A multivalued function, also known as a multiple-valued function (Knopp 1996, part 1 p. 103), is a "function" that assumes two or more distinct values in its range for at least one point in its domain. While these "functions" are not functions in the normal sense of being one-to-one or many-to-one, the usage is so common that there is no way to dislodge it. When considering multivalued functions, it is therefore necessary to refer to usual "functions" as single-valued functions.

While the trigonometric, hyperbolic, exponential, and integer power functions are all single-valued functions, their inverses are multivalued. For example, the function maps each complex number to a well-defined number , while its inverse function maps, for example, the value to . While a unique principal value can be chosen for such functions (in this case, the principal square root is the positive one), the choices cannot be made continuous over the whole complex plane. Instead, lines of discontinuity must occur.

The discontinuities of multivalued functions in the complex plane are commonly handled through the adoption of branch cuts, but use of Riemann surfaces is another possibility.

See also

Branch Cut, Branch Point, Complex Function, Function, Riemann Surface, Single-Valued Function

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References

Knopp, K. "Multiple-Valued Functions." Section II in Theory of Functions Parts I and II, Two Volumes Bound as One. New York: Dover, Part I p. 103 and Part II pp. 93-146, 1996.Morse, P. M. and Feshbach, H. "Multivalued Functions." §4.4 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 398-408, 1953.

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Multivalued Function

Cite this as:

Weisstein, Eric W. "Multivalued Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MultivaluedFunction.html

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