iv. 1). If certain constants which are the roots of an algebraical ec^uation are treated as known quantities the form of the integral can always be completely determined. But as the roots of such equations are not in general capable of explicit expression in finite terms, it is not in general possil)le to express the integral in an absolntel}^ explicit form, although our knowledge o't ii^finictvmal form is complete (iv. 2). We can always determine, by means of a finite number of elementary operations which can actually be performed, whether the integral is rational or not. If it is rational, we can determine it completely by means of such operations ; if not, Ave can determine its rational part (iv. 3. 4). The solution of the problem in the case of rational functions may therefore be said to be complete ; for the difficulty with regard to the explicit solution of algebraical equations is one not of inadequate kn(»wio (x), where A is a constant, and this is why logarithms can only occur in this form in the integrals of rational or algebraical functions. We have thus a general knowledge of the form of the integral of an algebraical function, ji/dx, when it is itself an elementary function. Whether this is so or not of course depends on the nature of the equation /(x, y) = which defines ?/. If this equation, when interpreted as that of a curve in the plane (x, y), represents a unicursal curve, i.e. a curve which has the maximum number of double points possible for a curve of its degree, or whose deficiency is zero, x and y can be expressed simultaneously as rational functions of a third variable t, and the integral can be reduced by a substitution to that of a rational function (v. 2 — 5). In this case, therefore, the integral is always an elementary function. But this condition, though sufficient, is not necessary. It is in general true that if f{x, y) = is not unicursal the integral is not an elementary function but a new transcendent, and we are able to classify these transcendents according to the deficiency of the curve. If, for example, the deficiency is unity, the integral is in general a new transcendent of the kind known as elliptic integrals, whose characteristic is that they can be transformed into integrals containing no other irrationality than the S(-[uare root of a polynomial of the third or fourth degree (v. 13 — 15). But there are in- finitely many cases in which the integral can be expressed by algebraical functions and logarithms. Similarly there are infinitely many cases in which integrals associated with curves whose deficiency is greater than unity are in reality reducible to elliptic integrals. Such abnormal cases have formed the subject of many exceedingly interesting researches, but no general method has been devised by which we can always tell, after a finite series of operations, whether any given integral is really elementary, or elliptic, or belongs to a higher order of transcendents (v. 12). 10 RATIONAL FUNCTIONS [iV When /(.r, ?/) = is iinicursal we can carry out the integration coihph'tehi in exactly tlie same sense as in the case of rational functions. In {(articular, if the integral is algebraical it can be found by means only of elementary operations which are always practicable. And it has been shown, nujre generally, that we can always determine by means of sucii operations whether the integral of any given algebraical function is algebraical or not, and evaluate the integral when it is algebraical. And although the general problem of determining whether any given integral is an elementary function, and calculating it if it is one, has not been solved, the solution in the particular case in which the deficiency of the curve / {x, ?/) = is unity is as complete as there is reaiion to suppose that any possible solution can be (v. 12). 3. The theory of the integration of transcendental functions (vi.) is naturally much less complete, and the number of classes of such functions for which general methods of integration exist is very small. These few classes are, however, of extreme importance in applications (vi. 2. 3). There is a general theorem concerning the form of an integral of a transcendental function (when it is itself an elementary function) which is (piite analogous to those already stated for rational and algebraical functions. The general statement of this theorem will be found in vi. (5) ; it shows, for instance, that the integral of a rational function of (say) x, e" and log x is either itself a rational function of those functions, or is the sum of such a rational function and of a finite number of numerical multiples of logarithms of similar functions. From this may be deduced a number of more precise results concerning more particular forms of integrals, such as [yt^dx, Jylogxdx, where i/ is an algebraical function of x (vi. 4. 6). IV. Rational functions. 1. It is proved in treatises on Algebra* that any polynomial can be exi)re.ssed in the form bo (X - a,)'"' {X - a^)'"^ . . . (.r - a,)"V, where nij, ... are positive integers whose sum is n, and a,, ... are real * See, e.g., Clirystal's Algebra, vol. i. pp. 151—162, 248—254. 1] RATIONAL FUNCTIONS 11 or complex quantities ; and that any rational function R {x), whose denominator is Q{x), may be expressed in the form A^T^ + A,x^'-'+... +A, + i jA]_ + ^^^ + ... + f^^'"'s \ . Hence f x^'^^ x^ \R ix) dx = Aq + Ai— + ... + A.,x + C J 2J+1 p ' + 2 ( A. 1 log (x - a^ - -fc - ... -■ //-'"^ ^ -] . From this we conclude that the integral of any rational function is an elementary function which is rational save for the possible presence of logarithtns of rational functions. In particular the integral will be rational if each of the quantities ^s, i is zero: this condition is evidently necessary and sufficient. A necessary but not sufficient condition is that Q (.r) should contain no simple factors. The integral of the general rational function may be expressed in a very simple and elegant form by means of symbols of differentiation. We may suppose for simplicity that the degree of P{x) is less than that of Q (x) ; this can of course always be ensured by subtracting a polynomial from R (x). Then = {ll{m, -1)1 {m, - 1) ! ... {m, - 1) l} D„,_,, .,_,...,«,_, ^^ , where Qo (x) = b^ (x - a^) {x - a,,) . . . {x - a,.), and Z>,„,_i,. ..,„(,._! represents the operation a'^-'yaai™!-! 8a/2-i ... da,."'r-\ Now Qo (^) s = l {'V - a.-) Qo (a.) ' and so \r (x) dx Hl/(>«.^l)!...(».-l)!i a„,,,-, a„,.,- -[i.g^)'°g(-^-°->]' For example / dx 8" f 1 , /x — a log {{x - a) {x - b)\'' dadb [a -b \x - b 12 RATIONAL FUNCTIONS [iV It has been assumed above that if then ■--= I ~<^^^ On J Oa dF df c^F da cx'da ' df d^F follows from the first that ;^ = .^-^^ what has reallv been assumed is that da Oa ex d^F ^ d^F cad.v dx'Ca' It is known tliat this equation is always true for .v = Xo, a = a„ if a circle can be drawn in the plane of {x, a) whose centre is Xq, oq and within which the differential coefficients are continuous. 2. From one point of view the preceding investigation is complete. From otliers, and notably from that of practical applicability, it is far from perfect, for the simple reason that the factors of the denominator cannot be found, as the roots of Q {.r) = are not in general explicit algebraical functions of the coefficients. The difficulty may be stated thus: the fu net io)Kt I form of the integral is completely determined, but it involves ro)i.'; 4- 7 and aiX' + ifiiX + yi are harmonically related. By a repetition of this argument we can prove that is rational if all the quadratics are harmonically related to any one of those in the numerator. 