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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.

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Can we *really* do algebraic operations involving roots on C?

With BSc in Maths and loads of grey hair, something has been on my mind for decades, and I couldn't quite enunciate it. Let me try. Root is inherently "multi-valued" operation. So $$ \sqrt{4}...
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Need help with evaluating a multivariable limit

Tried evaluating the following limit using the squeeze theorem: $$\lim_{(π‘₯,𝑦)β†’(0,0)} \frac{1-\cos(xy)}{xy^2}$$ $$0 \leq f(x,y) \leq \left| \frac{1-\cos(xy)}{xy^2} \right| \leq \left| \frac{1-\cos(...
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Multivariable taylor - why does $R_k(a) = o(||h||^k)$

Let $f : E \to \mathbb{R}$ be $C^k$ for $E \subset \mathbb{R}^n$ open. Choose $a \in E$ and $h \in \mathbb{R}^n$ such that $a + h \in E$. Then $\exists \theta \in (0,1)$ such that $f(a + h) = f(a) + (...
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Do equations $\sqrt{x}=-2$ and $\sqrt[3]{x}=-2$ have complex solutions?

According to the standard definition of a real-valued root, equations $\sqrt{x}=2$, $\sqrt[3]{x}=2$ and $\sqrt[3]{x}=-2$ have real-valued solutions, but $\sqrt{x}=-2$ doesn't. But when we move to the ...
Buckminster's user avatar
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Solving irrational equations over the field of complex numbers

Do irrational equations like $$x + \sqrt{x βˆ’ 1} = 7$$ have solutions over the field of complex numbers and which definition(s) of the complex $n$th root should we use to obtain solutions to such ...
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Multi valued function and lower semi continuity

I consider $X$ a metric space and $F_1,F_2$ two disjoint subsets of $X$. Let $T : X\rightrightarrows\mathbb{R}$ be a multi valued function defined by : $T(x) =\{0\}$ on $F_1$ $T(x)=\{1\}$ on $F_2$ $T(...
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Multi valued function and compacity

I consider $(X,d_{X})$ and $(Y,d_{Y})$ two metric spaces, the first being compact and $T : X \rightrightarrows Y$ a multi valued function which is upper semi continuous and with compact values (i.e $T(...
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Terminology for set-valued functions that are "included" in another function

I was wondering, if you have $f:A \rightarrow B$ a mapping onto sets, is there appropriate terminology for when, for a subset $C \subseteq A$, you have $g:C \rightarrow D$ that verifies $\forall e \in ...
Fluorine's user avatar
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Give an example of multivalued holomorphic function satisfying some condition.

I would like to find some examples of multivalued holomorphic function $f(z)$ satisfying the following conditions. $f(z)$ has only two branch points/singularities: $z=0,1$ on the whole complex plane $\...
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Clarification between multivalued, set-valued and vector functions

I tried to read as many notes and answers possible before asking this, but I find myself in need of some answer from you guys. So my confusion is about multivalued functions VS set valued functions VS ...
Heidegger's user avatar
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How to prove continuity of $g(y)=\max_{x\in K} f(x,y)$ when $K$ is compact?

I have a function $f$ that is continuous over the set of variables, $f: \ K \times M \to \mathbb{R}$, where $K$ is a compact domain of a metric space and $M$ is a metric space. And I want to prove ...
Silver_roof's user avatar
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How to invert system of multivariable polynomial equations for dependent variables

I was wondering if I could get some input on my strategy for inverting a set of multivariate polynomial equations for the independent variables. My functions are as follows: $$g_1(x,y) = a_0 + a_{1}x+...
Mr Phase Locked Loop's user avatar
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Chebyshev approximation for bivariate function

I read the paper. I am a litte bit confused regarding formulation of Chebyshev approximation for bivariate function(See photo). There is only one integral over variable x. Should it be in formula one ...
Masamune's user avatar
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Is there any use to a directional derivative with a vector that's not a unit vector.

I can't think of any reason as to why one would have a directional derivative using a vector that's not of unit length. It would always "mess up" the derivative by scaling it by the ...
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Prove whether or not there exists $N$ that satisfies following.

