**Contents:**

## Multivalued Function: Definition

A **multivalued function** (also called a *multiple-valued function*) has more than one distinct output for at least one input.

For example, the functions y = √ x and y = -√ x are two separate functions. They can be combined into one multivalued function y^{2} = x, y, which has two real values when x > 0.

## Multivalued Function Examples

Some of the most important multivalued functions in complex analysis are:

- √z,
^{n}√z,- √((z-a)(z-b))
- log z,
- z
^{a}, - sin
^{-1}z, - cos
^{-1}z.

## Multivalued functions are not actually functions

These functions work differently from the usual single-valued functions you come across in calculus, and they aren’t your everyday functions (i.e. they are not one-to-one or many-to-one). You can only perform calculus on a part of the function—a branch: the real-numbered part, imaginary part, modulus or argument. The place where these individual parts branch out from each other are called *branch cuts*. For example, the negative real axis is a branch cut for the two parts of the square root function (y = √ x and y = -√ x).

To complicate things further, these parts themselves might be multivalued. For example, log z has the multivalued imaginary part z^{1/3}.

## Branch Point

A **branch point** is a point “z” where a multivalued function equals zero or infinity. It’s the point where the function’s single-valued branches are “tied” together.

The following image shows what happens to the multivalued logarithm function around the origin:

A slightly more technical way of saying the same thing is that a branch point happens when a multivalued function w(z) is discontinuous upon traversing a small circuit around a particular point.

Finding the exact location for a particular function’s branch points can be **challenging** for uncommon functions. Branch points for some familiar functions are well known.

## Branch Points of Specific Functions

The complex-valued root function, w = z^{1/n} and the logarithm function, w = log z, have two branch points: z = 0 and, in the extended plane (a complex plane with a point at infinity attached), z = ∞. As a variable z in this function travels around the point z = 0, **it doesn’t return to its original starting point. **

## Example in Terms of Real-Valued Function

To visualize what happens at a branch point in a multivalued complex function, let’s take a look first at a graph of a real-valued function, y = *R*(x).

Any point around point “A” (where the derivative is zero) is two-to-one. Any point around point “B” is one-to-one, and has a derivative.

**A very similar thing happens with complex functions**, except on a 4-dimensional plane.

## Why does the function behave strangely around a branch point?

Anywhere there is a derivative on the complex plane, the neighborhoods around the point have one-to-one correspondence, just like the points around B in the real-valued function. But at points like A, where the first derivative is zero, that’s when things get a little strange: the **complex function starts to behave like the exponential function z ^{m},** orbiting “m” times as fast around the point and creating “m” preimages (

*preimages*are the set of function arguments that correspond to a subset in the range).

If this sounds a little nonsensical, that’s because it *is*: as an example, the logarithmic function has a branch point of infinity—it’s undefined at that point and has no meaning. So, in a way, the behavior of a function around a branch point has no meaning either!

## References

Ablowitz, M. et al. (2003). Complex Variables: Introduction and Applications. Cambridge University Press.

Brand, L. (2013). Advanced Calculus: An Introduction to Classical Analysis. Courier Corporation.

Craig, D. (2007). Complex functions, single and multivalued. Retrieved December 15, 2019 from: file:///C:/Users/brit6/Downloads/multifun.pdf

Haber, H. Branches of Functions. Retrieved December 15, 2019 from: http://scipp.ucsc.edu/~haber/ph116A/ComplexFunBranchTheory.pdf

Jensen, V. et al. (1977). Mathematical Methods in Chemical Engineering. Elsevier.

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.

Pham, T. Multivalued functions via Mathematica. Retrieved December 15, 2019 from: http://people.oregonstate.edu/~phamt3/Courses/S19-Math-483-583/MathematicaGuide-4.pdf

Paliouras, J. & Meadows, J. (2014). Complex Variables for Scientists and Engineers: Second Edition. Courier Corporation.

Branch point 3D image: Leonid 2 [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)]

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