5. It appears from the preceding paragraphs that we can always find the rational part of the integral, and can find the complete integral if the roots of Q {x) = can be found. The question is naturally suggested as to the maximum of information which can be obtained about the logarithmic part of the integral in the general case in which the factors of the denominator cannot be determined explicitly. For there are polynomials which, although they cannot be completely resolved into such factors, can nevertheless be partially resolved. For example ^14 _ 2a^ - 2.r^ - X* - 2af + 2x+l^{x' + a^-l) {x' - x- - 2x - 1), x"^ - 2x^ - 2x' - 2x^ - Ax" - x^ + 2x + 1 = {x' + x" J2 + A- ( V2 - 1) - 1 } {.r - x" J2 -x{j2 + \)-l}. The factors of the first polynomial h.ave rational coefficients : in the language of the theory of equations, the polynomial is reducible in the rational domain. The second polynomial is reducible in the domain formed by the adjunction of the single irrational ^/2 to the rational domain*. We may suppose that every possible decomposition of Q (.r) of this nature has been made, so that Then we can resolve R (x) into a sum of partial fractions of the type * See Cajori, An introduction to the Modern Theory of Equations (Macmillan, 1904). 4-5] RATIONAL FUNCTIONS 17 aud so we need only consider integrals of the type / Q ^^' ,vhere no further resolution of Q is possible (in technical language Q is irreducible by the adjunction of any algebr-aical irrationality). Suppose that this integral can be evaluated in a form involving only constants which can be explicitly expressed in terms of the constants which occur in PjQ. It must be of the form ^ilogXi + ... +.ltJogXi., where the ^'s are constants and the A^'s polynomials. We can suppose that no X has a multiple root : if e.g. X^ had one we could determine it rationally in terms of the coefficients of Xi and the corresponding factor (x — a)'" could be removed from Xi by inserting a new term mAi\og{.v - a) in the expression of the integral*. For a similar reason we can suppose that no two A^'s have any common factor. Now — J Ali . X2 A -^k or px,x, ...Xu = Q %A ,x^ . . . x_iX;a;+, . . . X, . All the terms under the sign of summation are divisible by Xi save the first, which is prime to A"]. Hence Q must be divisible by A'l : and similarly, of course, by Xo, X3, ..., X^. Since F is prime to Q, X1X2 . . . Xk is divisible by Q : hence I Q = Jl 1X2 •■• A fc save for a constant factor. But e,r hypothesi Q is not resoluble into factors which contain only explicit algebraical irrationalities. Hence all the A"'s save one must reduce to constants, and so P must be a constant multiple of Q', and jgdx=-A log Q, where A is a constant. Unless this is the case the integral cannot be expressed in a form involving only .constants explicitly expressed in terms of the constants which occur in P and Q. * If A'l had more than one multiple root of the same order we might not be able actually to determine them rationally in terms of its coefficients (e.g. Xi = (x-a)(.v'^-x-a)-), but we could so determine the factor corresponding to all these roots, so that the argument would not be affected. 2 18 ALGEBRAICAL FUNCTIONS [V Thus, for instance, the integral /. dx x^ + ax-\-h I cannot be expressed in a form involving only constants explicitly expressecl in terms of a and h ; and J x^ + ax + h can be so expressed if and only if c = a. We thus confirm an inference formed before (iv. 2) in a less rigid way. Before quitting this part of our subject we may consider one further problem ; mider what circumstances is j \R{x)dx=A\ogRi{x) ' where A is a constant and Rx rational? Since the integral has no rational part it is clear that Q {x) must have only simple factors, and that the degree of P {x) must be less than that of Q (x). We may therefore use the formula / R (x) dx = \og U {{x - a«)^('^»)/<^'("«^}. The necessary and sufficient condition is that all the quantities P {ag)lQ' (a,) must be commensurable. If e.g. \x-a){x-^y (a — y)l{a - ^) and (/3 — -y)/(/3 - a) must be commensurable, i.e. (a - y)/(/3 - y) must be a rational number. If the denominator is given we can find all the values of y which are admissible : for y = {aq - ^p)l{q —p) where p and q are integers. V. Algebraical Functions. 1. We shall now consider the integrals of algebraical functions, explicit or implicit. The theory of the integration of such functions is far more extensive and difficult than was the case with the rational functions, and we can only give here a brief account of the most important results and of the most obvious of their applications. If yi, y-i, ■•■, yn are algebraical functions of x any algebraical function z of x,yi, •..,yn is an algebraical function of x. This is obvious if we confine ourselves to explicit algeliraical functions. In the general case we have a number of equations of the ty^& P>,o i-r) y.'"" + Pm (x) 7/,'""-^ + . . . + P., „„ (x) = (v-l,2, ..., «), and i i\{^,y^, ■■■yn)z"' + ••• + Pm(.A\yx, ...,yn) = 0, where the F's represent polynomials in their arguments. The as 1-2] ALGEBRAICAL FUNCTIONS 19 elimination of ?/i, y.,, ..., i/„ between these e(iuations gives an eciuation for ;:;, whose coefficients are polynomials in .t only. The importance of this from our present point of view lies in the' fact that we may consider the standard algebraical integral under any of the forms (a) jy dx, where / (.r, y) = ; {b) JE (.r, y) dx, where/ (.r, y) = and R is rational ; {c) jE (.r, ;/i , . . . , y„) dx, where ./; (x, y) = 0,..., f„ (x, y,,) = 0. It is, for example, much more convenient to treat such an irrational X- J(x+ 1)- J(x- 1) 1 + jIx+1)+ s/(x-1) as a rational function of x, y^, y., where yi = s/{x+ 1), yo= J(x- 1), yi^ = X + 1, y.2^ = x — 1, than as a rational function of x and y=J(x+ 1) + J{x-l), so that y - Axy^ + 4 = 0; while to treat it as a simple irrational y, so that our fundamental equation is {x-yy~4x{x-yf{l+yf+4.{l+yy^0 is evidently still more inconvenient. Before we proceed to consider the general form of the integral of an algebraical function it will be convenient to consider one most important case in which the integral can be immediately reduced to that of a rational function, and therefore is always an elementary function itself. It will perhaps be well at this point to emphasize two points which we have already mentioned (ii. 3) : viz. (i) that our defining relation /{x, y) = is always supposed to be irreducible and (ii) that we confine our attention to one of its roots. 2. The class of integrals alluded to immediately above is that covered by the following theorem. 1/ there is a variable t connected ivith x and y (or yi, y^, ■■■, yn) by rational relations x^E.it), y = E,{t) (or y, = E4'^ (t), y. = i?./-' (t), ...) the integral JE (.r, y) dx (or JE (x, ?/i, ... , 3/„)f/.r) can be evaluated in finite terms by means of elementary functions. 2—2 20 ALGEBRAICAL FUNCTIONS [V This is practically obviou s, since all the capital letters denoting rational functions. It is to be observed that from our present point of view it is quite immaterial whether an integral be transformed by a real or by an imaginary substitution. For the equation ^/{^■)dj; = lf{{t)}(l>'{t)dt, dF(x) ., , means simply that it — -5 — =/ {x), then ^^j^=f{<^{t))<\>'{t\ and it is of no importance whether is a real function or not. It is of the utmost importance, of course, when we are de? are any quantities such that yf = «^- + 2g^ + c. We may for instance suppose ^ = 0, 1^ = Jc ; or t; = 0, while ^ is a root of the equation a^" + 2g$ + c = 0. 