Multivalued function

This diagram does not represent a "true" function, because the element 3 in X is associated with two elements, b and c, in Y.

In mathematics, a multivalued function is a total relation; i.e. every input is associated with one or more outputs. Strictly speaking, a "well-defined" function associates one, and only one, output to any particular input. The term "multivalued function" is, technically, a misnomer: true functions are single-valued. However, a multivalued function from A to B can be represented as a function from A to the power set of B.

Examples

• Each real or complex number except 0 has two square roots. Each complex number has three cube roots.

• Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic. We have

Consequently arctan(1) may be thought of as having multiple values: π/4, 5π/4, −3π/4, and so on.

• The natural logarithm function from the positive reals to the reals is single-valued, but its generalization to complex numbers (excluding 0) is multiple-valued, because the natural exponential function exp(z) (evaluated at complex arguments z) is periodic with period 2πi. Denoting this multi-valued function by "Log", with a capital "L" to distinguish it from its single-valued counterpart defined only for positive real arguments, the values assumed by Log(e) are 1 + 2πin for all integers n.

Multivalued functions of a complex variable have branch points. For the nth root and logarithm functions, 0 is a branch point; for the arctangent functions, the imaginary units i and −i are branch points.

See also

• partial function
• branch point