Elliptic integrals, general form According to https://mathworld.wolfram.com/EllipticIntegral.html, Elliptic integrals are integrals of the form of equation (1).
I am trying to convert an integral of the form
$$\int_{|x_+|}^\infty \frac{dx}{\sqrt{(x^2-x_-^2)(x^2-x_+^2)}}$$
To express as a function of elliptic integrals, where $|x_+|>|x_-|$, we can take $x_+>0$ for simplicity. We can easily see that this integral converges, and that it is real.
I have trying following the method outline in the webpage I linked, with no success, because of the arising of imaginary roots in the process. However, plugging this expression into mathematica with $x_-=1$, $x_+=2$, yields me the expression :
$$\int_{2}^\infty \frac{dx}{\sqrt{(x^2-1)(x^2-4)}}=\frac{i}{2}K(\frac{3}{4})+K(4)$$
Where $K(m) = \int_0^{\pi/2}\frac{1}{\sqrt{1-m\sin^2(\phi)}}d\phi$. Now clearly the integral I'm looking to re-express is well-defined, and real, but it would appear I get an complex number as an answer. However, looking closer, $K(4)$ will be complex because the parameter $m>1$. Thus, if we numerically compute the expression the imaginary parts cancel and we are left with a real answer, which is correct.
So my question is as follows : when we say that Elliptic integrals (as defined in my link) are expressible as a function of the three "kinds" of elliptic integrals (+ elementary functions), do we allow "complex" elliptic integrals ? In other words, is there any way to express my example as a function of "well defined" elliptic integrals, in the sense that we only integrate real numbers ?
I am unsure if I expressed myself clearly, I hope it makes sense.
 A: Byrd and Friedman 215.00 gives the answer as
$$\frac1{x_+}K\left(m=\frac{x_-^2}{x_+^2}\right)$$
By the constraints on the variables, the argument always lies in $[0,1]$. This expression is related to the one in the OP's answer by a Gauss transformation.
Nevertheless, elliptic integrals may be evaluated at complex numbers with no hassle, and indeed they are required in a few applications like parametrising minimal surfaces.
A: One way to express this integral as a function of the Elliptic Integral K, is as follows. Do the change of variables $x^2 = \frac{x_0+^2}{1-t^2}$. This converts the integral into :
$$
\frac{1}{\sqrt{x_+^2-x_-^2}}\int_0^1 \frac{dt}{\sqrt{(1-t^2)(1+\frac{x_-^2}{x_+^2-x_-^2}t^2)}}
$$
Which can be written formally as $\frac{1}{\sqrt{x_+^2-x_-^2}}K(-\frac{x_-^2}{x_+^2-x_-^2})$. This is fine, but there is an abuse of notation. Usually Elliptic functions are defined for $K(m), 0<m<1$, but here we have a negative $m$. Nonetheless, the integral is defined and convergent, but in principle it is not an Elliptic integral according to the definition.
