[Python] How to understand the type of Legendre function defined in both mpmath and Mathematica

I am working on a problem which involved the associated Legendre function of the first and second kind, of order one and half integer degree, denoted as P_{n-1/2}^1(z) and Q_{n-1/2}^1(z), here n is integer and goes from 0 to a certain positive integer.

For my current problem, z is always bigger than 1.

I tried with both mpmath and Mathematica and they deliver the same results. BUT the result is complex numbers which is out of expectation because I am calculating some real physical quantites.

I was stuck for a while and today I read there is actually a hidden parameter for both P and Q in mpmath and Mathematica, that is the type of the branch cut.

https://reference.wolfram.com/language/ref/LegendreP.html

https://mpmath.org/doc/current/functions/orthogonal.html

I tried to use type 3 and now the result is essentially real numbers with negligible imaginary part. So my question is what is this branch cut and how to choose the correct type?

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