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Fresnel Integral Functions #938

Merged
merged 37 commits into from
May 12, 2012
Merged

Fresnel Integral Functions #938

merged 37 commits into from
May 12, 2012

Commits on May 12, 2012

  1. First implementation of Fresnel integrals

    Raoul Bourquin committed May 12, 2012
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  2. Complex expansion for Fresnel integrals

    Raoul Bourquin committed May 12, 2012
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  3. Asymptotic series expansions for Fresnel integrals

    Raoul Bourquin committed May 12, 2012
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  4. Leading term expansion for Fresnel integrals

    Raoul Bourquin committed May 12, 2012
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  5. Numerical evaluation for Fresnel integrals using mpmath

    Raoul Bourquin committed May 12, 2012
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  6. Series expansions for Fresnel integrals

    Raoul Bourquin committed May 12, 2012
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  7. Documentation and examples for Fresnel integrals

    Raoul Bourquin committed May 12, 2012
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  8. Better asymptotic series expansions for Fresnel integrals

    Raoul Bourquin committed May 12, 2012
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  9. Examples in an IPython notebook

    Raoul Bourquin committed May 12, 2012
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  10. Fixed numerical evaluation of Fresnel integrals.

    Raoul Bourquin committed May 12, 2012
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  11. Updated examples in an IPython notebook

    Raoul Bourquin committed May 12, 2012
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  12. Renamed Fresnel integrals to 'fresnels' and 'fresnelc'

    Raoul Bourquin committed May 12, 2012
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  13. Add testcases for the Fresnel integrals

    Raoul Bourquin committed May 12, 2012
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  14. Adding Fresnel integrals to the hyperexpand Tables

    Raoul Bourquin committed May 12, 2012
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  15. Declared the Fresnel integrals to be unbranched.

    Raoul Bourquin committed May 12, 2012
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  16. Apply comments from pull request review

    Raoul Bourquin committed May 12, 2012
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  17. Added Meijer G representation of Fresnel Integrals

    Raoul Bourquin committed May 12, 2012
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  18. Caching 'taylor_term' functions

    Raoul Bourquin committed May 12, 2012
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  19. Improved Taylor terms of Fresnel integrals

    Raoul Bourquin committed May 12, 2012
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  20. Test integrals defining Fresnel functions

    Raoul Bourquin committed May 12, 2012
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  21. Addressing latest comments on code

    Raoul Bourquin committed May 12, 2012
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  22. Rules to rewrite fresnel{s,c} functions in terms of 1F2 hypergeometri… 

    …c functions
    Raoul Bourquin committed May 12, 2012
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  23. Fixed hypergeometric formula for Fresnel functions

    Raoul Bourquin committed May 12, 2012
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  24. Fix test_args test failures

    Raoul Bourquin committed May 12, 2012
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  25. Fix integration of Fresnel functions

    Raoul Bourquin committed May 12, 2012
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  26. Add tests for Fresnel integration

    Raoul Bourquin committed May 12, 2012
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  27. A different patch for the gamma pole issue (by ness)

    Raoul Bourquin committed May 12, 2012
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  28. Tests for the gamma pole patch

    Raoul Bourquin committed May 12, 2012
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  29. A new rule in hyperexpand to get nice results for the Laplace transfo… 

    …rm of Fresnel functions

Examples without this formula:

  laplace_transform(fresnels(t), t, s, noconds=True)

   ⎛         ⎛         │    4 ⎞        ⎛  2⎞        ⎛  2⎞⎞
   ⎜     ┌─  ⎜   1     │  -s  ⎟        ⎜ s ⎟        ⎜ s ⎟⎟
  -⎜2⋅s⋅ ├─  ⎜         │ ─────⎟ - π⋅sin⎜───⎟ - π⋅cos⎜───⎟⎟
   ⎜    1╵ 2 ⎜3/4, 5/4 │     2⎟        ⎝2⋅π⎠        ⎝2⋅π⎠⎟
   ⎝         ⎝         │ 16⋅π ⎠                          ⎠
  ────────────────────────────────────────────────────────
                           2⋅π⋅s

  laplace_transform(fresnelc(t), t, s, noconds=True)

            ⎛         │    4 ⎞           ⎛  2⎞           ⎛  2⎞
     3  ┌─  ⎜   1     │  -s  ⎟      2    ⎜ s ⎟      2    ⎜ s ⎟
  2⋅s ⋅ ├─  ⎜         │ ─────⎟ - 3⋅π ⋅sin⎜───⎟ + 3⋅π ⋅cos⎜───⎟
       1╵ 2 ⎜5/4, 7/4 │     2⎟           ⎝2⋅π⎠           ⎝2⋅π⎠
            ⎝         │ 16⋅π ⎠
  ────────────────────────────────────────────────────────────
                                2
                             6⋅π ⋅s

And with this formula we can expand the 1F2 above:

  laplace_transform(fresnels(t), t, s, noconds=True)

       ⎛  2⎞                  ⎛  2⎞      ⎛  2⎞                  ⎛  2⎞
       ⎜ s ⎟         ⎛s⎞      ⎜ s ⎟      ⎜ s ⎟         ⎛s⎞      ⎜ s ⎟
    sin⎜───⎟⋅fresnels⎜─⎟   sin⎜───⎟   cos⎜───⎟⋅fresnelc⎜─⎟   cos⎜───⎟
       ⎝2⋅π⎠         ⎝π⎠      ⎝2⋅π⎠      ⎝2⋅π⎠         ⎝π⎠      ⎝2⋅π⎠
  - ──────────────────── + ──────── - ──────────────────── + ────────
             s               2⋅s               s               2⋅s

  laplace_transform(fresnelc(t), t, s, noconds=True)

     ⎛  2⎞                  ⎛  2⎞      ⎛  2⎞                  ⎛  2⎞
     ⎜ s ⎟         ⎛s⎞      ⎜ s ⎟      ⎜ s ⎟         ⎛s⎞      ⎜ s ⎟
  sin⎜───⎟⋅fresnelc⎜─⎟   sin⎜───⎟   cos⎜───⎟⋅fresnels⎜─⎟   cos⎜───⎟
     ⎝2⋅π⎠         ⎝π⎠      ⎝2⋅π⎠      ⎝2⋅π⎠         ⎝π⎠      ⎝2⋅π⎠
  ──────────────────── - ──────── - ──────────────────── + ────────
           s               2⋅s               s               2⋅s

This result is nice and correct.
    Raoul Bourquin committed May 12, 2012
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  30. Adapt tests to new Laplace transformations of Fresnels

    Raoul Bourquin committed May 12, 2012
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  31. Fix a remaining test failure

    Raoul Bourquin committed May 12, 2012
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  32. Rewrite Fresnel functions in terms of Meijer G

    Raoul Bourquin committed May 12, 2012
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  33. Tests for Fresnel rewrites

    Raoul Bourquin committed May 12, 2012
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  34. Fresnel function latex printing

    Raoul Bourquin committed May 12, 2012
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  35. Add integral example to Fresnel docstring

    Raoul Bourquin committed May 12, 2012
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  36. Addressed points from review discussion

    Raoul Bourquin committed May 12, 2012
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  37. Manually tuned rule for Laplace transform of Fresnels

    Raoul Bourquin committed May 12, 2012
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