separation constant

separation constant (plural separation constants)

1. (calculus) a constant that may be introduced upon separation of variables

   The partial differential equation ${\displaystyle a\sin ^{2}(\theta )\Phi (\phi )+\Phi (\phi )\cos(\theta ){\frac {\partial \Theta (\theta )}{\partial \theta }}+{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}=0}$ can be rewritten ${\displaystyle a\sin ^{2}(\theta )+\cos(\theta ){\frac {\partial \Theta (\theta )}{\partial \theta }}=-{\frac {1}{\Phi (\phi )}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}\equiv K}$, where ${\displaystyle K}$ is a separation constant that depends on neither ${\displaystyle \theta }$ nor ${\displaystyle \phi }$. This then yields two ordinary differential equations.