VARIABLE 73 logarithm varies continuously, the value of log(l + P) will not have changed by a multiple of 1m when we return to the point we set out from. But, since the point 2 = is i within the circle, log(a n n ) = nlog(a n "z) changes its argu ment l>y n. 2-i in consequence of the circuit. Hence log/fc) changes by Imn : that is, in the circle of conver gence whose radius is as large as may be required there are always n values for which the rational function a + a l z + a^ 2 + ... + a, L z n vanishes. Some of these may be coincident ; but in all cases the total number of the orders of the different vanishings is n. 13. If two given points z (} and ^Tbe joined by any curve of finite length, and if we take as many intermediate points as we please z v z.,, . . . ?. n - upon the curve and form the sum then the complex limiting value to which this sum tends as n is increased indefinitely is called the definite integral of the function f(z), formed from z to Z along the path prescribed by the equation of the curve. Having first examined, after the manner of the theorems regarding real integrals, under what conditions this sum has a determinate limiting value, we have next to take into account that a definite complex integral differs essentially from a real one in that the path along which the variable z may travel from one limit to the other is perfectly arbitrary. We have therefore to investigate this problem : When is the integral of a complex function a unique function of its superior limit independent of the path of integration 1 ? In order to answer this we must recall an important theorem concern ing the reduction of a double integral of a function to .simple integrals along the boundary curve, which is in fact a case of Green s theorem (see INFINITESIMAL CALCU LUS, vol. xiii. p. 58) when confined to two variables. 14. Suppose P and Q are two real functions of x and y which are everywhere within a certain boundary finite and continuous, the theorem is then that the double integral frf&Q &P 11 ( ?r- - ) dxdy taken over the entire region is equal to the simple integral (Pdjc, + Qdy) taken along the entire boundary of the region. Taking the positive directions of the axes as usual, we define the positive direc tion along the boundary to be that for which the bounded surface is on the left hand; then, i d we have to exclude any portion, c.y., a circular space altogether within the external boundary, so that the points within it are to be regarded as not included in the re gion, the positive direc tion of the boundary is that indicated by the arrows in fig. 4. In all the simple integrals the inte gration must be effected in the positive direction thus explained. ^ To integrate^/ -^-dxdy with respect to ^, let us split up the region into elements by parallels to the axis of . . Select any one and, reading from left to right, denote the values of Q where the element crosses the boundary at the entrances by () p Q.,, etc., and at the exits by rSj/} </, Q", itc. ; then I ^ c .>- ^ -Q l + Q - Q, + Q" - etc., Now in each of these integrals y goes through all its values from the least to the greatest, therefore dij is always to be taken positively. But, denoting by dy v dy.^ etc., and by dy , dy", etc., the projections on the axis of y of the arcs of the boundary cut by the element as above, when we take the positive direction of the boundary into account, we have dy dt/ l - dy., = etc. = + dy = + dy" = etc. ; tlius where this is taken along the entire boundary in the positive direction. In like manner, dividing the region into elements parallel to the axis of y, and denoting the values of P at the entrances, proceeding from below up wards, by P v P. 2 , etc., and at the exits by P, P , etc., we have JJ* -fatted y = - I Idx + lpL<:- lP 2 djc + &c., where dx is positive. Hence, as before, taking account of the positive direction of the boundary, dj: = + dj. = + (],<-., = etc. = -<7,/=- d.c" = etc., and consequently JTj-djcdy .r - fp, l.c., - etc. = - jPds, taki takin the integral in the positive direction along the entire boundary. Accordingly, putting these both together, we have the surface integral ff( -^ - j-d.nly = I (Pd.r + Qdy), taken positively along the entire boundary. This proposition may be extended to complex values : when P = P + iP" and Q = Q + iQ"-, as the proposition is already proved for real /T/&O 8P /r/SO 8J" values, Ave can put // * ---- ? djcd y = // ( -^ ---- = dxdy JJ ox by/ J JJ ojc oy / y ) then, applying it to each of these, JT(jr - j) djcdy = I (Pels + Q dy) + i l(P"dx + Q"dy)
oJJ oy / J J
= I (Pdx+ Qdy). We assumed that there were no branch points within the region, or other points at which P or Q are discontinuous. If there were such, we should have to surround them with actual closed lines, as small as we please, and thus exclude them, introducing these lines as parts of the boundary of the region. 15. We can now let P=f(z), Q = if(z). We see that, if f(z) be an analytic function of z without exception in any simply connected region, //(;)</? = when taken all round the boundary of that region. If Ave take any two paths from ~ to Z in this region, since these paths from Z Q through A t Z and back through B to include such a region, we have I(z A ZHz ) = or I(z A Z) + I(Zlh^ = 0. But, as the value of the integral I(ZBz^)= - I(z^BZ) along the same path, the last equation becomes I(z AZ} = I(^BZ), or the value with which the integral starting from ? arrives at Z is inde pendent of the path travelled under the conditions sup posed : it is a function of the upper limit only. The path z^BZ is said to be reconcilable with the path z^AZ. If the region within which f(z) is an analytic function be multiply connected, u{z)dz vanishes when taken positively all round the boundary. The integral has a definite value along each separate boundary curve. For instance, if within the region we take a closed curve which includes a simply connected region, as a in fig. 5, the integral along this curve vanishes. But for a curve which includes a region along with a boundary curve, as /? in fig. 5, the in tegral has the same value as it has for this boundary curve. Whence, if a function f(z) which has to be integrated lose the character of an analytic function in isolated places,
XXIV. 10
