Mathematics Stack Exchange
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Questions tagged [multivalued-functions]

In Mathematics, set theory, a Multivalued function is defined as a left-total relation (that is, every input is associated with at least one output) in which at least one input is associated with multiple (two or more) outputs.

In Mathematics, set theory, a Multivalued function is defined as a left-total relation (that is, every input is associated with at least one output) in which at least one input is associated with multiple (two or more) outputs. Reference: Wikipedia.

Synonyms: many-valued function, set-valued function, set-valued map, point-to-set map, multi-valued map, multimap, correspondence, carrier.

252 questions
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A function is convex if and only if its gradient is monotone.

Let a convex $ U \subset_{op} \mathbb{R^n} , n \geq 2$, with the usual inner product. A function $F: U \rightarrow \mathbb{R^n} $ is monotone if $ \langle F(x) - F(y), x-y \rangle \geq 0, \forall x,y \in \mathbb{R^n}.$ Let $f:U \rightarrow…
user286485
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Complex towers: $i^{i^{i^{...}}}$

If $w = z^{z^{z^{...}}}$ converges, we can determine its value by solving $w = z^{w}$, which leads to $w = -W(-\log z))/\log z$. To be specific here, let's use $u^v = \exp(v \log u)$ for complex $u$ and $v$. Two questions: How do we determine…
Stephen Herschkorn
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Multivalued Functions for Dummies

I have been studying complex analysis for a while, but I still cannot "get" how multivalued functions work. Despite having it explained to me many times, my brain cannot process it. For example, I do not know / understand what $f(z) = \sqrt{z^2}$,…
Braindead
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Are multi-valued functions a rigorous concept or simply a conversational shorthand?

In Brown and Churchill's book, the concept of multivalued functions is not discussed in a very rigorous way (if at all). But I can see that branch cuts have importance in complex analysis, so I want to clarify my understanding of multivalued…
AmadeusDrZaius
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If a function can only be defined implicitly does it have to be multivalued?

What is the general reason for functions which can only be defined implicitly? Is this because they are multivalued (in which case they aren't strictly functions at all)? Is there a proof? Clarification of question. The second part has been…
vfiddlestix
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Why are vector valued functions 'well-defined' when multivalued functions aren't?

I'm looking for an 'intuitive' answer here, because I have no formal mathematical training but find myself in a comparatively math-heavy PhD (visual perception; lots of neuroscientists on the one side and CS folk on the other). Only functions which…
benxyzzy
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What is a branch point?

I am really struggling with the concept of a "branch point". I understand that, for example, if we take the $\log$ function, by going around $2\pi$ we arrive at a different value, so therefore it is a multivalued function. However, surely this…
user122316
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Do any authors take the sheaf-theoretic viewpoint on multivalued functions and/or indefinite integrals?

It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves. For example: The real square-root function can be viewed as the sheaf $\mathcal{F}$ defined on the open sets of $\mathbb{R}$ by letting…
goblin GONE
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Books/Notes recommendation request: Multivalued functions/Riemann surfaces

I'm trying to read a document that applies Riemann-Roch left, right and center. I don't know this theorem or the theory it comes from so I need to build up a bit more background before I can tackle this. Can you please recommend good books or…
anon
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How to choose the right branch to find the roots.

I want to find the roots of $$f(z)=\left[a+zg(z)\right]^2+g(z)^2=0$$ Where $a$ is real number and: $$ g(z)=\frac{1}{2\sqrt{z^2+1}}\ln\left(\frac{z+\sqrt{z^2+1}}{z-\sqrt{z^2+1}}\right) $$ It is known that $f(z)=0$ has double complex roots when…
an offer can't refuse
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why is the integer power of a complex number not multi-valued too?

my textbook [H. A. Priestley - Introduction to Complex Analysis] states about the argument of a complex number raised to a power : 'Only when $\alpha$ is an integer does $[z^{\alpha}]$not produce multiple values: in this case $[z^{\alpha}]$…
user3203476
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Confusion between function and multivalued function.

"What is a function?" can be answered as "Single-valued relations are called functions". But how can "What are the multi-valued function?" be answered? Will someone clarify my doubt why multi-valued functions are not violating the classical…
Styles
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Is the Gamma Function multivalued??

Consider the definition of the Gamma function $$ \Gamma(s) = \int_{0}^{\infty}\left[x^{s-1}e^{-x} \right] dx $$ Clearly: $x^{s-1}$ may have multiple defined values for $s$ if $s-1$ is rational or even infinitely many if $s-1$ is irrational. Does…
Sidharth Ghoshal
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Why isn't Euler's formula multivalued?

So it seems that all complex exponential functions are multivalued except for ones with base $e$. Why? Shouldn't all exponentials be multivalued?
dfg
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Branch point-what makes a closed loop around it special?

I am having difficulty understanding the concept of a branch point of a multifunction. It is typically explained as follows:branch point is a point such that the function is discontinuous when going around an arbitrarily small circuit around this…
Sangeeta
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