Week 2 3. Multivalued functions or dependences ۞ A multivalued function, or dependence, f(z) is a rule establishing correspondence between a value of z and two or more values of f. In real calculus, an n-valued function can be ‘separated’ (like Siamese twins) into n single-valued functions. These functions are called the branches of the dependence.

Example 1: The dependence where x and f are real, defines a two-valued function – which can be separated into two single-valued branches:

In most cases (like example 1), a real dependence can be separated into branches by requiring that each branch yields values of a certain ‘type’ (e.g. +tive or –tive). There’s, however, a subtler way. We can require for each branch to yield a certain value at a certain point x0, and then assume that the branch changes continuously when x moves away from x0. In example 1 we can require that and then assume that f1,2(x) are changing continuously when x is moving away from x0 = 1. The word “continuously” implies that we disallow ‘branch hopping’.

Example 2: The equation defines a two-valued dependence, which can be separated into two single-valued branches: and

The approach based on continuity doesn’t work for example 2, as one can continuously ‘slide’ from one branch to the other through the origin (which belongs to both branches). Thus, if we are going to use this approach, we’ll have to resort to ‘surgery’ – i.e. remove the point (0, 0) from the two branches. In example 2, the ‘surgery’ yields (1) and (2) Observe that the ranges of functions (1), (2) don’t include x = 0.

The situation with complex functions is more complicated. Example 3: Consider the complex dependence (3) where z ≠ 0 and require (4) Does condition (4) single out a branch from dependence (3)? To answer this question, consider the circle of unit radius centred at the origin and trace the value of f(z) while we move z along the circle. Will we return to the point z = 1 with the same value of f(z)? And the answer to this question is “no”.

To separate branches of a multivalued function of complex variable, one needs to remove from the complex plane more than just isolated points. One typically needs to remove curves. Example 4: Consider dependence (3) and condition (4) from Example 3, and assume that we have ‘cut out’ from the complex plane the non-positive part of the real axis (i.e. the negative part + the origin):

The contour considered in Example 3 is now ‘disallowed’ The contour considered in Example 3 is now ‘disallowed’... but perhaps we can’t return to the point z = 1 with a different value of f(z) through another contour? The answer to this question is “no”. All contours not enclosing the origin can’t make our function multivalued – whereas the contours that do enclose the origin are disallowed by the cut we made in the complex plane. ۞ A cut or a set of cuts on the complex plane, ‘breaking up’ a multivalued function into a set of single-valued functions (branches) is called a branch cut. Note that branch cuts are, typically, non-unique. In Example 4, for instance, a cut along any semi-infinite ray from the origin to infinity would do the job.

There’s a price to pay for introducing branch cuts, however. Example 5: Let f(z) be the single-valued function described by (3)–(4) on a complex plane with a cut described in Example 4. Consider two semi-circles of unit radius centred at the origin, both going from z = 1 to z = –1: P1 goes through the upper semi-plane and P2 goes through the lower one. With what values of f(z) will we arrive in z = –1?

Different limits of the function when approaching the cut from above and below don’t give rise to a contradiction if we allow functions to have different values at the upper and lower edges of the cut. The upper/lower edges of the branch cut of Example 4 are typically denoted by where the term ±i0 symbolises an infinitesimal imaginary correction (positive or negative) to the real number x. Thus, single-valued functions originating from a multivalued function typically involve a discontinuity (jump) along the branch cut. ۞ A branch point is a point such that the function is discontinuous along an arbitrarily small circle around this point.

Example 6: Consider the dependence (5) on the complex plane with the non-negative part of the imaginary axis removed, and require (6) (a) Argue that (5)–(6) and the proposed branch cut describe a single-valued function. (b) Find f(2i + 0) and f(2i – 0). The answer: f(2i + 0) = 21/3 (√3 + i)/2, f(2i – 0) = –21/3 i.