Elliptic Function/Examples/First Kind
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Examples of Elliptic Functions
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:
\(\ds x\) | \(=\) | \(\ds \sn u\) | ||||||||||||
\(\ds \sqrt {1 - x^2}\) | \(=\) | \(\ds \cn u\) | ||||||||||||
\(\ds \sqrt {1 - k^2 x^2}\) | \(=\) | \(\ds \dn u\) |
Also see
Compare with:
- $x = \sin u$
where:
- $\ds u = \int_0^x \dfrac {\d t} {\sqrt {1 - t^2} }$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): elliptic functions
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): elliptic functions