Elliptic integral
An elliptic integral is any function f which can be expressed in the form
- <math> f(x) = \int_{c}^{x} R(t,P(t))\ dt </math>
where R is a rational function of its two arguments, P is the square root of a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.
Particular examples include:
- The complete elliptic integral of the first kind K is defined as
- <math> K(x) = \int_{0}^{1} \frac{1}{ \sqrt{(1-t^2)(1-x^2 t^2)} }\ dt </math>
- <math> K(x) = \int_{0}^{1} \frac{1}{ \sqrt{(1-t^2)(1-x^2 t^2)} }\ dt </math>
- and can be computed in terms of the arithmetic-geometric mean.
- The complete elliptic integral of the second kind E is defined as
- <math> E(x) = \int_{0}^{1} \frac{ \sqrt{1-x^2 t^2} }{ \sqrt{1-t^2} }\ dt </math>
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Upon thy princely state here,
From year to year
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A PRESENT, BY A CHILD
Go, pretty child.html">child, and bear this flower
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Made of a clean straight oaten reed,
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Then never take a second on,
*256*
GRACE FOR A CHILD
Here, a little child, I stand,
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Amen.
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When the Judgment is reveal'd,
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*258*
TO DEATH
Thou bidst me come away,
Than for to shed some tears
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And next, to take a bit
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And so to travel hence
.
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