# Elliptic integrals

1. Aug 24, 2008

### koolmodee

$$\int(1-x^{4})^{-1/2}dx$$
and the integral goes over [0,1]

The book I'm reading says this is an elliptic integral, meaning that it is the integral of a rational function of x and y in which y² is a polynomial in x, of degree 3 or 4, having simple roots.

What does the author possibly mean?

What y? Why y² polynomial in x? Why degree 3 0r 4? And yes, why having simple roots?

Could someone give some hints?

thank you

2. Aug 24, 2008

### HallsofIvy

Staff Emeritus
Actually, I am not familiar with that definition! An "elliptic integral" is a generalization of the kind of integral you get trying to calculate the circumference of an ellipse. In any case, here if you take y2 to be 1- x4, a fourth dergee polynomial with obviously simple roots, then $(1- x^4)^{-1/2}= \frac{1}{\sqrt{1- x^4}}= \frac{1}{\sqrt{y^2}}= \frac{1}{y}$. a rational function of y.

3. Aug 24, 2008

### tiny-tim

4. Aug 25, 2008

### koolmodee

many thanks!

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