(ax + b) aig' II. Rational functions of x and — cx -1-d lax+b\ GIP' ax +b cx-1-4 cx+d Put — =wit, L.M.C. of /3, fr,.... Herewith the f becomes rational in u.
III. Rational functions of x and V ax'+2bx+c.
If we think of as a conic, and y—d3=u(x---a) as a secant through a point (a, /3) of the conic, then we may express both x and y rationally through u, which reduces this case to I; (a, /3) may be taken variously. Generally we bring y' to the form of sum or difference of two squares by putting s=z+ h.
IV. Rational functions of x and y, these being co-ordinates of a unicursal F(x, y) =0. We shall then have x-='0(u), y-0(u), where # and # are rational, whereby these Abelian Integrals reduce to I.
The binomial X'n ( a+ bxs )P can be reduced to H, and hence integrated, not generally, but in these important cases, by putting u=x92.
m + 1 r 1. p integral; if =T (r and s integers), Vu put or = z.
2. m -1- 1 s r integral; if p= - (r and s integers), put 4/a+biz.
m+1 3. — +p integral; if (r and s in- n tegers), put a+ bu= If f(x) be rational in x,Vax+b,\:Fb,put which reduces it to 3.
Of Transcendentals.— I. Rational func tions of sin x and cos x. Put u=tan, a very important substitution.
2. Rational functions of eax. Put u=eax.
3. Rational Integral functions of x, eax, ebs, ... sin mx, sin nx, ..., cos mx, cos nx, Express the sines and cosines through imagi nary exponentials. In the result express the imaginaries through sines and cosines. Here are Included rational functions of the hyper sine and -cosine.
4. Rational Integral functions of x and log x, or x and Put or u.
If f (x)=R (x, VT), T of 3d or 4th degree in x, we cannot rationalize but must introduce Higher Transcendents. Let dx f dx z V (x--ea) (x—e2) Here x and u are functions of each other and it seems natural to take a as function, x as dx argument; but in l= V sin / is a much simpler (periodic) function of I than I is of x; hence we may suspect that .r above is a simpler (periodic?) function of it. than u of x. Hence Abel thought the theory might be simplified by inverting the dependence before him assumed— one of the greatest div ination in mathematical history. We write dx r=1)(u) (Weierstrass), st= fr(u) V T =9 (x).
Hence Up-- 1 and so on. Now just 711kuil as sine and cosine have one period 2r, so has two periods, 2 w and it is an Elliptic or Doubly periodic function. the Theory of such Functions, one of the most august creations of the last century, is conspicuous in Analysis. Of Hyper-elliptic Integrals there is no space to.
speak The integral of an Infinite Series may be found by integrating term by term only when the series converges uniformly within an inter val comprising the extremes of the integration.
It is seen that the integrable forms are absolutely many, relatively few, the integra tion generally giving rise to'a new function.
Thus far we have raised no question as to Integrability, the Integrand being supposed unique, continuous, finite and therefore in tegrable, in [a, b]. But when, if ever, may we let one or more of these conditions fall? As to continuity, Riemann has discussed pro foundly, In what cases is a function integrable, and in what not! and still further precision has been attained by Du Bois Reymond and Weierstrass.• It is of particular interest to know whether 2ydx will vary finitely with varying modes of divisions of [a, b]. Riemann calls the subintervals di, d, . On; the greatest fluctuation of function-value in each dk he calls Dk; then must IttkEok be infinitesimal. Thence it follows that when, as each clk sinks in definitely toward 0, the sum of subintervals, in which D is >a, itself is infinitesimal, then the Sum has a definite limit, the same how ever [a, b] be subdivided. Hence the integral I Lk (xo) die= f (x) dx exists when f (x) (T,-o =o is finite and unique in [a, b], and when for every infinitesimal positive e there is also a posi tive d such that e when each f3k< d.
Plainly such is the case (1) for f(x) continu ous throughout [a, b]; but also (2) when f (x) is finitely discontinuous at a finite number of points in [a, b], and when f(x) has an ap of maxima and minima, or is quite undeter mined (though finite) at a finite number of points in [a, b), as sin at 1 and 2; (3) even when f(x) is discontinuous or finitely indeterminate at an .0 of points and has an ca of maxima and minima in the vicinity of an ea of points, provided only all these points of finite function-fluctuation form not a linear but only a discrete mass of points (Punktmenge) — the function is then said to be only point wise (punktirt) discontinuous dis crete mass or manifold of points is an be of points in a finite interval [a—h, so distributed at subintervals that the sum of these subintervals may be made small at will by enlarging at will the number of subintervals. Otherwise, the mass is linear. Functions linearly discontinuous are not integrable. For Cantor's more comprehensive theory of Derived Masses ((Math. Ann.,' XVII, 358f), see As semblages.