Multivalued Function - Examples

Examples

  • Every real number greater than zero or every complex number except 0 has two square roots. The square roots of 4 are in the set {+2,−2}. The square roots of 0 are described by the multiset {0,0}, because 0 is a root of multiplicity 2 of the polynomial x².
  • Each complex number has three cube roots or, in general, n nth roots.
  • The complex logarithm function is multiple-valued. The values assumed by log(1) are for all integers .
  • Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic. We have
\tan\left({\textstyle\frac{\pi}{4}}\right) = \tan\left({\textstyle\frac{5\pi}{4}}\right) = \tan\left({\textstyle\frac{-3\pi}{4}}\right) = \tan\left({\textstyle\frac{(2n+1)\pi}{4}}\right) = \cdots = 1.
Consequently arctan(1) is intuitively related to several values: π/4, 5π/4, −3π/4, and so on. We can treat arctan as a single-valued function by restricting the domain of tan x to -π/2 < x < π/2 – a domain over which tan x is monotonically increasing. Thus, the range of arctan(x) becomes -π/2 < y < π/2. These values from a restricted domain are called principal values.
  • The indefinite integral is a multivalued function of real-valued functions. The indefinite integral of a function is the set of functions whose derivative is that function. The constant of integration follows from the fact that the difference between any two indefinite integrals is a constant,


These are all examples of multivalued functions which come about from non-injective functions. Since the original functions do not preserve all the information of their inputs, they are not reversible. Often, the restriction of a multivalued function is a partial inverse of the original function.

Multivalued functions of a complex variable have branch points. For example the nth root and logarithm functions, 0 is a branch point; for the arctangent function, the imaginary units i and −i are branch points. Using the branch points these functions may be redefined to be single valued functions, by restricting the range. A suitable interval may be found through use of a branch cut, a kind of curve which connects pairs of branch points, thus reducing the multilayered Riemann surface of the function to a single layer. As in the case with real functions the restricted range may be called principal branch of the function.

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