ABSTRACT
A function w = f(z) is multivalued (Section 7.1) if there are at least some values of z to which correspond several values of w, viz. each of the values specifies one branch of the function. One branch may be selected as the principal branch, for example, the branch with smallest phase in modulus. A multivalued function may be treated as single-valued if the same branch is always chosen, for example, the principal branch (Section 7.3). A complex function w = f(z) transforms curves in the z-plane into curves in the w-plane for each of the branches; each of the branches lies on sheet, and different sheets may be connected to form a Riemann surface (Sections 7.1 and 7.2). Some paths in the z-plane will pass from one sheet to another in the Riemann surface of w, leading to a change of branch of the function. To avoid changing to another branch, the lines joining the different sheets in the Riemann surface are never to be crossed; these lines correspond to branch-cuts (Section 7.4) in the z-plane. The branch-cuts join between themselves (Section 7.8) or to infinity (Section 7.5), branch-points where the branches coincide, and in whose neighborhood they separate. Thus, picking the principal branch of a multivalued function, and performing appropriate branchcuts, will always retain the same branch if the cuts are not crossed. To draw the branchcuts (Section 7.9) it is necessary to locate the branch-points (Sections 7.6 and 7.7), and join them between themselves or to infinity so as to avoid passing to other branches; the principal branch is usually discontinuous across the two sides of a branch-cut (Section 7.4), since continuity could only be preserved by going over to another sheet of the Riemann surface. Whereas the representation of a single-valued function requires only one sheet, the connection between sheets of a multivalued function may have a meaning: if the branches of a multivalued function represent distinct solutions of the same problem or different states of the same system, then their connections correspond to possible or impossible choices or transitions.
