Multivalued function
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, a
logical contradiction, true functions are single-valued.
The above diagram does not represent a "true" function; because, the element 3, in X, is associated with two elements b and c, in Y.
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 tan(π/4) = tan(5π/4) = tan(−3π/4) = ... = 1 . Consequently arctan(1) may be thought of as having multiple values, among them π/4, 5π/4, −3π/4, etc.
- The natural logarithm function is single-valued, but its generalization to complex numbers 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 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