How Do You Solve the Cosmological Distance Integral Using Elliptic Functions?

[Nilupa]

Please help me to find this integral.

∫(1/sqrt(1-(2/x)-((x^2)/3)))dx

 

[DonAntonio]

Please help me to find this integral.

∫(1/sqrt(1-(2/x)-((x^2)/3)))dx

This looks mean, but if you don't write it with LaTeX as this forum offers, then I won't waste my time trying to solve something that may be wrong because I misunderstood...

DonAntonio

 

[Nilupa]

this looks mean, but if you don't write it with latex as this forum offers, then i won't waste my time trying to solve something that may be wrong because i misunderstood...

Donantonio

 

[jackmell]

I believe it's formatted unambiguously Don. I'd rather Latex too however:

$$\int\frac{1}{\sqrt{1-\frac{2}{x}-\frac{x^2}{3}}}dx$$

I'll try a little to solve it without Mathematica. I couldn't so I input it into Mathematica and it returned the antiderivative in terms of special functions, in this case elliptic functions. Usually that means it has no simple antiderivative but not always.

 

[uart]

I agree with jackmel. I don't think that it has an anti-derivative that can be expressed in terms of standard functions.

Is there any context to this integral Nilupa. Do you have a particular definite integral in mind. Can you use a numerical solution?

 

[Ray Vickson]

This looks mean, but if you don't write it with LaTeX as this forum offers, then I won't waste my time trying to solve something that may be wrong because I misunderstood...

DonAntonio

He/she wrote it clearly, with all necessary parentheses, unlike so many posters herein.

That said, the problem is still "almost un-doable": the final result involves a horrible expression involving Elliptic functions.

RGV

 

[uart]

He/she wrote it clearly, with all necessary parentheses, unlike so many posters herein.

I hear what you're saying Ray. Having redundant parentheses is better than omitting essential ones (which, as you say, so often happens here). Still, it would have been a bit easier on the eye if the integrand was written with a few less parentheses.

1 / sqrt( 1 - 2/x - (x^2)/3 )

 

[clamtrox]

This is actually exactly the kind of integral I had to calculate once to find some distances in cosmology. It's not terribly difficult to find the solution in terms of elliptic integrals, all you need is to change variables to y=1/x and then do some scaling to get the proper form. Unfortunately there is no easy way to evaluate the elliptic functions, so unless you really need to spend as little cpu time as possible on doing the integral, you should just do it numerically from the beginning.