section_path: Multi-valued functions
section_name: Multi-valued functions
paragraph_text: Multi-valued functions
  
  Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a neighbourhood of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point x 0, {\displaystyle x_{0},} there are several possible starting values for the function.
  
  For example, in defining the square root as the inverse function of the square function, for any positive real number x 0, {\displaystyle x_{0},} there are two choices for the value of the square root, one of which is positive and denoted x 0, {\displaystyle {\sqrt {x_{0}}},} and another which is negative and denoted − x 0. {\displaystyle -{\sqrt {x_{0}}}.} These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single smooth curve. It is therefore often useful to consider these two square root functions as a single function that has two values for positive x, one value for 0 and no value for negative x.
  
  In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the implicit function that maps y to a root x of x 3 − 3 x − y = 0 {\displaystyle x^{3}-3x-y=0} (see the figure on the right). For y = 0 one may choose either 0, 3, or − 3 {\displaystyle 0,{\sqrt {3}},{\text{ or }}-{\sqrt {3}}} for x. By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval and the image is; for the second one, the domain is and the image is (−∞, −1 ]. As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single multi-valued function of y that has three values for −2 < y < 2, and only one value for y ≤ −2 and y ≥ −2.
  
  Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions. The domain to which a complex function may be extended by analytic continuation generally consists of almost the whole complex plane. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets i for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets − i. There are generally two ways of solving the problem. One may define a function that is not continuous along some curve, called a branch cut. Such a function is called the principal value of the function. The other way is to consider that one has a multi-valued function, which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the monodromy.