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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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 ...
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Will branch cut choice affect branches?
Will brach cut affect branches? Eg. $z^{\frac{1}{3}}$ there are three branches for $-\pi$ to $\pi$ branch cut: $r^{\frac{1}{3}} e^i \frac{\theta}{3}$, $r^{\frac{1}{3}} e^i \frac{\theta}{3}+i2\frac{\...
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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 ...
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what would the graph of this complex function would look like and the values for x if ζ(x,1)=1 i have attched the equation image below
zeta equation image
I made an attempt to calculate the value of the entire function, as indicated in the attached image. Specifically, I sought to determine the value of x at which ζ(x,1)=1 However, ...
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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 $(z+w)^{1/T}$:
$$I=\sum_{i}\binom{1/T}{i}z^{1/T}z^{-i}w^...
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Understanding the function $f(z) = \left(\frac{z}{z + x_0} \right)^{\kappa}$
I have the following complex function:
$$f(z) = \left(\frac{z}{z + x_0} \right)^{\kappa}$$
where $x_0 \in \mathbb R$ and $x_0 > 0$. $\kappa$ is a parameter for which we look at three cases:
case: $...
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Question about the equivalence of three versions of Closed Graph Theorem
I am studying set-valued analysis, and I saw three version of this Closed Graph Theorem.
Version 1: (what I was taught in class)
Let $\Gamma: X \to Y$ be a correspondence. If $\Gamma$ is closed-...
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Prove the (path-) connectedness of the graph of a compact- and convex-valued upper hemi-continuous correspondence.
Let $\Gamma: [0, 1] \to \mathbb{R}$ be a compact- and convex-valued, upper hemi-continuous correspondence. Prove that the graph of $\Gamma$ is a connected set. Is it path-connected?
This is what I ...
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The existence of continuous approximate selection
I'm looking for a condition of the existence of continuous approximate selection for a (multi-valued) minimiser mapping.
That is, let's say a continuous function $f: X \times U \to \mathbb{R}$ is ...
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Find all partial derivatives
We have a funcion of 2 variables $z = g(x,y)$. And $x = cos(t)$,$y = sin(t)$. Find all partial derivatives
is it right that
$$\frac{∂z}{∂x}=\frac{∂}{∂x}g(x,y)*(x)'_x=\frac{∂}{∂x}g(x,y)$$
$$\frac{∂z}{∂...
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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}^...
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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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Expressing correspondence via inequalities
Consider a correspondence $C: X \rightrightarrows Y$, that is defined as $C(x) \doteq \{ f_s(x) \mid s \in S \}$ where for all $s \in S$, $f_s$ is a concave function.
Can I re-express this ...
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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 ...
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Matrix derivative over non-scalar function
I'm attempting to obtain a derivative for the following function by matrix $U$:
$$R = \sum_{i=1}^n||ƒ(s ⋅ U) ⋅ V ⋅ M_i - f(t ⋅ U) ⋅ V|| ^ 2$$
where $s \in \mathbb{R}^{1\times3}$, $t \in \mathbb{R}^{1\...
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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{...
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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:
$r = 1$...
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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 $\nabla^2 f(x_0)$...
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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
$$ (x_1,y_1)=(5,1) \quad \text{we have} \quad f(x_1,y_1)=5(23-\sqrt{5})$...
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If $f$ is a non constant entire function which is real on the real axis then $\arg f(\overline{z})=-\arg f(z) $
If $f$ is a non constant entire function which is real on the real axis then prove that $$\arg f(\overline{z})=-\arg f(z) $$
where $z=x+iY$, $0<x<1$ and $f$ is non zero on the horizontal line ...
user1084306
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Function arrow notation in multivariable functions
How can I implement the function arrow notation in a multivariable function?
I know that $f:\mathbb{R}\rightarrow\mathbb{Z}$ means that the domain is $\mathbb{R}$ and the range is $\mathbb{Z}$ for ...
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If $D \subset \mathbb C\setminus\{0\}$ is an open connection....
a) If $D \subset \mathbb C\setminus\{0\}$ is an open connection. Show that if $\theta_1$ and $\theta_2$ are argument branches in $D$, then there is $k \in Z$ such that $\theta_1(z) = \theta_2(z) + 2k\...
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Calculate total derivative directly.
Calculate directly (not via partial differentiation) the total derivative of the function $f(x_1,x_2)=x_1^2-10x_2.$ You may wish to use the fact that $\sqrt{x^2+y^2}\geq\frac{x+y}{2}.$
$$$$
For the ...
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How to solve equations involving multivalued functions in complex domain?
Does this equation have a solution in the complex domain?
$$
\sqrt{x+3} = 3 + \sqrt{x}
$$
Squaring both sides gives $\sqrt{x}=-1$, which suggests the solution to be $x=1$. But:
How can the complex ...
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Alternative to showing that $J(m,n)=\frac{1}{2}[(m+n)^2+3m+n], \ J:\mathbb{N}^2\to \mathbb{N}$ is bijective
I came up with the following proof, but it seems too complicated so I was wondering if anyone else has a simpler idea.
Proof. The idea is to notice that $J(m,n)$ can also be expressed equivalently as
$...
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Clebsch variables
I should find if it is true that I can formalize the velocity in BE-condensate with a clebsch decomposition.
We know that $$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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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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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!...