More types of infinity transformations

[DanielVandH]

Currently the only type of transformation used for handling infinite bounds is $u \mapsto a + t/(1-t)$ and $u \mapsto t/(1-t)^2$. It could be nice to eventually support other transformations.

In some of the integrals I've dealt with in my work, I've found that transformations like

$$ s = -\cot\left[\frac{\left(\pi + 2\arctan(a)\right)(\xi+1)}{4}\right], \quad -1 < \xi < 1, $$

and

$$ s = -\cot\left[\frac{\left(\pi - 2\arctan(a)\right)(s-1)}{4}\right], \quad -1 < \xi < 1, $$

could be useful, giving the (rather complicated..) results

$$ \int_{-\infty}^a g(s)\mathrm{d}s = \frac{1}{4}\left(\pi + 2\arctan(a)\right)\int_{-1}^1 \csc^2\left[\frac{1}{4}\left(s+1\right)\left(2\arctan(a) + \pi\right)\right]g\left(-\cot\left[\frac{\left(\pi + 2\arctan(a)\right)(s+1)}{4}\right]\right)\mathrm{d}s, $$

$$ \int_a^\infty g(s)\mathrm{d}s = \frac{1}{4}\left(\pi - 2\arctan(a)\right)\int_{-1}^1 \csc^2\left[\frac{1}{4}\left(s-1\right)\left(\pi - 2\arctan(a)\right)\right]g\left(-\cot\left[\frac{\left(\pi - 2\arctan(a)\right)\left(s-1\right)}{4}\right]\right)\mathrm{d}s $$

which typically gave better results when applying Gauss-Legendre quadrature afterwards. These integrals I dealt with had issues with oscillations and singularities, etc., so the currently used transform is still a good default. Another useful transform is $t \mapsto (2/\pi)\arctan(t)$, giving

$$ \int_{-\infty}^\infty g(t)\mathrm{d}t = \frac{\pi}{2}\int_{-1}^1 \sec^2\left(\frac{\pi t}{2}\right)g\left(\tan\frac{\pi t}{2}\right)\mathrm{d}t. $$

[ChrisRackauckas]

We can add an option to the problem type.

[lxvm]

Note that in #241 I've added an internal API that should make it easier to add these infinity transformations and other changes of variables.