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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Why do we need simultaneous limits? [closed]

Can someone suggest me some book to understand the concept of simultaneous limits of multivariate functions better i do have some knowledge about it but according to the definitions in my university ...
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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 ...
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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 $...
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Rouché's theorem and multivalued functions

It's a problem with the application of Rouché's theorem when the algebraic equations contain multivalued terms. Take an example, $3z^2+z^{1/2}+z=0$, and the circle is $|z|=1$. We will check the number ...
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Do Polylogarithms Always Converge

I have been reading the Wikipedia page dedicated to polylogarithms to understand the following paper. I am trying to understand the first term in equation (6) (reproduced below): $$ P(\lambda) = \...
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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\...
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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 ...
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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, ...
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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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Geometric Indicators in Solutions to Different Types of First Order PDEs

I have been graphing solutions to two input, first order PDEs as three-dimensional surfaces with characteristic curves printed upon them, in order to better understand the solution logic in this ...
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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 («a one dimensional complex manifold ...»), I am trying to get an example, as regards the complex logarithm. In the Wikipedia ...
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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 ...
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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 ...
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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+...
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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 ...
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Let $G, F$ be a multivalued map and $I$ a homogeneous (monovalued) map, prove:

Let $G, F$ be a multivalued map and $I$ a homogeneous (mono-valued) map, prove: a) $\operatorname{graph}(G \circ F)=(F \times I)^{-1} \operatorname{graph} G=(I \times G) \operatorname{graph} F$. b) $(...
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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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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 ...
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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 ...
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Definition of directional derivative: Why does it work?

The definition of the directional derivative in my textbook is $$ \nabla_{\vec{v}} f = \lim\limits_{h\to 0}\frac{f(\vec{x} + h \vec{v} )-f(\vec{x})}{h} $$ with $\vec{x} = (x_1, x_2)$ and $\vec{v} = (...
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Compare the limits of $(Z^2-1)^{\frac{1}{2}}$ above and below its branch cut.

This question is taken from Q7 of example sheet 1 of the Complex Methods course at the university of Cambridge. The link for the sheet can be found below: http://www.damtp.cam.ac.uk/user/examples/B7a....
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Let $f(z)$ be a complex function whose value is $\sqrt{z^2-1}\in\Bbb R$ when $z\in \Bbb R$ and $z>1$, and $f(z)$ is holomorphic if $1<|z|<\infty$

Let $f(z)$ be a complex function whose value is $\sqrt{z^2-1}\in\Bbb R$ when $z\in \Bbb R$ and $z > 1$, and $f(z)$ is holomorphic if $1<|z|< \infty$. Then, what is the complex integral $$I=\...
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Find the joint distribution and covariance with exponential density

Let $X_1, \dots , X_n$ be independently distributed with exponential density $$ f(x) = (2θ)^{−1}e^{−x/2θ}, x \geq 0 $$ and let the ordered $X$’s be denoted by $X_{(1)} \leq X_{(2)} \leq \cdots \leq ...
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The definition of surjective multivalued

I am confused by the question of how to define surjectivity for multivalued mapping( if this definition has already existed).
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Putnam Problem A3 2010: if $h=a\frac{\partial h}{\partial x}+b\frac{\partial h}{\partial y}$ and $h$ is bounded then $h\equiv0$

I found a possible solution to the following Putnam problem from the 2010 edition of the competition, and I was wondering whether my pproach/solution is correct or whether I have made some assumptions ...
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How to find the modulus of a complex components?

Firstly I am not a student of mathematics just because right now I am doing a course related to complex variable I am having interest on learning things in a proper way . Suppose, $Z=1+i$ and $C=1-i$ ...
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Lipschitz constant estimation of continuous selection of upper hemicontinuous multivalued function

I'm reading a book, Set-Valued Analysis, written by Aubin. I'd like to estimate the Lipschitz constant of the continuous selection of upper hemicontinuous multivalued function by Theorem 9.2.1 of the ...
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How can I calculate $\log(e^{e^i})$?

I'm studying complex analysis and I'm wondering how to calculate the following multivalued function (using the expression $\log(z)=\ln|z| + i\operatorname{Arg(z)}$): $$\Large \log(e^{e^i})$$ Thank you!...
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Singularity structure of a multivalued function

Consider the function $$f(z) = \frac{1}{z} \ln \left( \frac{1-z}{1+z} \right).$$ This is clearly multivalued. There has to be two branch points at $\pm1$. Are there any other singularities, such as ...
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Proof that $\int_0^\infty\frac{\ln x}{x^3 - 1} \, dx = \frac{4 \pi^2}{27}$

I realise this question was asked here, but I'm not able to work with any of the answers. The hint given by my professor is Integrate around the boundary of an indented sector of aperture $\frac{2 \...
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Does the dilogarithm function (which is multi-valued) have a single-valued inverse?

