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

[rwst]

Description changed:

--- 
+++ 
@@ -1,7 +1,7 @@
 The SymPy-Sage interface raises an error because there are no Fresnel integral functions in Sage: 
 
 ```
-integrate(sin(x^2), x, algorithm='sympy')
+sage: integrate(sin(x^2), x, algorithm='sympy')
 ...
 AttributeError: 
 
@@ -13,3 +13,6 @@
 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

[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?