Fresnel integrals

[rwst]

The SymPy-Sage interface raises an error because there are no Fresnel integral functions in Sage:

sage: integrate(sin(x^2), x, algorithm='sympy')
...
AttributeError: 

see:

sage: import sympy
sage: x = sympy.Symbol('x')
sage: sympy.integrate(sympy.sin(x^2), x)
3*sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*x/sqrt(pi))*gamma(3/4)/(8*gamma(7/4))

They are holonomic so they should be in Sage.

• https://en.wikipedia.org/wiki/Fresnel_integral
• http://mathworld.wolfram.com/FresnelIntegrals.html

Depends on #24425

CC: @mforets

Component: symbolics

Author: Marcelo Forets

Branch/Commit: f49ce18

Reviewer: Ralf Stephan

Issue created by migration from https://trac.sagemath.org/ticket/24212

[mforets]

Commit: e52fb2e

[mforets]

Branch: u/mforets/fresnel_integrals

[mforets]

comment:2

This is a tentative implementation of sine Fresnel integral. The numerical evaluation relies on mpmath. I doubted about how to write the custom evaluation _eval_ method. In fact I have two failing doctests. I expect:

sage: fresnel_sin(0)
0
sage: fresnel_sin(x).subs(x==0)
0

whereas i get fresnel_sin(0) in both cases. Any hints?

[mforets]

comment:3

Oh, and the beta(... --> beta... is because the missing parentheses were confusing my text editor, so that the entire file from that line on was shown as comments. I should remember to undo this change :)

[rwst]

comment:4

Returning None (or just passing) in _eval_ amounts to returning a held expression. _evalf_ is only used for inexact arguments or .n():

sage: fresnel_sin(0.)
0.000000000000000
sage: fresnel_sin(0).n()
0.000000000000000

For results to exact or symbolic arguments you need to implement them in _eval_.

[rwst]

comment:5

You would just need to handle 0 and oo in _eval_, and handle oddness before it, i.e. if arg.is_negative(): return -fresnel_sin(-arg).

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

f158660

symbolics in fresnel_sin eval

[sagetrac-git]

Changed commit from e52fb2e to f158660

[mforets]

comment:7

ok, that fixed the 0, oo and -oo evaluations. i added two new tests for the I*oo cases, but this is still wrong. since

sage: SR(I*oo).is_infinity()
True

probably i could use x.is_real() too.

[rwst]

comment:8

You mean unsigned_infinity?

[mforets]

comment:9

i should relate unsigned_infinity to test if the argument is complex and infinite? the doc of unsigned_infinity is precarious:

sage: unsigned_infinity?
Type:           UnsignedInfinity
String form:    Infinity
File:           ~/Tools/sage-master/local/lib/python2.7/site-packages/sage/rings/infinity.py
Docstring:         Initialize "self".
Init docstring:    Initialize "self". 

i'm tempted to add a ticket to explain this object more.

[rwst]

comment:10

Replying to @mforets:

i should relate unsigned_infinity to test if the argument is complex and infinite?

No, I was not sure what you had in mind. If you want to test if the argument is complex and infinite then do: if arg.is_infinity() and not arg.is_real(). Technically a False for arg.is_real() just means "I don't know" so you get True for SR(unsigned_infinity).is_real(), but the correct behaviour must wait until we have is_complex(). Just use not arg.is_real() for now.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

ada47e4

handle complex infinities

[sagetrac-git]

Changed commit from f158660 to ada47e4

[sagetrac-git]

Changed commit from ada47e4 to 869020b

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

869020b

add cosine Fresnel integral

[mforets]

Author: Marcelo Forets

[mforets]

comment:14

oh, there's no conversion back to sage:

sage: fresnel_sin(x)._sympy_() # ok
fresnels(x)
sage: fresnel_sin(x)._sympy_()._sage_() # not working!

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

e930137

add sympy -> sage conversion, and doctest

[sagetrac-git]

Changed commit from 869020b to e930137

[rwst]

comment:16

See how easy it is to add conversions now.

[mforets]

comment:17

See how easy it is to add conversions now.

awesome! it took me what, 15 minutes? and it was the first time i was doing that.

[rwst]

comment:18

Patchbot has relevant doctest errors.

[rwst]

Dependencies: #24425

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

a17755c	24425: Fix inherently failing random_expr doctest
96521a4	Merge branch 't/24425/fix_inherently_failing_random_expr_doctest' into t/24212/24212-2

[sagetrac-git]

Changed commit from e3cd31a to 96521a4

[rwst]

comment:44

Dependency needs review.

[sagetrac-git]

Changed commit from 96521a4 to e4f1592

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

e4f1592

Merge branch 'develop' into t/24212/24212-2

[mforets]

comment:46

The dependency has been approved and merged. Thanks.

I've merged with the latest development version and now i'm checking if the branch applies.

[mforets]

Changed commit from e4f1592 to none

[mforets]

Changed branch from u/rws/24212-2 to t/rws/24212-2

[mforets]

Changed branch from t/rws/24212-2 to u/mforets/24212-2

[mforets]

Commit: 61b2f76

[mforets]

New commits:

e3cd31a	24212: Fresnel integrals
96521a4	Merge branch 't/24425/fix_inherently_failing_random_expr_doctest' into t/24212/24212-2
e4f1592	Merge branch 'develop' into t/24212/24212-2
61b2f76	Merge branch 'develop' into t/24212/24212-2

[mforets]

comment:48

$ ./sage -t src/sage/calculus/calculus.py src/sage/functions/*
...
Git branch: t/24212/24212-2
Using --optional=mpir,python2,sage
...
----------------------------------------------------------------------
All tests passed!
----------------------------------------------------------------------
Total time for all tests: 106.7 seconds
    cpu time: 111.3 seconds
    cumulative wall time: 104.6 seconds

[tscrim]

comment:49

Quick question: do you want to change the test in calculus.py? In particular, do you still want to test that Sage handles the case when something in sympy gives something Sage doesn't know how to handle?

[mforets]

comment:50

Alright. I found this one:

sage: laplace(cos(-1/t), t, s, algorithm='sympy')
...
AttributeError: Unable to convert SymPy result (=meijerg(((), ()), ((-1/2, 0, 1/2), (0,)), s**2/16)/4) into Sage

(side note: i didn't read the syntax of sympy's meijerg function, but here is a quick comparison to WA's answer)

[rwst]

comment:51

Replying to @mforets:

sage: laplace(cos(-1/t), t, s, algorithm='sympy')
...
AttributeError: Unable to convert SymPy result (=meijerg(((), ()), ((-1/2, 0, 1/2), (0,)), s**2/16)/4) into Sage

For that, there is #17970.

[sagetrac-git]

Changed commit from 61b2f76 to f49ce18

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

f49ce18

add doctest with attributeerror

[rwst]

comment:54

When we have #17970 we'll likely convert the G expression to the Fresnel expression in Sage.

[vbraun]

Changed branch from u/mforets/24212-2 to f49ce18