Multivalued function

The function from the element "3" takes two values

The multi-valued function is a generalization of the concept of a function , allowing for the presence of several function values for a single argument ^[1] .

Formally, a multivalued function from the set $X$ ${\ displaystyle X}$ $X$ in the set $Y$ ${\ displaystyle Y}$ $Y$ - binary relation $F$ ${\ displaystyle F}$ $F$ between sets $X$ ${\ displaystyle X}$ $X$ and $Y$ ${\ displaystyle Y}$ $Y$ such that for any $x\in X$ ${\ displaystyle x \ in X}$ $x \ in X$ there is such $y\in Y:\ x{\mathrm {F} }y$ ${\ displaystyle y \ in Y: \ x {\ mathrm {F}} y}$ ${\ displaystyle y \ in Y: \ x {\ mathrm {F}} y}$ .

A many-valued function is also considered as a subset- valued: each $x\in X$ ${\ displaystyle x \ in X}$ $x \ in X$ set in correspondence ${\mathrm {F} }(x)\subset Y:\ {\mathrm {F} }(x)=\{y\in Y|\ x{\mathrm {F} }y\}$ ${\ displaystyle {\ mathrm {F}} (x) \ subset Y: \ {\ mathrm {F}} (x) = \ {y \ in Y | \ x {\ mathrm {F}} y \}}$ ${\ displaystyle {\ mathrm {F}} (x) \ subset Y: \ {\ mathrm {F}} (x) = \ {y \ in Y | \ x {\ mathrm {F}} y \}}$ By definition, non-empty. Normal functions considered as multifunctions have set values consisting of exactly one element.

Content

In complex analysis and algebra

A typical example of multivalued functions is some analytic functions in complex analysis . Ambiguity arises during the analytic continuation along different paths . Also, many-valued functions are often obtained by taking inverse functions .

For example, the square root function has two values, differing only in sign.

In complex analysis, the concept of a multivalued function is closely related to the concept of a Riemann surface — a surface in a multidimensional complex space on which a given function becomes single-valued.

Note

↑ G. Korn, T. Korn. Handbook of mathematics. For scientists and engineers. M., 1973 Chapter 4. Functions and limits, differential and integral calculus. 4.2. Functions. 4.2-2. Functions with special properties . ( a ), p.99.

Literature

Lavrentyev MA , Shabat B.V. Methods of the theory of functions of a complex variable. - 4th ed .. - M .: Science , 1972 .
Shabat B.V. An Introduction to Complex Analysis. - M .: Science , 1969 . - 577 s.
F.-C. Mitroi, K. Nikodem, S. Wąsowicz, Hermite-Hadamard inequalities for convex set-valued functions, Demonstratio Mathematica, Vol. 46, Issue 4 (2013), pp.655-662.

[1] G. Korn, T. Korn. Handbook of mathematics. For scientists and engineers. M., 1973 Chapter 4. Functions and limits, differential and integral calculus. 4.2. Functions. 4.2-2. Functions with special properties . ( a ), p.99.

Multivalued function

Content

In complex analysis and algebra

See also

Note

Literature

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