822 Another formula is r.*T (1 - a-) FUNCTION - , or as this may also be written, n (x - 1) n ( - a-) = g -g^ . It is to be observed that the function n serves to express the product of a set of factors in arithmetical progression, we have (x+a)(x+2a) . . a"n ( - + m )-i-n-. We can consequently express by means of
Ctt J CL
it the product of any number of the factors which present them selves in the factorial expression of sin u. Starting from the form ?<n"( 1 + ) n" ( 1 - ) , where n is here as before the sign of l Sir J 1 SirJ a product of factors corresponding to the different integer values of s, this is thus converted into or as this may also bo written which in virtue of becomes n(-+n)-T-nroIlH. m and n are hero large and positive, and calculating the second factor by means of the formula for Hx, in this case we have the before-mentioned formula The gamma or II function is tho so-called second Eulerian integral ; the first Eulerian integral/ arP- J (l - x)i- l dx, = TpTq ~ T (p + q), J o is at once expressible in terms of T, and is therefore not a new function to be considered. 15. We have the function defined by its expression as a hyper- a./3 a.a+l.j8./8 + l geometric series r.(a, 0, y, u) = l + ^ u + ^ ^ - ^?i 2 + &c., i.e., this expression of the f unction serves ns a definition, if the series be finite, or if being infinite it is convergent. The function may also be defined as a definite integral ; in other words, if in the integral ~ l-ux) we expand the factor (1 - ux) ^ in powers of ux, and then integrate each term separately by the formula for the second Eulerian integral, the result is which is _ r<x . rB ( , a .y a. a + l. y . 7 + ! 2 1 ~T(a + B ) I a + .l * a + B -a + B + l -1 .2 * +> J or writing a, /} , y = a, y - a, B respectively, this is = J~ a - r (> & 7. )> so that tho new definition is T (a, 0, 7, ) = / x (1 - a-) (1 - ?w) Pfc ; but this is in like
- T / o
manner only a definition under the proper limitations as to the values of a, ft 7, u. It is not here considered how the definition is to be extended so as to give a meaning to the function r(o, B, y, u) for all values, say of the parameters o, B, y, and of the variable u. There are included a large number of special forms which are either algebraic- or circular or exponential, for instance r (a, B, B, u) = (1 - u) ~ a , &c., or which arc special transcendents which have been separately studied, for instance Bcssel s functions, the Legendrian functions X B presently referred to, scries occurring in the develop ment of the reciprocal of the distance between two planets, &c. 16. There is a class of functions depending upon a variable or variables x, y . . and a parameter n, say the function for the para meter n is X n such that the product of two functions having the same variables, multiplied it may bo by a given function of the variables, and integrated between given limits, gives a result = 0or not = 0, according as the parameters are unequal or equal; /UX W X n dxdy . . - , but fJX n *djs not = ; the admis sible values of tho parameters being either any integer values, or it may be the roots of a determinate algebraic or transcendental equation ; and the functions X M may be either algebraical or transcendental. For instance, such a function is cos nx ; >. fK and n being integers, we have / cos mx . cos nx dx 0. but J o /7T cos 2 nx dx = ir Assuming the existence of the expansion , of a function fx, in a series of multiple cosines, we thus obtain at once the well-known Fourier series, wherein the coeificient of t~tt cos mx is = iir / cos mx . fx dx. The question whether the pro cess is applicable is elaborately discussed in Eiemann s memoir (1854), Ucbcr die Darstcllbarkcit cincr Function durch cine trirjono- mctrische Rcihc, No. xii. in the collected works. And again we have the Legendrian functions, which present themselves as the coefficients of the successive powers of a in the development of (1 -ax + a?}~^, X 1, Xj = x, X 3 =f (z 2 -^), &c. : here m, n /~l r " being any positive integers/ X M n dx = 0, but / X n dx= . . *^ 1 ^/ 1 n + 1 And we have also Laplace s functions, &c. Functions in General. 17. In what precedes a review has been given, not by any means an exhaustive one, but embracing the most important kinds of known functions ; but there are questions to be considered in regard to functions in general. A function of x + iy has been built up by means of analytical operations performed upon x + iy, (x + iy) 2 = ar + jy 3 + i. 2xy, &c., and the question next referred to has not arisen. But observe that, knowing x + iy, we know x and y, and therefore any two given functions < (x, y), ty (x, y) of x and y : we therefore also know ( x > y) + z $ ( x > 2/)i an( l the question is, whether such a function of x, y (being known when x + iy is known) is to be regarded as a function of x + iy; and if not, what is the condition to be satisfied in order that <f> (x, y) + i ty (x, y) may be a function of x + iy. Cauchy at one time considered that the general form was to be regarded as a function of x + iy, and he introduced the expression "fonction monogene," monogenous function, to denote the more restricted form which is the proper function of x + iy. Consider for a moment the above general form, say x + iy = <j> (x, y) + i l/ (x, y}, where <p, ty are any real functions of the real variables (x, y) ; or what is the same thing, assume x = (x, y), 2/ = ^ (x, y) ; if these functions have each or either of them more than one value, we attend only to one value of each of them. We may then as before take x, y to be the coordinates of a point P in a plane n, and x , ij to be the coordinates of a point P in a plane n . If for any given values of x, y the increments of <f> (x, y), j/ (x, y) corresponding to the indefinitely small real increments h, k of x, y be Ah + Ek, Ch + Dk, A,B,C,D being functions of x, y, then if the new coordinates of P are x + h, y + k, the iiew coordinates of P will be x + Ah + ~Bk, y + Ch + Dk ; or P, P will respectively describe the indefinitely small straight paths at the inclinations tan-l- t tan-l h ively ; calling these C + D tan Ah + Kk angles 0, 6 , x to the axes of x, oi respect- wo have therefore tan tf** A + B tii & a y ^ e = ^ x + *y)* a function of x + iy, tho condition to be satisfied is that the incre ment of x + iy shall be proportional to the increment h + ik of x + iy, or say that it shall )e = ( + i/j.)(h + ik), , fj. being functions of x, y, but independent of h, k ; wo nmst therefore have Ah + B, Ch + i)k = h - p.k, fji + k respectively, that is A, B, C, D = A, - /u, ju, A. respectively, and the equation for tan tf thus becomes ^ + A tan p. tan ff = - a > ncnco writin - = tan o, where a is a func- a > . tan a + tan 6 tion of x, y, but independent of h, k, we have tan ff = i _ tan g tan that is tf = a + 6 ; or for the given points (x,y), (x ,y }, the path of P being at any inclination whatever 6 to the axis of x, the path of P is at the inclination 6 + constant angle a to the axis of x ; also (AA- M fr) 2 + (^ + A/t) 2 = (A 2 + M 2 )(ft. 2 + fc> ! ), i.e., the lengths of the paths are in a constant ratio. The condition may be written (da! what is the same thing + i dif ^r J daf iSy) . dy + ^ ^ J x < . (to+%), that is, - + i- = (+t/)i dx dx 1 di/ j consequently^- + * dx 1 + dy * dy 0;thatis, dx } dif df dx . !_ jr - ;[> ~j rf~~ i as the analytical conditions in order that x + iy may be a function of x + iy : they obviously imply dx*~ + rf^" = da* + dx*~ = ; and tf x be a fimction of x > y
