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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96 views

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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Complex integration of multivalued function

Can anyone help me with this the integral $$\int_{0}^{1}\frac{dx}{\left(a-bx\right)\sqrt{x\left(1-x\right)}},$$ where $0<b<a$? I have to solve it using the technique of complex integration of ...
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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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Branch cut of two Riemann sphere to integrate elliptic integral

To integrate elliptic integral, we often cut riemann surface and glue them together, and gain a torus. We do this to avoid indeterminacy of integral, in other word, integral is not path independent. ...
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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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Example based on the Hartogs's theorem.

I am reading an INTRODUCTION TO COMPLEX ANALYSIS (PART-2), FUNCTIONS OF SEVERAL COMPLEX VARIABLES all by myself. I have came across an example which is based on Hartogs's theorem in Shabat and I am ...
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1answer
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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

I found a possible solution to the following Putnam problem from the 2010 edition of the competition, and I was wondering whether my approach/solution is correct or whether I have made some ...
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1answer
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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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Does parametric function or multi-output function have Gradient?

I know what gradient is. It is just a pack of all the partial derivatives of a function. But how do you pack all the partial derivatives of a multi-output or vector-valued function?
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Defining critical points of a 2-variable function with boundary

I have a continuous function $f(x,y)$ where $x \in [a,+\infty); y \in [b, c]$. I want to find the global maximum of $f$ by comparing its values at critical points and encounter some problems. ...
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How to solve this equation in the entire complex plane?$z^z=t$

I ask wolfram for this question.solve 2^z=z^2 But I have some doubts. First,why $n$ is only belonging to $\{-1,0,1\}$,instead of belonging to $\mathbb N$ in this solution $e^{W_n{(\mathrm{Ln(t)})}}$. ...
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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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Finding branch points and branches for $(1 - |z-1| - 2 i \Im{\sqrt{1-z}})^{1/2}$

I made many many attempts to solve this problem, but there are some subtle confusions for me. I went like this. I substitute $1-z = r e^{i\theta + i 2 n \pi}$ and then $\sqrt{1-z} = \sqrt{r} \exp\left(...
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2answers
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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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1answer
100 views

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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1answer
80 views

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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2answers
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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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1answer
40 views

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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How to find the single valued branch for a complex multi-valued function : $f(z)=\sqrt{z^4-z^3}$?

I need to show that such function $f(z)=\sqrt{z^4-z^3}$ admits a single-valued branch on $C-[0,1]$. So how should I start with this kind of question, separate it into $\sqrt{z}*\sqrt{z}*\sqrt{z}*\sqrt{...
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Is this function from $\mathbb{R}^3 $ to $\mathbb{R}^4$ locally invertible at $(x,y,z)$?

Is this function from $\mathbb{R}^3 $ to $\mathbb{R}^4$ invertible at $(x,y,z)$? $k(x,y,z) = (x+y+z, e^x \cos z, e^x \sin z, \cos z)$ at $(x,y,z)$ I have only studied the Inverse Function Theorem ...
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1answer
37 views

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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38 views

Product and set division in multivalued function Complex Analysis

i know that $a^{\frac{1}{n}}=\lbrace z\in \mathbb{C}: z^n=a \rbrace$, now consider the following: $[(-1)^{\frac{1}{4}}]^2=[\lbrace 1, i, -1, -i\rbrace]^2=\lbrace 1,-1\rbrace$ the question is Why? ...
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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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1answer
58 views

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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1answer
45 views

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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27 views

What is the set of pairs of points in $\mathbb C$ generated from one pair by the AGM iteration and its inverse?

The arithmetic mean of two complex numbers $a,b$ is $\tfrac12(a+b)$, and the geometric mean is $\pm\sqrt{ab}$, which is two-valued. The arithmetic-geometric mean is the limit of the pair of sequences ...
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48 views

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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Best book on multivalued functions [duplicate]

We know that a multivalued function is a function that takes two or more values for the same value of z. This happens due to the same value of the argument of z as one goes counter clockwise about 0 ...
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3answers
293 views

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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1answer
68 views

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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1answer
115 views

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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1answer
50 views

marginal expectation of joint probability density function (difference in order of integration)

I am trying to compute $E[X_2]$ of a joint pdf with random variables $X_1$ and $X_2$ shown below: $f_{{1},{2}}(x_1,x_2)=\frac{1}{4}(x_1-x_2)e^{-x_1}, 0<x_1<\infty, -x_1<x_2<x_1 $, and $...
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2answers
40 views

Proving the continuity of a function using the definition

Proving the continuity of a function using the definition. The function is: $f(x,y) = e^x + sen(y)$ I don’t know where to start. usually I have a point in where I have to find if the function is ...
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1answer
37 views

Showing that the sequence $\left\{x_\lambda\right\}_{\lambda > 0}$ is bounded.

(Part of the following can be found at page 36 of "Non-linear Differential Equations of Monotone Types in Banach Spaces" of Viorel Barbu.) Note for the following, that $A$ is a coercive and maximal ...
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34 views

How do you call a function that is the gradient of a function?

Let us consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that there exists a real-valued function $F: \mathbb{R}^n \rightarrow \mathbb{R}$ that satisfies $\nabla F(x) =f(x)$. How would you ...
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91 views

A Poincare-Miranda theorem for multivalued functions

The Poincare-Miranda theorem is an extension of the intermediate value theorem to multi-dimensional functions. It is considered to be equivalent to Brouwer's fixed-point theorem. Both these theorems ...
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1answer
31 views

Limits of multivariable functions hints

I have tried switching to polar coordinates to no avail, I have tried using a gazillion path tests to no avail, I tried sandwiching to no avail and now I am really desperate, I even attempted an $\...
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19 views

Reference for Measurable Selection

Let $F(t,a):T\times A \to \mathbb{R}$, continuous in $a$ for each $t$ and measurable in $t$ for each $a$. Is there a selection theorem for the case $$ I(t) = \{a\in A: F(t,a) = B(t)\} $$ I found a ...

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