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. […]

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 […]

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?

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 […]

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 $\log(z^2)=2\log(z)$?

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 […]

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 […]

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