3. It musb not be imagined that this general method is always practically the best for the integration of \R (.r, ■Jax^ + 2gx + c) d.v. In practice we proceed as follows. Let y = ^fX= '>Jax^ + 2^A- + c. Then R{x^ y) is of the form P{x\ y)jQ {x, y), where P and Q are polynomials. By means of the equation y'^ = a.v^ + 2gx + c, R{.v,y) may be I'educed to the form A + B^X _ {A+BJX){C-DJX) C+D^IX C^-D^X where A, B, C, D are polynomials in .v ; and so to the form 3f+X,^fX, where J/ and X are rational, or (what is the same thing) the form where P and Q are rational. In the cases of most frequent occurrence in practice a, g, c, .sjaa;^ + 2gx + c and the coefl&cients which occur in P and Q are real. The rational part may be integrated by the methods of iv., and the integral j -pr^ dx may by the theory of partial fractions be made to depend J V ^i upon a number of integrals of functions of the forms 1 1 {x-p)JX' {x-pYs'X' {aa^ + 2^x+y) JX ' (a.r2 + 2/iJ.i- + yf ^IX ' 22 ALGEBRAICAL FUNCTIONS [v where /?, ^, r;, n, /3, y are real constants and r a positive integer. The result is generally required in an exi)licitly real form : and as further progress depends on transformations involving p (or a, /3, y) it is generally not advisable to break up a quadratic factor a^^ + 2/3.i'+y whose roots are imaginary into its constituent linear factors. The integrals which involve powers of x —p or ax^ + 2fix + y higher than the first may be deduced from those which involve only the first powers by differentiations with respect to p or y. The integral /, . ,„ , may be evaluated in a variety of manners. (i) We may follow the general method described above, taking ^=Py Ti = J(ap^ + 2gp + c)*. Eliminating y from the equations f' = ax^ + 2gx + c, y~r} = t(x-^), and dividing by x — ^, we obtain f-{x-^) + 2r}t-aix + i)-2(/ = 0, J 2dt dx dx ana so — s = , ., — = — . f—a ((.r-S+, y But {fi-a){x-^) = 2a^ + 2g-2rit; and so [ dx f dt 1 , , » I (^^pT) " - 1 ^4TI^t - -, '°« <"« +* - "> If ap'^-\-2gp-\-c<0 the transformation is imaginaryf. Suppose, e.g. ,{a)y = ^f{x + 1), p = 0, (b) i/ = J{x -l),p = 0. We find (") lxJ.v+l) = ^^Sit-^\ where i-x + 2t-l = 0, or t = {-l + \/x + l)/x, * Jordan, Cours d^Aiialyse, t. ii. p. 21. t We have supposed that p is not a root of the equation X = 0. If it is, the integral is, as we sliall see later (v. 9(i)), algebraical, andean be determined by a series of .elementary algebraical operations which are always practicable. Otherwise the integral is purely transcendental. A factor of the denominator of Q which is also a factor of A' can be found liy elementary methods, and the algebraical part of / -j^ lie can always be determined completely by such methods. This result is quite analogous to that already proved in the case of rational functions. 3] ALGEBRAICAL FUNCTIONS 23 the positive sign of the radical corresponding to the case in which y=+^/(.^•+l): (^) /^l)4^«S(^-^-i), where fl.v + 2it-l = 0. Neither of these results is expressed in the most convenient form, the second in particular being very inconvenient. (ii) The most straightforward method of procedure is to use the substitution commonly used in text-books on the Integral Calculus. We then obtain dz f dx _ f which is a well known form reducible by a substitution of the type 2 = ^+ ^- to one of the three standard foruLS [ dt f dt f dt These forms may of course be rationalised, as e.g. by the respective substitutions _ 2mw 2)nu m(l+'W^) ^^1+^2' ^ = 13^2' ^= 2^ • but it is more convenient to use the transcendental substitutions t = 7n sin 0, i = 7n sinh (f), t = m cosh 0. And it is often convenient when dealing with more complicated algebraical integrals, containing only one irrationality of one of the types to reduce it to a transcendental form not involving roots by means of one of these three substitutions. As alternative substitutions t = VI tanh 0, t = m tan 0, t = m sec 0, are often useful. Prof. Bromwich points out that the forms usually given in the text-books for these three standard integrals, viz. sin- 1 (.t^/o), sinh - * {xja), cosh ~ i {xja), are not entirely accurate. It is obvious, for example, that the first two of these functions are odd functions of a, while the corresponding integrals are even functions of a. The correct formulae are sin" i (.r/| a j), sinh-i(.r/|a |), and ±cosh-i(|.r|/|a|) = log(.r+ v^A-'-^-a^), where the ambiguous sign is the same as that of x, as the reader will easily verify. In some ways it is more convenient to use the equivalent forms / X \ / ^' \ . ^ ,/\/^"''-«^\ 24 ALGEBRAICAL FUNCTIONS [V (iii) The most elegant method of integration is unquestionably that associated with the name of Prof. Greeuhill* who uses the transformation V=^-^ . ^ x—p It will be found that r dx _ f dz which is one of the three standard forms written above. When we are dealing with the integral h ^dx, .(!)• (a^2 + 2)307 + y)s/Jr (which will naturally only be the case when the roots of ax- + 2^x + y=0 are imaginary) by far the most convenient method of procedure is to use Prof. Greenhill's substitution say. I f J= (a^ - go) x^ - (ca — ay) x+gy- c/3, 1 dz _ J z dx A'Xi The maximum and minimum values of z are given by J=0. Agani z^-\ = - ^ i wherein the numerator will be a perfect square if K= (ay - /32) X2 - (ay + ca- 2^/3) X + ac -g^ = 0. It will be found by a little calculation that the discriminant of this quadratic and that of J=0 difl'er from one another and from (ll-^l')(l2-^l')(^.-^2')(l2-^2'), where ^i, ^o ^^'^ the roots of A'=0 and I/, ^-Z those of ^'1 = 0, only b}'^ a constant factor which is always negative. Since ^1' and ^o ^^^ conjugate imaginaries this product is positive, and so .7=0 and A''=0 have real roots. We denote tlie roots of the latter by Xi, X2 (Xi>X2). Then X , - 2'- = {••^V(^i«-«) + V(^iy-g)}'' ^ {'mx+7if A I Xi ,2 , _ {^x/(«-X2a) + v/(c-X2y)l^ {m'x + ny * '^2— y — V' {^) say. Further, since *- - X can vanish for two equal values of x only if X is equal to Xi or X2, i.e. when z is a maximum or a minimum, J^can only differ from (;mx + n) {m'x + )i') * A. G. Greenhill, A Chapter in the Integral Calculits (1888, Francis Hodgson), p. 12 : Differential and Integral Calculus, p. 399. 3-4] ALGEBRAICAL FUNCTIONS 25 by a constant factor ; and by comparing coefficients and using the identity (Xia-a)(a-X2a) = (a/3-^a)2/(ay-^2), we find that J=^{ay — 0^){mx-\-n){m'x+n') (3). Finally we can write ^x-\--q in the form A {mx + 71) + B {m'x + n'). Using equations (1), (2), (2'), (3) we find that j x-jx"^-'-] J ^^^^^^^ = ^ f dz B f dz - s/iay -0')] v/(Xi - Z^) ^ J {ay - ^^) J ^{z^ - \^) ' and the integral is expressed in terms of real standard forms*. 