Prove whether or not there exists a positive integer $N$ such that for all $\epsilon\gt0,$ $\vert f^{(n)}(x_k)-f^{(n)}(x_1)\vert<\epsilon$ if $n\gt N$. Function $f$ is defined as follows. For ...
user_A's user avatar
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Two binomial expansions of a fractional power of z+w as multivalued functions

Let's treat $z^{1/T}$ and $w^{1/T}$ as multivalued complex functions. Then, what does $(z+w)^{1/T}$ exactly mean? We have two ways to interprate $(z+w)^{1/T}$: $$I=\sum_{i}\binom{1/T}{i}z^{1/T}z^{-i}w^...
S.Gau at Math's user avatar
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1 answer
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Understanding the function $f(z) = \left(\frac{z}{z + x_0} \right)^{\kappa}$

I have the following complex function: $$f(z) = \left(\frac{z}{z + x_0} \right)^{\kappa}$$ where $x_0 \in \mathbb R$ and $x_0 > 0$. $\kappa$ is a parameter for which we look at three cases: case: $...
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Question about the equivalence of three versions of Closed Graph Theorem

I am studying set-valued analysis, and I saw three version of this Closed Graph Theorem. Version 1: (what I was taught in class) Let $\Gamma: X \to Y$ be a correspondence. If $\Gamma$ is closed-...
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5 votes
1 answer
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Prove the (path-) connectedness of the graph of a compact- and convex-valued upper hemi-continuous correspondence.

Let $\Gamma: [0, 1] \to \mathbb{R}$ be a compact- and convex-valued, upper hemi-continuous correspondence. Prove that the graph of $\Gamma$ is a connected set. Is it path-connected? This is what I ...
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The existence of continuous approximate selection

I'm looking for a condition of the existence of continuous approximate selection for a (multi-valued) minimiser mapping. That is, let's say a continuous function $f: X \times U \to \mathbb{R}$ is ...
Jinrae Kim's user avatar
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Find all partial derivatives

We have a funcion of 2 variables $z = g(x,y)$. And $x = cos(t)$,$y = sin(t)$. Find all partial derivatives is it right that $$\frac{βˆ‚z}{βˆ‚x}=\frac{βˆ‚}{βˆ‚x}g(x,y)*(x)'_x=\frac{βˆ‚}{βˆ‚x}g(x,y)$$ $$\frac{βˆ‚z}{βˆ‚...
Nick Schemov's user avatar
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1 answer
90 views

Total derivative for functions with higher dimensional codomain.

I've been learning some multivariable calculus and I ran into the concept of total derivative. I think I grasp the idea and I've seen several examples of how to calculate it for functions $\mathbb{R}^...
Alooffi's user avatar
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Representation of concave multivalued/point-to-set maps

Given a point-to-set map $C: X \rightrightarrows Y$ defined by some vector valued-function $\mathbf{g}: X \times Y \to \mathbb{R}^n$ such that $C(x) \doteq \{y \in Y | g_1(x,y), …, g_n(x,y) \geq 0 \}$,...
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1 answer
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composite function of several variables

recalling functions of single variable ,when writting \begin{gather} (f\circ g) (x)=f(g(x)) \end{gather} this means the $x$ in the domain of $g$ and $g(x)$ in the domain of $f$,that makes the ...
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Given several values of x, y and z in the relation z=ax+by+c, determine a, b and c

I have inherited a cloud pricing mechanism that essentially uses two variables to derive prices. No-one in the organisation knows the original derivation, but I have about 16 sets of datapoints from ...
length's user avatar
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1 vote
1 answer
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Matrix derivative over non-scalar function

I'm attempting to obtain a derivative for the following function by matrix $U$: $$R = \sum_{i=1}^n||Ζ’(s β‹… U) β‹… V β‹… M_i - f(t β‹… U) β‹… V|| ^ 2$$ where $s \in \mathbb{R}^{1\times3}$, $t \in \mathbb{R}^{1\...
Lu4's user avatar
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Trace of matrix logarithm for two invertible matrices

Consider an unitary matrix $U$ and an positive definite, invertible and diagonalizable matrix $\rho$ . Then, if the following identity holds (i.e., if there are additional $2\pi I$ factor), \begin{...
Richard's user avatar
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Is there a function analytic in $D(0,r)$ such that $(f(z))^2 = e^z + z$ for all $z\in D(0,r)$?

