What does it mean for a solution to be regular (Legendre equation)?

Question

For example, Wikipedia says:

When $n$ is an integer, the solution $P_{n}(x)$ that is regular at $x = 1$ is also regular at $x =−1$, and the series for this solution terminates (i.e. it is a polynomial).

What is 'regular' in this context?

Answer 1

Regular means that the function is not singular in the point mentioned; singular points being the ones where you divide by zero. Of course there are singular points of functions that may be resolved, for instance $f(x)=\frac{x}{x}$, but other functions have singular points that cannot be resolved, like $f(x)=\frac{1}{x}$ or indeed the other solutions of the Legendre differential equation, for example $Q_0(x)=\frac{1}{2}ln(\frac{1+x}{1-x})$, which is singular (not regular) in $x=1$