4. We may now proceed to consider the general case to which the theorem of iv. 2 appHes. It will be convenient to recall two well known definitions in tlie theory of algebraical plane curves. A curve of degree n can have at most h{n -1) (n — 2) double points t. If the actual number of double points is v the number p=hin-l)(n-2)-v is called the deficievcyX of the curve. If the coordinates x, yof ^"he poiiits on a curve can be expressed rationally in terms of a parameter t by '^n nations x = R,{tl y = B,(i), we shall say that the curve is unicursal. In this case \\o have seen that we can always evaluate jR (.r, y) dx in finite terms. The fundamental theorem in this part of our subject is 'J curve whose deficiency is zero is unicursal, and vice versa.' Suppose first that the curve possesses the maximum number of double points§. Since 2 {n - 1) (« -2) + n-3=^h(n-2)(n + l)-l, * The reader should refer to Prof. Greenhill's writings quoted above and to Chrystal's Algebra, vol. i. pp. 464 et seq. Prof. Greenhill gives interesting numerical examples. t Salmon, Higher Plane Curves, p. 29. J Salmon, ibid. p. 29. French genre, German Geschlecht. § We suppose in what follows that the singularities of the curve are all ordinary double points. The necessary modifications when this is not the case are not dillficult to make. It has been shown that an ordinary multiple point of order k may be regarded as equivalent to hk(k-l) ordinary double points (Salmon, loc. cit. p. 28, 26 ALGEBRAICAL FUNCTIONS [V and h(n-2)(n + 1) points are just sufficient to determine a curve of degree n-2* we can, through tlie |(w - l)(n-2) double points and w-3 other points chosen arbitrarily on the curve draw a simply infinite set of curves of degree n - 2, which we may suppose to have the equation g{x,y) + th{x,y) = 0, where /" is a variable parameter. Any one of these curves meets the given curve in n (n-2) points of which (n - 1) (w - 2) are accounted for by the |(w-l)(w-2) double points and w-3 by the n-S arbitrarily chosen points. These (w - 1) (w - 2) + n - 3 = w (w - 2) - 1 points are independent of t ; and so there is but one point of intersec- tion which depends on t. The coordinates of this point are given by g {a\ y) + tk (.r, y) = 0, /(.r, y) = 0. The elimination of y gives an equation of degree n (n - 2) in a; whose coefficients are polynomials in t, and but one root of this equation varies with t. The elimiuant is therefore divisible by a factor of degree n(n-2) -1 which does not contain t. There remains a simple equation in .r whose coefficients are polyi^'^"' "'* '" - ^^us nit; ^-coordinate of the variable point is-, ^ 7*^^'^^^ ^s a rational function f. , ,, y . ^ -'^'liiJarly determined. 01 t, and the Tz-coorduiate^'^^ • We may therefoi-- ^ = ^Ai), y = R.{t). •^3(0' ^~Mf) (1), .encKU bt of degree n ; none of them can be of Basset, Quartn-l,, Journal, xxxvi n 300^ A ordinary multiple point of order ^-UeauhalLTT ^"^'"' " ^^^"^^ ^^' ^" points) is therefore unicursal. (^I^'^^^^"* to i ,« _ i) („ _ 2} ordinary double havJt Tz:i i!:tZ'zzi::r:'' ^" '^-^'^- p-^^-^- -.« ..ie,. and to give a satisfactor/accTunt of a ' "T "' ^"^ °'"°- conventions, sometimes by no means easy. The LvLti T^'T '*^"^"""^^ ^'"8"'-^^^ i as essentially occupied with\he ;;::rcaTe " '''''''' '""^^^'^ '""^^ '^ -^-^ed Salmon, he. cit. p. lo. k 4-5] * ALGEBRAICAL FUNCTIONS 27 higher degree, and one at least must be actually of that degree, since an arbitrary straight line A..^' + /x,?/ + V = must cut the curve in exactly n points*. "We shall now prove the second part of the theorem. If x:y: 1 :: ^^{t) : ,{t): cl>,{f), where ^j, ^o, 4>s are polynomials of degree n, the line ua; + V1/ + W-0 will meet the curve in n points whose parameters are given by u(t>,{t) + v,{t) + if>,(t) = 0. This equation will have a double root t^ if «<^l (^o) + V2 (to) + i (to) = 0, «/ (Q + V(t>o' (to) + i^/ (to) = 0. Hence the equation of the tangent at the point to is .T y 1 «^l(^o) 2{h) 4>z{t,) =0. -2 (to) ^3 (to) - «/>•; (^o) ^3 (to) is of degree 2)i-2 in t^,, the coefficient of /o""~^ obviously vanishing. Hence in general the number of tangents which can be drawn to a unicursal curve from a fixed point (the class of the curve) is 2n - 2. But the class of a curve whose only singular points are 8 double points is known f to be n {n - 1) - 28, Hence the number of double points is l{7i(n-\)-(2n-2)] = \{n-\)(n-2). 5. The preceding argument fails if n < 3, but we have already seen that all conies are unicursal. The case next in importance is that of * See Niewenglowski's Geometrie Anahjtique, t. ii. p. 103. By way of illustra- tion of the remark concerning particular cases in the footnote (§) to page 25, the reader will do well to consider the example given by Nieweuglowski in which t" t- + 1 equations which appear to represent the straight line 2.c=(/ + l (part of the line only, if we consider only real values of t). t Salmon, Higher Plane Curves, p. 54. 28 ALGEBRAICAL FUNCTIONS [V a cubic with a double point. If the double point is not at infinity we can, by a change of origin, reduce the equation of the curve to the form {(LT + by) {ex + dy) = pr^ + 'iqx-y + Srxy- + sy^, and by considering the intersections of the curve with the line y = tx we find _ {a + bt) (c + dt) _ t{a + bt) (c + dt) X — „ . „ ,o ) y — p + 3qt + Srf +ps' " p + Sgt + 3rf +ps ' If the double point is at infinity the equation of the curve is of the form (a.r + f3y)- (yx + 8y) + ex + ly + 6 = (the curve having a pair of parallel asymptotes), and by considering the intersection of the curve with the line ax + /Sy = t we find St' + Ct + ISO yf +et + ae ^~ {I3y - aS) t" + e(S - aC ^ ~ {(Sy - a8) f + e{3 - a^' (i) The case next in complexity is that of a quartic with three double points. (a) Tiie lemiii.scate {x- + y-)- = a- {x- — y-) has three double points, the origin and the circular points at infinity. The circle x-+y- = t{x-y) passes through these points and one other fixed point at the origin, as it touches the cvn-ve there. Solving we find (h) The curve 2ay3 _ 3^22^2 = ^^ _ ^a^^a has the double points (0, 0), (o, a), ( - a, a). Using the auxiliary conic x^--ay=tx{y-a) we find a- = |(2-3^2), y = ^{2-2fi){2-f-). (ii) The curve _?/» = a-" + a.r" ~ 1 has a multiple point of order n-\ at the origin, and is therefore unicursal. In this case it is .sufiicieut to consider the intersection of the curve with the line y = tx. This may be harmonised with the general theory by regarding the curve as passing through each of the \ {n- \){n- 2) double points collected at the origin and through 7t- 3 other fixed points collected at the point y = 0, x=-a. 5-7] ALGEBRAICAL FUNCTIONS 29 The curves y» = A« + «.i-H-i (1)^ y" = l+«2 (2), are projectively eqiiivalent, as appears by rendering their equations homo- ! geneous by the introduction of quantities z = l in (1) and .>;=1 in (2). We conckide that (2) is unicursal, having the maximum number of double points at infinity. In fact we may put and \R{z, ;:,l{\ + az)]dz is integrable in finite terms. (f) The curve is unicursal if and only if either (i) fior v = or (ii) ^x. + v = in. Hence \R [x, y{x-aY{x-bY] dx, is integrable in finite terms for all forms of R in these two cases only ; of course it is integrable for special forms of R in other cases*. 6. There is of course a similar theory connected with unicursal curves in space of any number of dimensions. Consider for example the integral jR {.V, sJia-r + h), sj{cx + d)] dx. A linear substitution x = Ix + m reduces this to the form lR,\y, J{y + 2),J{y-2)]dy, and this can be rationalised by taking The curve whose Cartesian coordinates t, Vi ^ are given by $:r,:^:l::t' + l:t(t- + l):t{f'-l): f, is a unicursal tw'isted quartic, the intersection of the parabolic cylinders It is easy to deduce that \mx + nj V Vw^.^; + nj) can always be evaluated in finite form. 7. When the deficiency of the curve f{x, y) = is not zero the integral JB (x, y) dx is in general not an elementary function ; and the consideration of such integi-als has consequently introduced a whole series of classes of * Ptaszycki, Bull, des Sciences Mathematiques, xii. p. 263 : Appell and Goursat, Theorie des Fonctioiis Algebriques, p. 245. fB{., yi 30 ALGEURAICAL FUNCTIONS [v new transcendents into analysis. The simplest case is that in which the deficiency is unity : in this case, as we shall see later on, the integrals are expressible in terms of elementary functions and certain new transcendents known as elliptic integrals. When the deficiency rises above unity the integration necessitates the introduction of new transcendents of groAving complexity. But there are infinitely many particular cases in which integrals, associated with curves whose deficiency is unity or greater than unity, can be expressed in terms of elementary functions, or are even algebraical themselves. For instance the deficiency of is unity. But / X +1 dx ^ . {l+x)--^ sjjl+x") x-2 J{l + a^)~ ^^{\+xf + ^J{l+a^y 2-01? dx 2x I: l+ar'^il+a?) J{1 + a?)' And, before we say anything concerning the new transcendents to which integrals of this class in general give rise, we shall consider what has been done in the way of formulating rules to enable us to identify such cases and to assign the form of the integral when it can be expressed in finite terms. It will be as well to say at once that this problem has not been completely solved. 