Articles of multivalued functions

Why are vector valued functions 'well-defined' when multivalued functions aren't?

I’m looking for an ‘intuitive’ answer here, because I have no formal mathematical training but find myself in a comparatively math-heavy PhD (visual perception; lots of neuroscientists on the one side and CS folk on the other). Only functions which map some number of inputs to a single output are considered ‘true’ or ‘well-defined’ functions. […]

Are multi-valued functions a rigorous concept or simply a conversational shorthand?

In Brown and Churchill’s book, the concept of multivalued functions is not discussed in a very rigorous way (if at all). But I can see that branch cuts have importance in complex analysis, so I want to clarify my understanding of multivalued functions. Is there a rigorous development of the definition of a multivalued function […]

Why do I get one extra wrong solution?

I’m trying to solve this equation: $$2-x=-\sqrt{x}$$ Multiply by (-1): $$\sqrt{x}=x-2$$ power of 2: $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$ But $1$ is not a correect solution to the original equation. Why have I got it? I’ve never got a wrong solution to an equation before. What is so special here?

Complex towers: $i^{i^{i^{…}}}$

If $w = z^{z^{z^{…}}}$ converges, we can determine its value by solving $w = z^{w}$, which leads to $w = -W(-\log z))/\log z$. To be specific here, let’s use $u^v = \exp(v \log u)$ for complex $u$ and $v$. Two questions: How do we determine analytically if the tower converges? (I have seen the interval […]

Is $\log(z^2)=2\log(z)$ if $\text{Log}(z_1 z_2)\ne \text{Log}(z_1)+\text{Log}(z_2)$?

For any $z1, z2$ in $\mathbb{C} \setminus {0}$, $\log(z_1 z_2)=\log(z_1)+\log(z_2)$, but in general $\text{Log}(z_1 z_2)\ne \text{Log}(z_1)+\text{Log}(z_2)$. Is $\log(z^2)=2\log(z)$?

If a function can only be defined implicitly does it have to be multivalued?

What is the general reason for functions which can only be defined implicitly? Is this because they are multivalued (in which case they aren’t strictly functions at all)? Is there a proof? Clarification of question. The second part has been answered with example of single value function which cannot be given explicitly. The third part […]

Branch point-what makes a closed loop around it special?

I am having difficulty understanding the concept of a branch point of a multifunction. It is typically explained as follows:branch point is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point. What I am unable to understand is what is special about the set of points […]