From a mathematical point of view, how do you determine the number of solution of a function?
I'll explain:
If I have a generic equation with a function (not considering polynomials):
$$f(x)=c$$
Depending on the type of function I can have multiple results:
$f(x)$ is a direct function
$$e^x=c \ \Rightarrow \ x=\ln(c)+2k\pi i \qquad k\in\mathbb{Z}$$ $$\sin(x)=c\ \Rightarrow \ x=2\pi k+\pi-\arcsin(c), x=2k\pi+\arcsin(c)\qquad k\in\mathbb{Z}$$ $$x e^x=c\ \Rightarrow \ x=W_{k}(x)\qquad k\in\mathbb{Z}$$
In this case the equation has an infinite number of solutions
$f(x)$ is an inverse function
$$\ln(x)=c \ \Rightarrow \ x=e^c$$ $$\arcsin(x)=c\ \Rightarrow \ x=\sin(c)$$ $$W(x)=c\ \Rightarrow \ x=c e^c$$
In this case the equation has only 1 solution
Therefore given a generic function $$f(x)=\sum_{n=n_0}^{\infty} a_n x^n$$
How can I determine if the equation $$f(x)=c$$
has 1 or infinite solutions?
I don't know if it's a trivial question, but it doesn't seem obvious to me. I know that for example the logarithm is defined as the inverse function of the exponential, but if I started by first defining the logarithm as the derivative of $1/x$ I would go so far as to say that the exponential is its inverse.