User:Tohline/Math/EQ Toroidal05
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<math>~ \int_a^b\biggl[(\nu - \sigma)(\nu + \sigma + 1) + (\rho^2 - \mu^2)(1 - z^2)^{-1} \biggr] w_\nu^\mu ~w_\sigma^\rho ~dz </math> |
<math>~=</math> |
<math>~ \biggl[ z(\nu-\sigma) w_\nu^\mu ~w_\sigma^\rho + (\sigma+\rho) w_\nu^\mu ~ w_{\sigma-1}^\rho - (\nu + \mu) w_{\nu - 1}^\mu ~w_\sigma^\rho \biggr]_a^b </math> |
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A. Erdélyi (1953): Volume I, §3.12, p. 169, eq. (1) |
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| where, <math>~w_\nu^\mu(z)</math> and <math>~w_\sigma^\rho(z)</math> denote any solutions of Legendre's differential equation |