More types of infinity transformations
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Description
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
$$
\int_{-\infty}^\infty g(t)\mathrm{d}t = \frac{\pi}{2}\int_{-1}^1 \sec^2\left(\frac{\pi t}{2}\right)g\left(\tan\frac{\pi t}{2}\right)\mathrm{d}t.
$$
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Research direction
Start by reviewing the existing infinite-bound transformations in Integrals.jl and the current integration entry points that select them. Compare the proposed cotangent and arctangent transforms with the existing default, then define the supported transformation API and validation needed for the new options; done means the alternatives are usable without displacing the current default.
Written by the indexing model from the issue text.
Assessment
- Tech stack
- julia
- Domain
- backend
- Issue type
- Feature
- Difficulty
- 5/5
- Estimated time
- Over a week
- Activity status
- Stale
- Clarity
- Needs clarification
- Newbie friendliness
- 35/100