content_image/331972.jpg content_image/331976.jpg content_image/331977.jpg content_image/331973.jpg content_image/331975.jpg
with 

Differentiating, we get 

Using the fact that Jacobi elliptic functions reduce to trigonometric functions as ν ′ → 0 and substituting 2 

we infer from Eq. (D7) 

and 

that confirms the extra π/ 2 in Eq. (51) for an ellipse in comparison with a rectangle. 

<img src='content_image/331972.jpg'>

<img src='content_image/331976.jpg'>

<img src='content_image/331977.jpg'>

<img src='content_image/331973.jpg'>

<img src='content_image/331975.jpg'>

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