Numerical Evaluation
The duplication theorem can be used for a fast and robust evaluation of the Carlson symmetric form of elliptic integrals and therefore also for the evaluation of Legendre-form of elliptic integrals. Let us calculate : first, define, and . Then iterate the series
until the desired precision is reached: if, and are non-negative, all of the series will converge quickly to a given value, say, . Therefore,
Evaluating is much the same due to the relation
Read more about this topic: Carlson Symmetric Form
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