I'm stuck on the following problem: Let $D = \{z\in\mathbb{C}:|z|<r\}$, $r>0$. Is there a function analytic in $D$ such that $(f(z))^2 = e^z + z$ for all $z\in D$? Look at the two cases: $r = 1$...
kimjueun's user avatar
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1 answer
80 views

The limit of $ \frac{(x^2+y^2)^2}{x^2 - y^2} $ at the origin

I calculated it as follows, if we use the polar coordinates $x=r\cos\theta$ and $y=r\sin\theta$ as $r$ goes to zero, we get $ \frac{((r\cos\theta)^2+(r\sin\theta)^2)^2}{(r\cos\theta)^2 - (r\sin\theta)^...
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Determine complex exponent to make multi-valued function negative

I have the following complex function and I would like to make it so that $g_k(z=20) = -\frac{1}{75}$ where $z+z_0=\rho \exp(j(\theta+2 k \pi))$. $$g(z)=\frac{1}{\sqrt{z+5} (z-5)}$$ I know that $z_0=-...
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If $\Delta f\geq0$ for $f\in C^2(D)$ where $D\subset\mathbb{R}^n$ is convex, show that $f$ has no local maximum in $D$ unless it is constant.

So the Laplacian is the trace of the Hessian matrix and also if $f$ attains local maximum at for example $x_0\in D$ then $\nabla^2 f(x_0)\leq0$. This concludes that all eigenvalues of $\nabla^2 f(x_0)$...
barbatos233's user avatar
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Is there any hope to find the general function $f(x,y)$ if we know the function value at two points as given?

I am looking for a general function $f(x,y)$ for which I have $$ (x_0,y_0)=(3,1) \quad \text{we have} \quad f(x_0,y_0)=12 $$ and $$ (x_1,y_1)=(5,1) \quad \text{we have} \quad f(x_1,y_1)=5(23-\sqrt{5})$...
sara96's user avatar
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2 answers
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If $f$ is a non constant entire function which is real on the real axis then $\arg f(\overline{z})=-\arg f(z) $

If $f$ is a non constant entire function which is real on the real axis then prove that $$\arg f(\overline{z})=-\arg f(z) $$ where $z=x+iY$, $0<x<1$ and $f$ is non zero on the horizontal line ...
user avatar
2 votes
1 answer
222 views

Function arrow notation in multivariable functions

How can I implement the function arrow notation in a multivariable function? I know that $f:\mathbb{R}\rightarrow\mathbb{Z}$ means that the domain is $\mathbb{R}$ and the range is $\mathbb{Z}$ for ...
Agente 156's user avatar
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123 views

If $D \subset \mathbb C\setminus\{0\}$ is an open connection....

a) If $D \subset \mathbb C\setminus\{0\}$ is an open connection. Show that if $\theta_1$ and $\theta_2$ are argument branches in $D$, then there is $k \in Z$ such that $\theta_1(z) = \theta_2(z) + 2k\...
Cel's user avatar
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2 answers
134 views

Calculate total derivative directly.

Calculate directly (not via partial differentiation) the total derivative of the function $f(x_1,x_2)=x_1^2-10x_2.$ You may wish to use the fact that $\sqrt{x^2+y^2}\geq\frac{x+y}{2}.$ $$$$ For the ...
Techlover's user avatar
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How to solve equations involving multivalued functions in complex domain?

