# 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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### 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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### 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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### 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 ...