12. When in the definition of § 1 more than one value is assigned to each point of S, the first thing to be done is to group these values in such a way that they form several single-valued functions. Consider for example the function V—z. Setting e'14, we have R=V—r, 0=0, 40+ 7r.
If now we sever the plane along a line running from 2=0 to z=po, say along the positive axis of reals, we can then spread out two single valued functions each analytic in this region, namely, efri — to1=V r e2 and to, # and these two functions together just exhaust the values of the multiple-valued function 1/ s. Let us, therefore, consider two planes instead of one, each severed along the cut in question and lying one above the other like the leaves of a book, and assign the values tv, to the points of the one plane, w, to those of the other. We observe, furthermore, that the values of tv, along the upper side of the cut in its leaf and those of w, along the lower side of the cut in the second leaf agree with each other. Let these two edges be united, so that z, moving continuously in the surface thus formed, will pass from leaf I to leaf II without interruption when it crosses the positive axis of reals from the upper side. Similarly the values of w, along the lower edge of the cut in leaf I and those of w, along the upper edge of thc cut in leaf II correspond. Hence we shall unite these two edges. The final result is a closed surface of two sheets in which the function -Te is single-valued. The sheets cut through each other along a line; but the point z in crossing this line is not at liberty to pass into either sheet. z has not come to a fork in the road, but to a switch which has already been set, and its further course is definitely determined.
The point r..---- 0 is called a branch-point, the cut a junction-line.
More generally, let G(w,$)=-Ao(s)wIn+Ai(s)unn—i+ Am(s) ---- 0 be an algebraic equation. Then to each value of z correspond in general in distinct values uh, . . . , tom. Mark the points a, , .. , (finite in number) for which this is not the case. Draw a curve C, not cutting itself, through these points and continue C from the last point to s= . Then spread out m leaves over the
s-plane and cut each leaf along C. The m values . . . , wm of the function can now be assigned to the points of these leaves so as to form m single-valued functions analytic throughout these regions. Finally, correspond ing edges of the leaves are to be connected along each of the segments into which C is divided by the points al, . . . , aK,00 . The re sult is a closed surface on which w is a single valued function of z. If G(w, z) is irreducible, the surface will consist of a single piece, and conversely. A branch-point in which tg leaves are connected in cycle is said to be of order P — 1.
The surfaces here described are due to Rie mann (Gottingen, thesis for the doctorate, 1851) and are known as Riemann's Surfaces. Corresponding to any given analytic Junction there can always be constructed a Riemann's surface on which the function becomes single valued. In all ordinary cases the leaves are joined together in cycle in branch-points, and they always pass over into each other along junction lines, never merely at isolated points. See ANALYSIS SITUS.
13. Analytic Continuation.— If a function f(z) be single-valued and analytic in a region S, it may happen that the function can be 49 FIG. 4 defined in an adjacent region 5, in such a way that the extended function is analytic through out the enlarged region (S, Si). In this case f(z) is said to be continued analytically beyond the region S into the region Si. If S, overlaps S, it will in general be necessary to introduce a Riemann's surface for the extended function.
Consider, for example, a function given by the power series (4). Let so 4: 0 be an interior point of the circle of convergence K. Develop the function by Taylor's Theorem about the point so. The new power series, which pro ceeds according to powers of z — zo, will surely converge within a circle tangent internally to K, and it may converge throughout a larger circle In the latter case analytic continuation is possible, the region Si corresponding to the part of K, exterior to K.