8. The first general theorem deals with the case in which the integral is algebraical, and asserts that if u = jydx is an algebraical functioti of x it is a rational function of x and y. If u is an algebraical function of x it satisfies an equation ^\> {X, U) = 0, whose coefficients are polynomials in x. By means of the equation f{x, ?/) = we can introduce y into this equation and write it in the form <^ (■'', y, ") = 0, Avithout altering the degree of the equation in u. The succeeding proof depends essentially on the presence of y explicitly iu this equation. If /(.r, y) = Po (.»•) y" + . . . + P„ (.r) = 0, and A.^ is a term in P„(.r), it is obvious that A3^ = P{x,y\ 7-8] ALGEBRAICAL FUNCTIONS 31 P denoting a polynomial. If y\f (.r, u) contains a power of x as high as the kih. we can obviously introduce y at once by means of this equation : if not we must first multiply >//■ (,)■, u) by some power of x. We can suppose <^ {x, y, u) irreducible, fur if not we could replace it by some simpler equation. By differentiating /= 0, <^ = we obtain da: dt/ dx ' dx dy dx du dx ' and eliminating ->- we obtain an expression for -j- of the form du _ X {x, y, ti) dx fx (x, y, u) ' j where A and ju, are polynomials. In order that u shall be the integral of 3/ it is necessary and sufficient that -T-=y, i.e. that the equations ^ (.r, y, u) = 0, k{x,y,'u)-ytx{x,y,u) = 0, shall hold simultaneously. Now the equation ^ = has other roots th, u.2, ■■■, ih besides u (unless it is of the first degree, in wdiich case u is obviously a rational function of x and y\ and these roots must all satisfy the two equations ^ = 0, X - 2//X = 0. For otherwise we could determine the greatest common measure of ^ and X - yjx, considered as polynomials in u : this common factor would be a polynomial in x, y, u and divide <^ (x, y, u). But this is impossible, since ^ {x, y, u) is irreducible. Hence a, Ui, u->, ••• ttk are all integrals of y, and therefore -r~.(ii'+ Ui+ ■■■ + ih-) is an integral of y. But this function is a symmetric function of tiie roots of <^ {x, y, u) = 0, and is therefore a rational function of x and y. The theorem is therefore proved. Thus if the integral is algebraical, P and Q being polynomials. If ^i, v/o, ..., y„_i are the roots of/(x, y) = 0, other than y, [ _P{x,y)Q{x,y,) Q(x,yn-i) 32 . ALGEBRAICAL FUNCTIONS [V Tlie denominator is a symmetric function of y, jji, ...,i/,^-i and tliere- fore a rational function of .v. Moreover Q (^, ^i) Q (^. 2/2) ■■•Q(^, yn-i) is a symmetric function of the roots of the equation in ;:; z-y whose coefficients are polynomials in ,v and ;/ of which the first does not contain y. It is therefore a rational function of a' and y integral with respect to y, and so jydx consists of a sum of a number of terms of the type R„{x)y''. By means of the equation /(.r, ?/) = all such terms which involve powers of y higher than the ;^th can be eliminated. We thus arrive at the final conclusion that if jyd.v is algebraical it may be expressed in thefm-m B,^R,y^...-vIt,,-,y^'-^ where Bo, Bi, ■■■ are rational functions of x*. The most important case is that in which y=^J'B{^, where B{x) is rational. In this case dy ^B'{x) dx ny^~'^ ' But n-l y^Bo +Biy+...+B'n-iy + {B, + 2B,y+ ... + (/.- l)i?„_,/-=} f^ (1). Eliminating -^ between these equations we obtain an equation OT (x, y) = where ro (x, y) is a polynomial. In virtue of a theorem proved and used before this equation, and therefore the equation (1), must be satisfied l)y all the roots oi y^ = B (x). The same therefore holds of the equation jydx = Bo + B,y+...+B„-^y"-\ In this eciuation we may therefore replace y by (ay, w being any * For the preceding proof see Abel, (Euvres, t. i. p. -545 et seq., and Crelle, b. IV. p. 264; Liouville, Journal dc VEcoU Poly technique, t. xiv. p. 149; Bertrand, Calcul Integral, Ch. V. 8-9] ALGEBRAICAL FUNCTIONS 33 primitive nth. root of unity. Making this substitution and multiplying by w"-^ we obtain Ji/do! = (o»-ii?o + B,:i/ + wILi/ + ... + w"--'R,^_,f-\ and on adding the n e(iuations of this type we obtain ji/dw = E,i/. Thus in this case the functions 7?o, ^2, •••, ^«-: all disappear. It has been shown by Liouville that the preceding results enable us in all cases to obtain by a tinite number of elementary algebraical operations a solution of the problem 'to determine whether jydx is algebraical, and to find the integral ivhen it is algebraical.' 9. (i) It would take too long to attempt to trace in detail the steps of the general argument. We shall confine ourselves to a solution of a particular problem which will give an illustration sufficient for our present purpose of the general nature of the arguments which must be employed. We shall determine under what circumstances k dx {x-p) J{ax^ + 2gx + c) ' is algebraical. This question might of course be answered by actually evaluating the integral in the general case and finding when the integral function reduces to an algebraical function. We are now, howe\er, in a position to answer it without any such integration. In this case y = -pr^ , A'= {x - pf (a.r- + 2gx + c), sjA and if ji/dx is algebraical it must be of the form B {x)/s,fX. Hence y dx\s]x, or 2A'=2A'i?'-/Lr'. We can now show that R is a })olynomial in x. For if 11= Ui I', where U and Tare polynomials, I', if not a mere constant, must contain a factor {X + A)\ (a>0), and we can nut R = , W{x + Ay where 6^ and Tfdo not contain the factor x + A. Substituting this expression for R, and reducing, we obtain 2aUWX ^ ^ ^, ^^^^_ 2 ^. ^^,^ _ ^ ^^r^, _ 2 1,722' {x + A f. x-\-A Hence A' must be divisible by x-\-A. SupiJose then that X={x-\-AY X H. 3 34 ALGEBRAICAL FUNCTIONS [V where X is prime iox + A. Substituting in the equation last obtained we deduce ] x-\-A which is obviously imjjossible, since neither U, IF, nor X is divisible by x-\-A. Hence /" ^^ : ^M J {x—p) ^/(ax^ + 2gx + c) (x- p) J(ax^ + 2gx + c) where U{x) is a polynomial. Differentiating and clearing of radicals {{x-p) (U'-l)- U} {ax^ + 2gx + c)= U{x-p) {ax+g). If the first term in U is vIa'"' we find at once on equating coefficients of y;m + 2 -ti^at »i = 2. We may therefore take U=Ax''- + 2Bx + C, so that {{x-p) {2Ax + 2B-\) - Ax"' - 2Bx - C} (ax"' + 2gx+c) = (x -p) (ax+g) {Ax^ + 2Bx+C). Now ax^ + 2gx-\-c is not divisible by ax+g, as in that case it would be a perfect square. Hence either ax'^ + 2gx + c and Ax- + 2Bx-\-C differ only by a constant factor, or they have one factor ^— 5- in common, and x-p divides ax'^+2gx+c If x — t is the second factor of Ax^ + 2Bx + C we must have 2Ax + 2B-l = 2A{x-t). Dividing out by xl {x—p){x — q){x — t) we obtain a{2{x-p)- {x-q)}=ax+g, or a{q-2p)=g, i.e. 2q-Ap= —q-p, q=P, which is not the case. Hence the only possible case is that in which ^'iax^' + igx + c) h = const {x-p)sj{a.v^ + 2gx + c) ' x-p ' where ap- + 2gp + c = 0. It is easily verified that this equation is actually satisfied, the value of the constant being lU(g'^ — ac). The formula is equivalent to j{x-p)\l{x-p){x-q)~ q-pSj \x-pj ' (ii) The result of the preceding paragraph also enables us to supply a strict proof of the two fundamental theorems stated without proof in II. 5 ; viz. (a) e' is not an algebraical function of x : (6) log X is not an algebraical function of x. * Greenhill, A Chapter in the Integral Calculus, p. 18. The same method may f dx be applied to the integral 1 -. r — ; — - — (r>l). 