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The mean along the eccentric anomaly of an ellipse log distance to a point within the ellipse

Conjecture. Let $$ f(r,\alpha,p, \theta) = \ln\left(\left(r\sin\alpha-\sin\theta\right)^{2}\left(1-p\right)^{2}+\left(r\cos\alpha-\cos\theta\right)^{2}\left(1+p\right)^{2}\right). $$ Then for any ...
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