# Associated Legendre polynomials: complex vs real argument

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## Main Question or Discussion Point

I am having trouble understanding the relationship between complex- and real-argument associated Legendre polynomials. According to Abramowitz & Stegun, EQ 8.6.6,
$$P^\mu_\nu(z)=(z^2-1)^{\mu/2}\cdot\frac{d^\mu P_\nu(z)}{dz^\mu}$$
$$P^\mu_\nu(x)=(-1)^\mu(1-x^2)^{\mu/2}\cdot\frac{d^\mu P_\nu(x)}{dx^\mu}$$

Since the Legendre polynomials P_v(z) and P_v(x) don't differ by overall imaginary factors (EQ 8.6.18, Rodrigues' formula), it would seem that one could write
$$P^\mu_\nu(z)=\frac{P^\mu_\nu(x)}{i\,^\mu}$$

However, calculating the complex-argument polynomial from the real-argument polynomial this way gives numerically different values than using the complex formula directly. What am I missing in the relationship between these definitions? Thanks!

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

Some more info for clarification - from different sources, these expressions can be written (for $\nu=\mu=m$) as...

According to Abramowitz & Stegun, EQ 8.6.6,
$$P^m_m(z)=\frac{(z^2-1)^{m/2}}{2^m m!}\cdot\frac{d^{2m} (z^2-1)^m}{dz^{2m}}$$
According to Arfken 85, Section 12.5,
$$P^m_m(z)=\frac{(1-z^2)^{m/2}}{2^m m!}\cdot\frac{d^{2m} (z^2-1)^m}{dz^{2m}}$$
According to Wolfram MathWorld,
$$P^m_m(z)=(-1)^m\frac{(1-z^2)^{m/2}}{2^m m!}\cdot\frac{d^{2m} (z^2-1)^m}{dz^{2m}}$$

Testing some calculations arbitrarily for m=3, these come out to be...

Abram: $P^m_m(Cos(z))=-15\,i\,Sin^3(z)$
Arfken: $P^m_m(Cos(z))=15\,Sin^3(z)$
Wolfram: $P^m_m(Cos(z))=-15\,Sin^3(z)$
(note, Wolfram's appears to be real valued. Dividing by $i^m$ as in the OP would bring this one in line with Abramowitz.)

Clearly, all of these can't simultaneously be right. Something about the general state of the definition for $P^m_m(z)$ seems to have serious issues. Does anyone know what is going on here?

Supposedly, the $(-1)^m$ in Wolfram's definition is a phase which makes it differ from Arfken, but why is a phase included at all in something that is purely mathematical and not at all physical? Aren't the polynomials just solutions to a mathematical equation with no physical meaning?