Solving Elliptic Integrals to Understanding y² Polynomials in x

[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

 

[HallsofIvy]

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.

 

[tiny-tim]

Hi koolmodee!

For details, see http://en.wikipedia.org/wiki/Elliptic_integral

 

[koolmodee]

many thanks!