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Mathematics
General Math
Associated Legendre polynomials: complex vs real argument
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[QUOTE="avikarto, post: 5510377, member: 559932"] 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? [/QUOTE]
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Mathematics
General Math
Associated Legendre polynomials: complex vs real argument
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