So I was toying around with the idea of recursive functions, where there are two variables, and one of them is on both sides of the equation. I stumbled upon/came up with this function: y = x^(1/y), or "the y th root of x equals y"
Now I started wondering how this function would look like. There are the obvious coordinates like (4|2) or (1|1). Pretty much everything after x = 1 is also pretty clear because there is one definitive y coordinate. x=0 is obviously not defined. But what about negative x values and values and values between 0 and 1? What would the function look like there?
I entered it into wolframalpha (because no other function plotter would accept my function) and it showed me this:
https://i.stack.imgur.com/LZKqG.jpg
What the hell does that mean? What is this strange blue part in the range of 0-1? And why is there no negative part? Wouldn't there at least be an imaginary negative part?
Could someone explain this to me and help me with plotting the function and showing me what it would really look like?
The function
As @Daryl points out $$ \begin{align} % y &= x^{\frac{1}{y}} \\ % \ln y &= \frac{1}{y} \ln x \\ % y \ln y &= \ln x \\ % x &= y^{y} \\ % \end{align} $$
Therefore, to answer your question, study the complex function $$ f(z) = z^{z} $$ where $z=x+i y$. The resolution follows $$ \begin{align} % \text{Re } f &= \left(x^2+y^2\right)^{x/2} e^{-y \arg (x+i y)} \cos \left(\frac{1}{2} y \ln \left(x^2+y^2\right)+x \arg (x+i y)\right), \\ % \text{Im } f &= \left(x^2+y^2\right)^{x/2} e^{-y \arg (x+i y)} \sin \left(\frac{1}{2} y \ln \left(x^2+y^2\right)+x \arg (x+i y)\right). % \end{align} % $$
Frequent sightings
You have selected a beautiful yet dangerous function. A common problem comes from the indeterminate form $0^{0}$. For example, careful mathematicians write the exponential function as $$ \color{blue}{e^{x} = 1 + \sum_{k=1}^{\infty} \frac{x^{n}}{n!}} $$ because this form is well defined at $x=0$. The common form $$ \color{red}{e^{x} = \sum_{k=0}^{\infty} \frac{x^{n}}{n!}} $$ is not defined at $x=0$.
Another place where users can stub their toe is in polynomial approximation. The most careful statement is $$ \color{blue}{y(x) = a_{0} + \sum_{k=1}^{d} a_{k} x^{k}}, $$ but a common form is $$ \color{red}{y(x) = \sum_{k=0}^{d} a_{k} x^{k}}. $$ The former is defined at $x=0$, the latter is not. A computer code using the latter form will throw an error when $x=0$.
Gallery
Zooming in to the origin.
This is a 3D plot of Im$f$, over the same range as the top plot of the above sequence.