Elliptic Function/Examples
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Examples of Elliptic Functions
First Kind
Consider the incomplete elliptic integral of the first kind:
- $u = \ds \int_0^x \dfrac {\d t} {\sqrt {\paren {1 - t^2} \paren {1 - k^2 t^2} } }$
Then we have the following elliptic functions:
\(\ds x\) | \(=\) | \(\ds \sn u\) | ||||||||||||
\(\ds \sqrt {1 - x^2}\) | \(=\) | \(\ds \cn u\) | ||||||||||||
\(\ds \sqrt {1 - k^2 x^2}\) | \(=\) | \(\ds \dn u\) |