Legendre functions can be of first and second kinds; $P$, $Q$.

They can have order $\mu$ and degree $\nu$; $P^\mu_\nu$ $Q^\mu_\nu$

But do they also have Types? Some of the numerical software I am using defines "types" for Legendre. In fact, most if not all seem to.

Mathematica defines "types" 2 and 3

http://functions.wolfram.com/HypergeometricFunctions/LegendreP2General/02/ http://functions.wolfram.com/HypergeometricFunctions/LegendreP3General/02/

And the MPmath software I am using(via Sagemath) does the same.


The types do not agree for non-integer orders $\mu$. Confusingly, there does not appear to be a type 1 in either source.

So my question is: What are these types? Are they just numerical conventions, or are they important mathematical subtleties which can affect the solutions to Legendre's differential equation? (I'm particularly worried about the behaviour of the exponent m/2 given in the formulae for irrational m on negative numbers.)


They differ in the choice of branch cuts. You can find an explanation here: http://reference.wolfram.com/mathematica/ref/LegendreP.html

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