9-10] ALGEBRAICAL FUNCTIONS 35 If y = log^, x^ev, and if x is an algebraical function of y, y is an algebraical function of x. It is therefore sufficient to prove that [dx y-- is not algebraical. If y is algebraical it is a rational function of x and l/jr, i.e. of X. That is to say log.^' = A',/A'o (1) where X^ and Xo are polynomitils. It is not difficult to show by purely algebraical reasoning that the equation \_ X{X,-X^Xi X X^ obtained by diflferentiating (1), is impossible. But it is simpler to argue otherwise. The right-hand side of (1) either tends to a finite limit for x=cc or becomes infinite or vanishes like a power of x, viz. .r'""", where m and n are the degrees of A'l and X.^. On the other hand log.r tends to infinity with X, but more slowly than any power of x. Hence log .r is not rational, and therefore not algebraical. Not only is it impossible that log^fc- should be algebraical but also it is impossible that any sum of the form 2^4j.log(:r-aj.), where all the a's are diflferent, should be algebraical (and therefore, by v. 8, rational). The reader should by now be able to prove this for himself, or he can refer to Liouville's proof of this and a number of more general theorems in the memoir referred to on p. 5. It is this result which was assumed in iv. 3. (iii) If f-3y + 2x = the integral ^ydx is algebraical and equal to l{6xy-3f-)*. 10. The general theorem of 8 gives the first step in tlie rigid proof of Lcqdaces principle stated in iii. 2. On account of the immense importance of this principle we repeat Laplace's words — 'I'integrale dhtm fonctioii differentielle ne peut contenir d'autres qiiantites radicaux que celles qui entrent dans cette fonction! This general principle, combined with arguments similar to those used above (v. 9 (i)) in a particular case, enables us to prove without difficulty that a great many integrals cannot be algebraical, notably the standard elliptic integrals r dx [ If ^-^' \j { ^^ j^{(l _ ^) (1 _ y^v) ' ^ j \/ U - ^a?) '^' JV {^0? -g^- gz) ' which give rise by inversion to the elliptic functions. * This is easy to verify. A .synthetic proof following Liouville's general line of argument will be found in a memoir by Baffy {Annales de I'Ecule Nurmale Superieure, p. 185), 3—2 36 ALGEBRAICAL FUNCTIONS [V 11. We must now consider in a very summary manner the much more difficult question of tlie nature of those integrals of algebraical functions which are expressible in finite terms by means of the elementary transcendental functions. In the first place, no integral of ani) algebraical function can contain any exponential. Of this theorem it is, as we remarked before, easy to become convinced by a little reflection, as doubtless did Laplace, who certainly possessed no rigid proof. The reader will find little difficult}^ in coming to the conclusion that exponentials cannot be eliminated from an elementaiy function by differentiation. But we would strongly recommend him to study the exceedingly beautiful and ingenious proof of this proposition given by Liouville*. We have unfortunately no space to insert it here. It is instructive to consider particular cases of this theorem. Suppose for example that \ydx\ where y is algebraical, were a polynomial in o: and e^, say 22a,„,„A-»'e'« (1). When this expression is differentiated e^ must disappear from it : otherwise we should have an algebraical relation between x and e^. Expressing the con- ditions that the coefficient of every power of e^ in the differential coefficient of (1) vanishes identically we find that the same must be true of (1), so that after all the integral does not really contain e*. Liouville's 2)roof is in reality a development of this idea. The integral of an algebraical function (if expressible in finite terms) can therefore only contain algebraical or logarithmic functions. The next step is to show^ that the logarithms can only be logarithms of the first order, i.e. simple logarithms of algebraical functions, and can only enter linearly, so that the general integral must be of the type jydx = t + A log u + B log v+ ... + K log ic, where A, B,..., K are constants and t, u, v,.. ,w algebraical functions. Oidy when the logarithms occur in this simple form will dificrentiation eliminate them t. Lastly it can be shown t by arguments similar to those of 8 that t, n,..., w are rational functions of x and jj. Thus ^ydx, if a finite eleraentai-y function, is the sum of a rational function of x and y and of certain constant multiples of logarithms of such functions. We can suppose that no two of A, B,...K are commensurable, or indeed, more generally, that no linear relation Aa + B(3+ ... + Kk = 0, * Journal de VEcole Polytechnique, t. xiv. cahier xxiv. p. 46. The proof may also be found in Bertrand's Calcul Integral, p. 99. t Liouville and Bertrand {loc. cit.). 11-12] ALGEBRAICAL FUNCTIONS 37 with rational coefficients, holds between them. For if such a relation held we could eliminate A from the integral, writing it in the form Ji/dw ^t + B log (;vii '')+...+ ^ log {wu '^). The case of greatest interest is that in which y is a rational function of .V and JJT, where X is a polynomial. As we have already seen, 2/ can in this case be expressed in the form where P and Q are rational functions of a-. We shall suppress the rational part and suppose that y^Q/JX. In this case the general theorem gives J-^(?^ = ^'+^.+ ^ log {a + /3JX) + B log (y + SJX) + ..., where S, T, a, ft, y, 8, ... are rational. If we differentiate this equation we obtain an algebraical identity in which we can change the sign of JJT. Thus we may change the sign of JJT in the integral equation. If we do this and subtract we obtain (after writing 2 A, ... for y1,...) which is the standard form for such an integral. It is evident that we may suppose a, /?, y,... to hQ polynomials. 12. (i) By means of this theorem it is possible to prove that a number of important integrals, notably the integrals are not explicitly expressible in finite terms, and so represent genuinely new transcendents. The formal proof of this was worked out by Liouville*; it rests merely on a consideration of the possible forms of the differential coefficients of expressions of the form and the arguments used are purely algebraical and of no great theoretical difficulty. The proof is however too detailed to be inserted here. It is not difficult to find shorter proofs, but these are of a less elementary character, being based on ideas drawn from the theory of functions t. * Journal de VEcole Polytechnique, t. xiv. cahier xxni. p. 37. t The proof given by Laurent {Tniite (VAnalyse, t. iv. p. 1;33) appears at first sight to combine the advantages of both methods of proof but unfortunately will not stand a closer examination. 38 ALGEBRAICAL FUNCTIONS [v The general question-s of this nature which arise in connection with integrals of the form I -prr dx I or, more generally, / ;;;vj, dx \ are of extreme interest and difficulty. The case wliich has received most attention is that in which ?