Does this equation have a solution in the complex domain? $$ \sqrt{x+3} = 3 + \sqrt{x} $$ Squaring both sides gives $\sqrt{x}=-1$, which suggests the solution to be $x=1$. But: How can the complex ...
Mohammad Ali's user avatar
2 votes
0 answers
52 views

Alternative to showing that $J(m,n)=\frac{1}{2}[(m+n)^2+3m+n], \ J:\mathbb{N}^2\to \mathbb{N}$ is bijective

I came up with the following proof, but it seems too complicated so I was wondering if anyone else has a simpler idea. Proof. The idea is to notice that $J(m,n)$ can also be expressed equivalently as $...
Aaron Welson's user avatar
1 vote
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Clebsch variables

I should find if it is true that I can formalize the velocity in BE-condensate with a clebsch decomposition. We know that $$u=\nabla \phi.$$ with $\phi$ the phase of the madlung tranform $$\phi=\sqrt\...
Andrea's user avatar
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How to find a branch which is analytic on the exterior of the unit circle for $\sqrt(z^2 +1)$, $|z| > 1$

I know we can rewrite $\sqrt{z^2 +1}$ as $z^2 (1+z^{-2})$ and use this by looking at the principal branch of the function $\exp{\left(\frac{1}{2} \log(1+z^{-2})\right)}$. However I am struggling to ...
idkkkkkkk's user avatar
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Confusion about the change of variable $z \to \frac{1}{z}$ for a multivalued function

I'm currently struggling with something that came up in my studies. I'm trying to integrate a multivalued function like the square root on a given path, specifically a function with two branch points, ...
M4dMel's user avatar
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1 vote
1 answer
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How do we know that this function is multivalued?

So I have an integral $$ \int_{0}^{2\pi} \frac{1}{2}\left(e^{e^{ix}} + e^{e^{-ix}}\right) \text{ d}x$$ I am told that I am able to substitute $z=e^{ix}$ into this and convert it into a contour ...
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2 votes
0 answers
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Differences between "the Riemann surface" and "the imaginary part" of complex logarithm

Even if it's easy to find the formal definition of a Riemann surface (Β«a one dimensional complex manifold ...Β»), I am trying to get an example, as regards the complex logarithm. In the Wikipedia ...
BowPark's user avatar
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Can a compound functions of multi-valued functions result in a single-valued function?

For example, $f_1(z)$ and $f_2(z)$ are some multi-valued functions. Is it possible for there to exists a single-valued function $f(z)$ such that$$f(z) = f_1(f_2(z))$$ If so can you please give an ...
Ian Hsiao's user avatar
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2 answers
274 views

How to determine if $z+\sqrt{z-1}$ and $\frac{\sin{\sqrt{z}}}{\sqrt{z}}$ is a multi-valued function?

What I know: I understand that for a complex function to be a multi-valued funciton they must have some branch points. I understand branch points as this definition (translated into English from my ...
Ian Hsiao's user avatar
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REVISED Proving $-1,1$ are branch points of $\sqrt{z^2-1}$

On some lecture notes that I am working on there is an exercise to prove that $-1,1$ are branch points of the multi-function $\sqrt{z^2-1}$. I know that the branch $f=\sqrt{rs}e^{i\frac{1}{2}(\theta_1+...
jcneek's user avatar
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2 votes
1 answer
310 views

What is the inverse in the complex plane of $e^z + z$?

I want to decide the domain and range of $e^z + z$ and find some properties of its inverse function such as multivalued function. And I guess the domain and range should be whole complex plane ...
Ziqin He's user avatar
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2 votes
3 answers
141 views

Solving $C\cos(\sqrt\lambda\theta)+D\sin(\sqrt\lambda\theta)=C\cos(\sqrt\lambda(\theta+2m\pi)) + D\sin(\sqrt\lambda (\theta + 2m\pi))$

I want to solve $$C\cos(\sqrt\lambda \theta) + D\sin(\sqrt\lambda \theta) = C\cos(\sqrt\lambda (\theta + 2m\pi)) + D\sin(\sqrt\lambda (\theta + 2m\pi))$$ The solution must be valid for all $\theta$ in ...
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2 votes
1 answer
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Relation of solutions of differential equations around different singular points

In order to clarify my problem, I start with an example $$ y'-\frac{1}{2}\left(\frac{1}{x}+\frac{1}{x-1}\right)y=0 $$ which has two singular points, $0$ and $1$. The exact solution is not difficult to ...
user142288's user avatar
0 votes
1 answer
331 views

How to find principal branch of a complex multivalued function?

Suppose I have a complex multivalued function $\log(f(z))$, and I am required to find the principal branch of this function. The method I have learned says that the principal branch of $\log(z)$ is ...
AP666's user avatar
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