Integral representations of linear operators

Two numbers $ \sigma $ and $ \tau $ connected with rectangular Cartesian coordinates by the formulas

$$ x ^ {2} = \frac{( \sigma + a ^ {2} ) ( \tau + a ^ {2} ) }{a ^ {2} - b ^ {2} } , $$

$$ y ^ {2} = \frac{( \sigma + b ^ {2} ) ( \tau + b ^ {2} ) }{b ^ {2} - a ^ {2} } , $$

where $ - a ^ {2} < \tau < - b ^ {2} < \sigma < \infty $.

Figure: e035440a

The coordinate lines are (see Fig.): confocal ellipses ( $ \sigma = \textrm{ const } $) and hyperbolas ( $ \tau = \textrm{ const } $) with foci ( $ - \sqrt {a ^ {2} - b ^ {2} } , 0 $) and ( $ \sqrt {a ^ {2} - b ^ {2} } , 0 $). The system of elliptic coordinates is orthogonal. To every pair of numbers $ \sigma $ and $ \tau $ correspond four points, one in each quadrant of the $ xy $- plane.

The Lamé coefficients are

$$ L _ \sigma = \frac{1}{2} \sqrt { \frac{\sigma - \tau }{( \sigma + a ^ {2} ) ( \tau + b ^ {2} ) } } , $$

$$ L _ \tau = \frac{1}{2} \sqrt { \frac{\tau - \sigma }{( \sigma - a ^ {2} ) ( \tau + b ^ {2} ) } } . $$

In elliptic coordinates the Laplace equation allows separation of variables.

Degenerate elliptic coordinates are two numbers $ \widetilde \sigma $ and $ \widetilde \tau $ connected with $ \sigma $ and $ \tau $ by the formulas (for $ a = 1 $, $ b = 0 $):

$$ \sigma = \sinh ^ {2} \widetilde \sigma ,\ \ \tau = - \sin ^ {2} \widetilde \tau , $$

and with Cartesian coordinates $ x $ and $ y $ by

$$ x = \cosh \widetilde \sigma \cos \widetilde \tau ,\ \ y = \sinh \widetilde \sigma \sin \widetilde \tau , $$

where $ 0 \leq \widetilde \sigma < \infty $ and $ 0 \leq \widetilde \tau < 2 \pi $. Occasionally these coordinates are also called elliptic.

The Lamé coefficients are:

$$ L _ {\widetilde \sigma } = L _ {\widetilde \tau } = \ \sqrt {\cosh ^ {2} \widetilde \sigma - \cos ^ {2} \widetilde \tau } . $$

The area element is:

$$ d s = ( \cosh ^ {2} \widetilde \sigma - \cos ^ {2} \widetilde \tau ) d \widetilde \sigma d \widetilde \tau . $$

The Laplace operator is:

$$ \Delta \phi = \frac{1}{\cosh ^ {2} \widetilde \sigma - \cos ^ {2} \widetilde \tau } \left ( \frac{\partial ^ {2} \phi }{\partial \widetilde \sigma ^ {2} } + \frac{\partial ^ {2} \phi }{\partial \widetilde \tau ^ {2} } \right ) . $$

References

[a1]	G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Gauthier-Villars (1887) pp. 1–18