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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!

$$P^\mu_\nu(z)=(z^2-1)^{\mu/2}\cdot\frac{d^\mu P_\nu(z)}{dz^\mu}$$

$$P^\mu_\nu(x)=(-1)^\mu(1-x^2)^{\mu/2}\cdot\frac{d^\mu P_\nu(x)}{dx^\mu}$$

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!