Elliptic Integral - Incomplete Elliptic Integral of The Second Kind

The incomplete elliptic integral of the second kind E in trigonometric form is

Substituting, one obtains Jacobi's form:

Equivalently, in terms of the amplitude and modular angle:

Relations with the Jacobi elliptic functions include

The meridian arc length from the equator to latitude is written in terms of E:

where a is the semi-major axis, and e is the eccentricity.

Read more about this topic:  Elliptic Integral

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