Theory of Functions of a Complex Variable (Paperback)

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1893 edition. Excerpt: ... 4. Shew that, if the surface associated with the equation have /u boundary-lines instead of one, and if the equation have the same branch-points as in the foregoing proposition, the connectivity is 2-2n + fx + 2. 179. The consideration of irreducible circuits on the surface at once reveals the multiple connection of the surface, the numerical measure of which has been obtained. In a Riemann's surface, a simple closed circuit cannot be deformed over a branch-point Let &/ /q A be a branch-point, and let AE... be the branch-line (a(e/ having a free end at A. Take a curve...GED... crossing D D' the branch-line at E and passing into a sheet different Fi 60 from that which contains the portion CE; and, if possible, let a slight deformation of the curve be made so as to transfer the portion GE across the branch-point A. In the deformed position, the curve...G'E'Dr... does not meet the branch-line; there is, consequently, no change of sheet in its course near A and therefore ED'..., which is the continuation of...G'E cannot be regarded as the deformed position of ED. The two paths are essentially distinct; and thus the original path cannot be deformed over the branch-point. It therefore follows that continuous deformation of a circuit over a branch-point on a Riemann's surface is a geometrical impossibility. Ex. Trace the variation of the curve GED, as the point E moves up to A and then returns along the other side of the branch-line. Hence a circuit containing two or more of the branch-points is irreducible; but a circuit containing all the branch-points is equivalent to a circuit that contains none of them, and it is therefore reducible. If a circuit contain only one branch-point, it can be continuously deformed so as to coincide...