Elliptic Function/Examples/First Kind

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