VARIABLE 71 of j , with the complex we can travel along any path whatever leading from one point to the other. 5. When the values of one complex variable 10 = u + iv are determined by the values of another z = x + iy } in such a way that for each value of z within a definite region one or more values can be assigned to 10 by performing a defi nite set of mathematical operations on z, then 10 is said to be a function of the complex variable z. Functions are distinguished as one-valued and many-valued according to the number of values of w which belong to one value of z, as algebraic and transcendental according to the form in which the variables appear, and as implicit and explicit according as the relation is solved for w or not. The course of the function w will accordingly be illustrated by points in another plane or planes having the rectangular coordi nates 11 and ? , and its dependence on z will be geometri cally a transformation from one plane to the other. A very simple instance is when w= 1/z, where the transforma tion is equivalent to geometric inversion. If the explicit function n<=f(.-) is to be unrestrictedly continuous in the region for which it has definite values, there must be a finite connected area, however small, around any admis sible value of .?, to which corresponds a connected region of w : that is, u and v must vary continuously when x and y do so. Let z be increased by A? = A,r 4- i?y and put w + A?/> == (n + A?/) + i(i< + A; 1 ) =f(z + A*) ; then in whatever way A,/; and A// may converge to zero we must have lim A?< = and limAr = : in other words, iv is continuous for a value z when this value can be included in a region such that for every value z + Az within it the modulus of the difference mod(A>) = mod[/ (,2 + Az) -/(z)] = /A?i 2 + Ay 2 is less than any assignable small number. 6. We proceed to give some elementary examples of functions. Ex. 1. The power with positive integer exponent. w=z m is a one- valued (monotro2)e) function which is continuous for the entire plane, since u = p m cosmO, v=p m sinm0 are continuous functions of p and 6 which do not alter when the argument 6 is increased by multiples of 27T. When 2 becomes infinite so does w, and thus this value z is a singular one (pole] for the function, as contrasted with a regular one, which is such that a function has a finite value continuously changing with 2. Ex. 2. Any rational integer function of the nth degree in 2. /(,-.) = rt + 1 3 + rto~+ . . . +a n ~" is one-valued and continuous for the entire plane and has no singularity, except for z oo, for which it becomes infinite (pole). Ex. 3. The power with a rational fractional index. 2(0 + 2/,-Tr) + i sin^(0 + Ik* <! 1 where k takes all integer values from to q- 1. Each value of k determines at each spot the value of a branch of the function, which is therefore a many -valued (polytroj)e) one. At the points 2 and 2 oc all branches have the same value. Thus, if p/q>Q, at 2 = the value of w is ; at 2= oo its value is oo. These two points are called branch (critical) -points of the function. If we wish to contemplate the values of the function which belong to one branch, so as to have each branch by itself in general a continuous function, we employ the method of Cauchy : that is, we draw from the origin to the point infinity a curve which does not enclose any space, for instance, the positive part of the axis of x. Suppose we let p=l, y = 2, then w = pW )+ - k * ) i , where & = or 1. Taking 7,- = 0, the values of the independent variable and of the function along the prime axis for = are 2 = p and w = p* ; along the axis of y when
- = 7r/2 they are c = pi and w=p^ iri ; along the negative axis of x
when = ?r they are z - p and w = p$e^ wi ; along the negative axis of y, when O Sir/ Z, ,:= -p/and yr=pM rt ; and along the prime
- ixis, on completing a rotation, when d = 2ir, z = p and ic - p}.
Hence, for any value of p, w is continuous all round the circle, but its ultimate value is different from its initial value. Thus, as will happen in general with many-valued functions along their branching sections, the values be longing to points alongside of the prime axis having posi- tive ordinates, however small, differ by finite amounts from those for the corresponding points, however near that line, having negative ordinates. The branch of the function so constructed is therefore discontinuous along the prime axis. Accordingly we conceive the plane cut through along this prime axis, and the branch of the function is easily seen to be continuous in the connected surface, which con sists of the infinite plane so cut through. Riemann per fected this method of Cauchy and enabled us to keep all the branches simultaneously in view, and thus regard the function as unique and continuous upon all admissible paths without restriction. His process is in the case of a ^-valued function to let the variable z move upon q different plane leaves. Thus, as above, when p = l, g = 2, taking the case k = , let us suppose 2 to move in a second plane ; proceeding as before for the points of the prime axis, we have now 6 = 0, 2 = p, w = p%e ITl = - pi ; and, when 6 = 2ir, z = p and ; = pic 27ri = pi. Thus, though the rotation has brought w round continuously from the values for small positive ordinates to those for small negative ones, these last differ finitely from the first. Now these final values in the second plane are the same as the initial ones in the first plane ; moreover, the final values in the first leaf are the same as the initial ones in the second ; hence, if the second leaf be cut through / all along the prime axis, as the ,/ first was, and if we conceive the L. ^ leaves to cross one another (as & ** indicated in profile in fig. 3) all along the cut, we have a connected two-leaved surface, called by Riemann a winding surface of the first order, to each point of which corresponds a definite value of the function w = z$, and to each continuous curve in it, whether in the same leaf or passing through both, correspond values of the function which vary continuously ; every closed curve which crosses the branching section either an even number of times or not at all leads finally back to the initial value. This conception can evidently be extended to surfaces with a greater number of leaves corresponding to functions admitting of a greater number of values. 7. In considering infinite series we must explain the con ception of absolute convergence introduced by Cauchy : an infinite series of real quantities is absolutely convergent when the sum of its positive terms taken apart, and like wise that of its negative terms taken by themselves, are each finite. When these are not separately finite, a series which converges for some value of the variable is semi-converyent. A complex series ! (w 4- iv) converges absolutely when the u series and the v series are both absolutely convergent. Hence it can be shown that the necessary and sufficient condition that a complex series may converge absolutely is that the series of the moduli of its terms converges. Such a series presents the same value if the sequence of its terms is rearranged according to any law whatever. Thus an infinite series of ascending powers of z the complex vari able, a + rt r 2 + a.,: 2 -f . . . + z n + . . . =/(;)> Ay i tn complex coefficients, is absolutely convergent for any value of z for which the series A + A 1 p + A^y i + . . . + A n p n + . . . formed by the moduli of its terms is convergent ; and v-ice versa. In geometric language, if such a series be absolutely conver gent for the point corresponding to any value of z, it is abso lutely convergent for all other points equally distant from the origin of coordinates ; these are therefore situated on a circle. It is also absolutely convergent at all points within this circle, and therefore the region of convergence of this series is always a circle round the origin. The radius of the circle of convergence of a series is the greatest value of p for which the series of moduli converges. It is found by the con- t dition that as n increases indefinitely we must have j +1 < 1 /( ^ n As the terms of an infinite series of powers A or p >i+i are one-valued, the series itself, so long as it converges, is a one-valued function of the complex variable, which does not become infinite anywhere. It can also be proved that this function is continuous : that is, if z and z 8 be values of z for which it converges, then lim[ f(z 8) -/(?)] = 0, when 8 = 0. 8. When a variable quantity depends upon two others
in such a way that, if their values are given, its value is