Talk:Multivalued function

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As best I can tell, the relation discussed here is what economists call a correspondence. I've put a cross-reference in here, and added a mention of multivalued functions in the correspondence article. As far as I can tell these are two names for the same thing, used in different areas of math. Isomorphic 22:25, 29 July 2006 (UTC)

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[edit] Misnomer?

What this "misnomer" is supposed to mean? I do not know the formal mathematical definition of such a term. Usually the functions are assumed single-valued but the general definition of a function relates elements from one set to the elements of another (or same) one. Even the ordinary sqrt(x) is having two values (not to mention sin-1(z))! It's just for convinience that usually only one of the values is deliverately chosen. Or the implicit functions are also a "misnomer". -- Goldie (tell me) 22:19, 24 August 2006 (UTC)

'Implicit function' is a misnomer, in the large. A careful statement of the implicit function theorem will only give a local existence theorem. Charles Matthews 14:03, 12 October 2006 (UTC)

[edit] A graphical, interactive example of a multi-valued function

Go to [1] to see an example of a multi-valued function. This came from a class titled Complex Analysis. This demonstrates how a function can be analytic in a region, but not in the entire complex plane. The input is shown in black, and the three possible outputs are shown in red, green, and blue. As long as you don’t go around or through one of the "bad" points (shown in pink) you can view this as three ordinary functions.

For additional examples see [2].

The documentation is out of date. If you want to download TCL, you will need to go to [3].

[edit] Output: single multiset

The square root of 4 is the multiset {+2,−2). The square root of zero is the multiset {0,0}, because zero is a double root of the equation x2=0. Using the concept of a multiset, the term 'multivalued function' ceases to be a misnomer. Any comments? Bo Jacoby 16:33, 14 December 2006 (UTC)

[edit] Output: single multiset or single value

As far as I know, some authors accept that the codomain of a multivalued function is a set of sets or multisets, but many others interpret multivalued functions as functions which return a single (arbitrarily selected) value. For instance, many define the indefinite integral of f as one of the infinitely many antiderivatives of f. This is obviously convenient. Consider this question:

  • Does the square root of x return (1) all the numbers the square of which is x, or (2) any number the square of which is x?

In other words, is its output a multiset with two elements or a single (not uniquely determined, and arbitrarily selected) number? In other words, does the algorithm imply the process of "collecting all possible solutions" or the process of "arbitrary selection of only one solution"? I guess that some mathematicians will defend the second option.

It is quite intersting to notice that the second option implies what follows:

  • the square root is regarded as a multivalued function but, paradoxically, it has a single-valued output, and
  • its codomain is simply R (rather than a set of multi-sub-sets of R).

Note that, in both cases, the square root returns a single value (either a single multiset or a single number). THis example can be generalized to all multivalued functions.

Conclusion. It seems that we have only two options for defining a multivalued function:

  1. a multivalued fonction is single valued and uniquely determined.
  2. a multivalued fonction is single valued but nonuniquely determined.

Multivalued functions are actually single valued! Paolo.dL 21:29, 27 September 2007 (UTC)

[edit] Notation for multifunctions

I have seen [\![ and ]\!] used to donate multivalued functions, as in:

[\![\,\log z\,]\!] := \{\,\log |z| + i\theta : \theta \in [\![\,\arg z\,]\!]\,\},

[\![\,\arg z\,]\!] := \{\,\theta \in \mathbb{R} : z = |z| e^{i\theta}\,\}

and

[\![\,z^\alpha\,]\!] := \{\, e^{\alpha(\log |z| + i\theta)} : \theta \in [\![\,\arg z\,]\!]\,\}.

(Priestley, H. A. (2006). Introduction to Complex Analysis, Second Edition, Oxford University Press. Chapters 7 and 9.)

129.67.19.252 02:18, 26 October 2007 (UTC)