The $p$-logarithm is defined for $|z|<1$ by $$\text{Li}_p(z)=\sum_{n=1}^\infty\frac{z^n}{n^p}$$ and defined elsewhere in $\mathbb C$ by analytic continuation, though it may be multi-valued, ...
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Some doubts in the evaluation of: limit as $(x,y)\to(0,0)$ of $\frac{\sin xy}{x+y}$

I must evaluate $$\lim_{(x,y)\to(0,0)}\frac{\sin xy}{x+y}$$ My reasoning is the following, can someone tell me if this is correct? Since $|\sin t| \leq |t|$ for all $t\in\mathbb{R}$ and it is $|xy|\...
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Example of compact upper semicontinuous map

can any one give me an exemple of compact multivalued upper semicontinuous maps in $L^{2}$ ? thanks !
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How to calculate Multivariable Limits in general

Like an example, can I say that $\lim_{(x,y)->(0,0)}\frac{\sin(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}}$ = 1 I can substitute $\sqrt{x^2 + y^2}= z$ and then use the Single Variable calculus result. ...
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Where in my work is my math falling apart when solving for the partial derivative of Residual sum of squares (Linear Algebra)

I am given the following equation: $$RSS(B, \alpha) = \sum_{i=1}^{N} (y_{i} - B^{T}x_{i} - \alpha)^{2} $$ My steps are as follows. I provided the images to my work at the bottom and this is just a ...
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4 votes
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Find all functions $f$ such that $f(f(x, y), z) = f(x, yz)$

Find all functions $f:\mathbb{R}_{\geq{0}} \times \mathbb{R}_{>{0}} \to \mathbb{R}_{\geq{0}}$ such that for all $x \in \mathbb{R}_{\geq{0}}$ and all $y, z \in \mathbb{R}_{>{0}},$ $$ f(f(x,y),z)=...
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Is a circle a multivalued function?

I don't really understand multi-valued function. I hope one of you can make me understand it. What I've learned from google, I suppose that a multi-valued function is a binary relation that maps the ...
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What does this notation $(dy/dx)_f$ mean?

If we have : $f=f(x,y)$, then what does the following mean and how to compute it : $(dy/dx)_f$ ? Note : This was found in a mathematics textbook destined for physicists. If it is used differently by ...
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Calculate the limit, Euclidean norm, multivariable function $\lim_{x\to 0} \frac{(\ln(1+x_2)-x_2)(1-\cos(x_3))\tan(x_1)}{\|x\|^4} $

I have a problem with this limit: $$\lim_{x\to 0} \frac{(\ln(1+x_2)-x_2)(1-\cos(x_3))\tan(x_1)}{\|x\|^4} $$ where $\|\cdot\|$ indicates the Euclidean norm and $x\in\Bbb R^3$. I have used Taylor ...
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An exercise in high-dimensional chain rule and Jacobians

In my vector calculus class, we are studying multivariate vector-valued functions and I have come across this exercise Let $f: \mathbb{R}^2 \to \mathbb{R}^3 $, $g: \mathbb{R}^3 \to \mathbb{R}^2$, and ...
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If the line integral of a differential = 0 does this imply it is a total differential?

Assume a differential of the form $df(x,y) = X(x,y) \,dx + Y(x,y) \,dy$. If $\oint\ df=0$, it's easy to see that $\frac{\partial X}{\partial y} = \frac{\partial Y}{\partial x}$, which can be seen ...
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Find local extremum and saddle points of the function $f(x, y) = x^2y^3(6 − x − y)$

$\nabla \left(f\right) = \begin{bmatrix}y^3x\left(12-3x-2y\right)\\ x^2y^2\left(18-3x-4y\right)\end{bmatrix}$ and in $(0,a), (a,0), (2,3)$ points $\nabla \left(f\right) = 0$ for all real a. Hessian ...
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Limit of multivariable function $f(x,y) = {(x^2+y^2)}^{x^2y^2}$

$$f(x,y) = {(x^2+y^2)}^{x^2y^2}$$ I need to find the limit at (0,0) point I applied the exponent rule and got $$e^{x^2y^2ln(x^2+y^2)}$$ and now with chain rule, I need to find the limit of $${x^2y^2ln(...
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Intuition behind the coefficients in Taylor expansion of a function

" For example, the best linear approximation for f(x) is $$ f(x) \approx f(a) + f'(a) (x-a)$$ The linear approximation fits f(x) (shown in green below) with a line (shown in blue) through x=a ...
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2 votes
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Hemicontinuous compact-valued mapping on $\mathbb{R}$ with no continous selection

I was told that there exists a mapping $f \colon \mathbb{R} \to \mathcal{P}(\mathbb{R})$ such that $f(x)$ is non-empty and compact for every $x \in \mathbb{R}$, $f$ is both upper and lower ...
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Determine extrema of a multivariate function defined on a set

Given a function $$f(x,y)=2+2x^2+y^2$$ and the set $$A:=\{(x,y)\in\mathbb R^2 | x^2+4y^2\leq 1\}$$ and $f:A\to\mathbb R$ how do I determine the global and local extrema on this set? Normally I would ...
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Square Root as a multi-valued function

I was scrolling through this article on Wikipedia, and I was stumped when I came across this line: Every real number greater than $0$ has two real square roots, so that the square root may be ...
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Is $f(x,y)=x^{xy}$ continuous for positive $x$ and real $y$?

Can someone please explain to me why this function is continuous? $$f(x,y)=x^{xy},\quad x>0, \quad y\in \mathbb{R} $$ I have thought like this that we can rewrite the question in this form: $f(x,y)...
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Separation of variables in a multivariable function

I was trying to visualize the meaning of a certain properties of some multivariable functions. For simplicity, let's consider a two variables function, whose graph can be easily visualized: f(x,y). In ...
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Transform equation assuming $a$ and $b$ as new variables.

The task is to transform the following equation $x \frac{dx}{dz}+y \frac{dz}{dy} =0$ into new one with independent variables taking $\varphi=\varphi(a,b)$ as a new function, for $a=\frac{y}{x}, b=y, \...
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