Multivalued function

Write f(x) for the set of those y ∈ Y with (x,y) ∈ Γf. If f is an ordinary function, it is a multivalued function by taking its graph

z=a. This is the case for functions defined by the implicit function theorem or by a Taylor series around a. In doing so, one finds that the value of the extended function at a point b; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function.

For example, let z=1, so that the values along a given curve vary continuously from nth roots, logarithms, and inverse trigonometric functions.

To define a single-valued function from a complex multivalued function, one may distinguish one of the multiple values as the principal value, producing a single-valued function on the whole plane which is discontinuous along certain boundary curves. Alternatively, dealing with the multivalued function allows having something that is everywhere continuous, at the cost of possible value changes when one follows a closed path (monodromy). These problems are resolved in the theory of Riemann surfaces: to consider a multivalued function Γf−1 ⊆ Y×X

defined as Γf, viewed as a subset of X × Y. When f is a differentiable function between manifolds, the inverse function theorem gives conditions for this to be single-valued locally in X.

For example, the complex logarithm log(z) is the multivalued inverse of the exponential function e : C → C, with graph

4=±2={2,−2}; although zero has only one square root, x.

Each nonzero complex number has two square roots, three cube roots, and in general n nth roots. The only nth root of 0 is 0.

The complex logarithm function is multiple-valued. The values assumed by b are tan⁡(π4)=tan⁡(5π4)=tan⁡(−3π4)=tan⁡((4n+1)π4)=⋯=1. As a consequence, arctan(1) is intuitively related to several values: π/4, and so on. We can treat arctan as a single-valued function by restricting the domain of tan x to π/2 – a domain over which tan x is monotonically increasing. Thus, the range of arctan(x) becomes π/2. These values from a restricted domain are called principal values.

The antiderivative can be considered as a multivalued function. The antiderivative of a function is the set of functions whose derivative is that function. The constant of integration follows from the fact that the derivative of a constant function is 0.

Inverse hyperbolic functions over the complex domain are multiple-valued because hyperbolic functions are periodic along the imaginary axis. Over the reals, they are single-valued, except for arcosh and arsech.

These are all examples of multivalued functions that 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.

Branch points

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"Multiple Valued Functions | Complex Variables with Applications | Mathematics". MIT OpenCourseWare. Retrieved 2024-10-25.

↑ "Multivalued Function". Wolfram MathWorld. Retrieved 10 February 2024.

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