« = 2 and X is of the third or fourth degree, in which case the integral is said to be elliptic. An integral of this kind is cciUed psetido-elliptic if it is expressible in terms of algebraical and logarithmic functions. An example was given above (v. 7). General methods have been given for the construction of such integrals, and it has been shown that certain interesting forms are p.seudo-elliptic. In Goursat's Cours d' Analyse*, for instance, it is shown that if f{x) is a rational function such that then /' f(x)dx J^{x(l-x){l-k^xy\ is pseudo-elliptic. But so far no absolutely complete method has been devised by which we can always determine in a finite number of steps whether a given elliptic integral is pseudo-elliptic, and integrate it if it is, and there is reason to suppose that no such method can be given. And up to the pi'esent it has not, so far as we know, been actually and explicitly proved that the function /dx is not a root of an elementary transcendental equation ; all that has been shown is that it is not explicitly expressible in terms of elementary trans- cendents. The processes of reasoning employe.d here, and in the memoirs to which we have referred, therefore do not suffice to prove that the inverse function A'=sn u is not an elementary function of ?<. Such a proof must rest on the known properties of the function sn u, and would lie altogether outside the province of this pamphlet. The reader who desires to pursue the subject fiu-ther will find references to the original authorities in the Appendix. (ii) One particular class of integrals which is of especial interest is that of the binomial integrals jx"'{ax'' + b)Pdx, where m, n, p arc rational. Putting ax^=ht and neglecting a constant factor we obtain an integral of the form \tfi{\+t)>'dt where j) and q are rational. If ^) is an integer and q a fraction r/s this can be at once integrated by putting < = »», which rationalises the integrand. If q is an integer and p=^r/s we put 1 -f < = ?(«. If p + q is an integer, a,ud p = r/s, we [lut 1 -\-t = tu'. * Pp. '264-2GG. 12-13] ALGEBRAICAL FUNCTIONS 39 It follows from TcheLichef's researches (to which references are given in the Appendix) that these three cases are the only ones in which the integral can be evaluated in finite form. 13. In V. 4. 5 we considered in some detail the integrals connected with curves whose deficiency is zero. We shall now consider in a more summary way the case next in simplicity, tliat in which the deficiency is unity, so that the number of double points is Kw-l)(/i-2)-l = i«(w-3). It has been shown by Clebsch* that in this case the coordinates of the points of the curve can be expressed as rational functions of a j)arameter t and of tJie square root of a polynomial in t of the third or fourth degree. The fiict is that the curves y'^=a-\- hx + c.^■2 + dx^^ y'^ = a-'r bx + CA'- + d^ifl + e.^•*, are the simplest curves of deficiency 1. The first is simply the typical cubic without a double point. The second is a quartic with two doable points, in this case coinciding in a taciiode at infinity, as we see by making the equation homogeneous with z, then writing 1 for y and comparing the resulting equation with the form treated by Salmon on p. 215 of his Higher Plane Curves. The reader who is familiar with the theory of algebraical plane curves will remember that the deficiency of a curve is unaltered by any birational transformation of coordinates, and that any curve of deficiency 1 can be birationally transformed into the cubic whose equation is written above. The argument by which this general theorem is proved is very much like that by which we proved the corresponding theorem for unicursal curves. The simplest case is that of the general cubic curve. We take a point on the curve as origin : then the equation of the curve is of the form (a, b, c, d\x, yy + {e, f g\x, yf + (h, k^a; y) = 0. Let us consider the intersections of tliis curve with the secant ?/ = te. Eliminating y we see that w is given by («, b, c, dll, tfaf + (e,f glh t)Kr + (/^ ^^1, = 0. Hence the only irrationality which enters into the expression of .r, and so of y, is JiK^,/, 9lh OT-4(/^ m, t)[(a, b, c, dll, t)% A more elegant method has been given by Clebsclit. If we write the cubic in the form L3IN^F, * Crelle, h. G4, p. 210. t Hermite, Cours, pp. 422-425. 40 AI.GEHRAICAL FUXCTIONS [V where L, M, N, P are linear functions of x and y, so that X, M, N are the asymptotes, the hyperbolas LM= t will meet the cubic in four fixed points at infinity, and therefore in only two points which depend on t. For these points LM^t, P^tN. Thus if the curve is a? + y - ^(ixy +1 = 0, so that L = ox + (1^-1/ + a, 3/= w-.r + w?/ + «, N=x + y + a, F = d^-1, oi being an imaginary cube root of unity, we find that the line x + y + a = {a^- l)/t meets the curve in the points given by _ b-at JsT _ h-at _ VsT ^~ 2t - U ' ^~ 2t ^ 6t ' where b-a^-1 and T = W - da-f + Gabt - b'. In particular for the curve a^ + y' + 1 - 0, 14. It will be plain from what precedes that JJi {x, ^/(a + bx + cx^ + dx^)} dx, can always be reduced to an elliptic integral, the deficiency of the cubic y = a + bx + cx" + dx^ being unity. In general integrals associated with curves whose deficiency is greater than unity cannot be so reduced. But associated with every curve of (let us say) deficiency 2 there will be an infinity of integrals JR (x, y) dx reducible to elliptic integrals or even to elementary functions ; and there are curves of deficiency 2 for which all such integrals are reducible. For exami>le /^ (■?', J.i^ + «.r* + bx" + c) dx may be sj)lit up into tlie sum of the integral of a rational function and tw'O integrals of the type /' ^1 (.r-) dx f XR2 (.r-) dx \j{af + ax' + bx- + c) ' jJix^ + aa^ + bx' + c) ' I 13-15] ALGEBRAICAL FUNCTIONS 41 and each of these becomes elliptic on putting or = t. But tlie deficiency of 'if' = x^ + ax^ + bx^ + c is two. Another example is given by JR {x, slx^ + ax^ + bx^ + ex + d) dx*. 15. It would be beside our present purpose to enter into any detail as to the general theory of elliptic integrals, still less of the integrals (usually called Abelian) associated with curves of deficiency greater than unity. We have seen that if the deficiency is miity the integral can be transformed into the form jn (x, ^x) dx where JT = x* + ax? + bx" -^ cx + d^. It can be shoAvn that by a transformation of the type at + p ^~ yt + ^ this can be transformed into an integral JBit,JT)dt where T = t' + Af + B. We can then, as when T is of the second degree (v. 3) decompose this into two integrals of the forms [B (t) dt fR(t)dt, f^ of which the first is elementary while the second can be decomposed i into the sum of an algebraical term and certain multiples of the integrals 'dt (fdt [dt [ t'dt and of a number of integrals of the type dt k These integrals (v. 12 (i)) cannot in general be reduced to elementary functions, and are therefore new transcendents. * See Legendre, Traite des Fonctions Elliptiques, t. i. Cli. xxvi-xxvi., xxxii-xxxiii. ; Bertrand, Calcul Integral, p. 67, aiul Enneper, Elliptisclw Fttiik- tionen, Note 1, where abundant references are given. t There is a similar theory for curves of deficiency '2, in which A' is of the sixth degree. X V. e.g. Goursat, Coins iVAitahjse, t. i. pp. '2.55-207. •42 TRANSCENDENTAL FUNCTIONS [VI We will only add, before leaving this part of our subject, that the algebraical part of these integrals can be found by means of the elementary algebraical operations, as was the case with the rational part of the integral of a rational function, and with the algebraical part of the simple integrals considered in v. 3 and v. 9. VI. Transcendental functions. 1. The theory of the integration of transcendental functions is naturally much less complete than that of the integration of rational or even of algebraical functions. It is obvious from the nature of the case that this must be so, as there is no general theorem concerning transcendental functions which in any way corresponds to the theorem that any combination of algebraical functions, explicit or implicit, may be regarded as a simple algebraical function, the root of an equation of a simple standard type. One may almost say that there is no general theory: the theory reduces to an enumeration of the few cases in which the integral may be transformed by an appi-opriate substitution into an integral of a rational or algebraical function. These few cases are however of immense importance in the applications of the general theory of integration. 2. (i) The integral where F is an algebraical function, and a, b, ...,/• commensurable numbers can always be reduced to that of an algebraical function. In particular jRie'^, g^^, ...,(^)dx, where R is rational, can always be calculated in finite terms. In the first place a substitution of the type x = ay will reduce it to the form SR.{ey)dy, and then the substitution e" = z reduces this to the integral of a rational function. In particular, since cosh^ and sinh.r are rational functions of e', and cos.r and sin.r are rational functions of e'~', the integrals \R (cosh X, sinh .v) d.r, JR (cos x, sin x) dx can always be evaluated in finite form. In the case of the latter 1-2] TRANSCENDENTAL FUNCTIONS 43 I integral the substitution indicated above is imaginary, and it is ' generally more convenient to use the substitution tan ^a; = f, which reduces the integral to that of a rational function, since 1-^' . 2t , 2dt cos^ = - -, , sui a' = r. , aw 1 + ^ 1 + t-' 1 + f (ii) The integrals jR (cosh X, sinh x, cosh 2x, sinh mx) dx, jR (cos X, sin x, cos 2x, sin mx) dx, are included in the two standard integrals above. Let us consider some further developments couceruing the integral \R (cos X, sin x) dx*. If we make the substitution 2=e'^ the subject of integration becomes a rational fmiction II (z), which we suppose split up into (i) a constant and certain positive and negative powers of z, (ii) groups of terms of the type ^ I ^1 I , ^n (I) z-a'^ {z-ay- •■• "^(s -«)» + ! ^ ^' The terms (i) when expressed in terms of x give rise to a term 2 {c/c cos kx + dk sin kx). In the group (1) we put 2; = e*'*^, a = 6'", and using the equation J_=le-'■'^f-l-^•cot•^') z-a ■■ \ 2 / we obtain a polynomial of degree n + \ in cot \ {x - a). Since cot2 x=-\ , — , cot-^ ^^ = - cot .r - - ;i- (cot- .r), . . . dx ''i a.t this polynomial may be transformed into the f(jrm C+ Co cot i (.r - a) + (7i ^-^. cot |(a- - a) + . . . + C„ ^, cot ^ (.r - a). The function Ii (cos x, sin x) is now expressed as a sum of a number of terms each of which is immediately integrable. The integral is a rational function of cos a; and sin a- if all the constants C„ vanish ; otherwise it includes a niuuber of terms of the type 2fologsin i {x-a). * Hermite, Cours, pp. 320 et seq. 44 TRANSCENDENTAL FUNCTIONS [VI Let us suppose for simplicity that H{z) when split up into partial fractions contains no terms of the types C, 2"^, 2-^ \l{z-af (jD>l). Then R (cos Xy sin x) = Co cot |(^ - a) + -Dq cot ^ (.r - ^) + . . . , and tlie constants Cq, Dqj ••• ^re easily assigned by considering the behaviour of the function R for values of x very nearly equal to a, ^, It is often convenient to use the equation cot \ {x — a) = cot {x — a) + cosec (.r - a) which enables us to decompose the function R into two parts C (x) and V (x) such that U {x + 7r)=U (x), V {X + tt) = - F (x). If R has the period n, it is easy to see that V must vanish identically ; if it merely changes sign when x is increased by tt, (7 must vanish identically. Thus we find without difficulty that if ??i I 6 I we suppose a positive and use the transformation {a + b cos x) {a — b cosy) = d- - b-, which leads to dx _ dy a-\-bco&x ^{a^~b'^)' If I a I < I 6 1 we suppose b positive and use the transformation (6 cos X + a) {b cosh y - a) = b'' — cfi. dx The integral k- ', + b cos x-\-c sin x maybe reduced to this form by the substitution .7; + a=y, where cota = 6/c. The integrals dx f dx f ■b cos-r+csina?)" may be at once deduced by diflferentiation. The integral dx (A cos^ X + 2B cos .r sin x + C sin^ .r)" !• 2-4] TRANSCENDENTAL FUNCTIONS 45 is really of the same type, since A cos^ x-\-%B cos X sin x+C sin- x = h{A + C) + ^ {A - C) cos 2x + 5 sin 2.r. And similar methods may be applied to the corresponding integrals wliich contain hyperbolic functions, so that this type includes a large variety of integrals of common occurrence. (iv) The same substitutions may of course be used when tlie subject of integration is an irrational function of cos.r and sin.r, though sometimes it is better simply to use the sulistitutions co^x = t or sin.r = /. Thus J/i(cos.r, sin .r, ^IX)dx, where A'=(«, />, c, f\ g, /if cos x, sin.r, 1)^ is reduced to an elliptic integral by the substitution tan|.i' = <. The most important integrals of this type are fH (cos X, sin x) dx [ ^ (^*^s -^j si'^ ^) ^^ J v'( 1 - ^^ siii^ .v) ' _/ V(« + h cos X 4- c sin x) ' 3. The integral where «, h, ...,/■ are S' {x) is a rational function ; (ii) a number of terms of the type fe'dx a I , J .I'-a If all the constants a vanish the integral can be calculated in the finite form e^S^x). If they do not we can at any rate assert that the integral cannot be calculated m this form. For no such relation as J X- a j x — b J X- k Avhere T is rational (or even algebraical) is possible. To see this it is * V. Hermite, Com;-*' d' Analyse, p. 352 et seq. 4] TRANSCENDENTAL FUNCTIONS 47 only necessary to put x = a^h and to expand in ascending powers of h. For J x~ a J h = ae" (log h + h+ ...), and no logm-ithm can occur in any of tlie other terms. Consider, for example, the integral This is equal to ,j A. I L and snice 3 \^dx= h 3 \ — dx\ J x^ X J X ' J x-^ 2.r^ 2 _/ x^ 2x^ 2x 2 J x ' we obtain finally Similarly it will be found that /„.(i-iyS" = 0, 2S:x +T'^E, Tx + I r' + 2 Bklix - a^) = 0. Hence .S is a constant, say C, and -/(-^O- We can always determine by means of elementary operations (iv. 3) whether this integral is rational for any value of C or not. If not the given integral cannot be expressed in finite form. If the integral is rational we calculate T. Then --/f-^}^. must be rational, for some value of the arbitrary constant implied in T. "We [T can calculate the rational part of \ — dx: the transcendental part must be cancelled by the logarithmic terms 25fclog(.t'-a;t). The necessary and sufficient condition that the integral .should be an elementary function is therefore that R should be of the form where Ri is rational. That the integral is in this case such a function becomes obvious if we integrate by parts, for In particular (i) n^^^dx, (ii) I'^f^^^dx, ^ j x-a J (a' - «) ('■''' - b) are not elementary functions unless in (i) a— 0, and in (ii; 6 = a. If the integral is elementary the integration can always be carried out, with the same reservation as was necessary in the case of rational functions (iv. 5). It is evident that the problem considered in this paragraph is but one of a whole class of similar problems. The reader will find it instructive to formulate and consider such problems for himself. f)-"] TRANSCENDENTAL FUNCTIONS 51 7. It will be obvious by now tliat the number of classes of transcendentcal functions whose integrals are always elementary is very small, and that such integrals as j"/(.r, }^ LD 21A-60m-10,'65 (F7763sl0)476B p LD 21-100m-9,'48iB3998l6)476 General Li bran University of Calif Berkeley ..""ins^lfmM.* THE FUNDAMENTAL THEOREMS OF THE DIP FBRENTIAL CALCULUS, by W. 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