wrong sign with function Ei and limit

[Daniel Krenn]

We have

sage: (Ei(-Z)).limit(Z=oo)
Infinity

which is wrong, since it should be 0. It seems that there goes something wrong with the - sign. This can be also seen here:

sage: (Ei(-Z)).limit(Z=1000)   
-Ei(1000)
sage: (Ei(-Z)).limit(Z=1000).n()
-1.97204513714124e431

But

sage: Ei(-1000).n()
-5.07089306023517e-438
sage: -Ei(1000).n()
-1.97204513714124e431

which are the correct results.

---

Apply trac_13271-Ei-maxima.2.patch​ to Sage library.

[Karl-Dieter Crisman]

Just as a data point, we still get this after #11143, unsurprisingly given that that doesn't touch Maxima.

This is a little weird, since in both Maxima in current Sage and in Maxima 5.27

(%i5) limit(expintegral_ei(-z),z,inf);
(%o5)                                  0

Further,

sage: A = Ei(-Z)._maxima_()
sage: A.limit(Z,'inf')
0

Using the Maxima lib version is ok too.

I have a feeling that the following is the problem,

sage: sage.interfaces.maxima_lib.sr_to_max(SR(Ei(-Z)))
<ECL: ((%GAMMA_INCOMPLETE) 0 ((MTIMES) |$z| -1))>

combined with

(%i1) limit(gamma_incomplete(0,-Z),Z,inf);
(%o1)                              infinity

So somehow this is completely bypassing the symbol table. I am out of my depth in our ECL conversions...

[Karl-Dieter Crisman]

Or maybe I'm not.

sage.functions.other.Ei : lambda X : [[max_gamma_incomplete], 0, X]

which is ​here. I was kind of wondering about this in #11143, but since that didn't touch that I figured it was okay. ​Mathworld seems to indicate that this conversion isn't quite right (where by "isn't quite right" I mean "wrong"), and who knows what this would imply for complex Z. Yikes.

[Karl-Dieter Crisman]

In fact, this makes no sense whatsoever. We should change the two lone references to max_gamma_incomplete to use the appropriate thing for Ei instead. This dates from the very start of the ECL conversions at #7377. Unless the Maxima expintegral_ei is somehow less powerful than the incomplete gamma?

I'm going to try just eliminating all that.

[Karl-Dieter Crisman]

Huh.

Apply

sage/interfaces/maxima_lib.py

@@ -1165 +1165 @@
 max_psi=EclObject("$PSI")
 max_array=EclObject("ARRAY")
 mdiff=EclObject("%DERIVATIVE")
-max_gamma_incomplete=sage_op_dict[sage.functions.other.gamma_inc]
 max_lambert_w=sage_op_dict[sage.functions.log.lambert_w]
 
 def mrat_to_sage(expr):
@@ -1358 +1357 @@
     sage.functions.log.polylog : lambda N,X : [[mqapply],[[max_li, max_array],N],X],
     sage.functions.other.psi1 : lambda X : [[mqapply],[[max_psi, max_array],0],X],
     sage.functions.other.psi2 : lambda N,X : [[mqapply],[[max_psi, max_array],N],X],
-    sage.functions.exp_integral.Ei : lambda X : [[max_gamma_incomplete], 0, X],
     sage.functions.log.lambert_w : lambda N,X : [[max_lambert_w], X] if N==EclObject(0) else [[mqapply],[[max_lambert_w, max_array],N],X]
 }

to get

sage: var('Z')
Z
sage: (Ei(-Z)).limit(Z=oo)
0
sage: (Ei(-Z)).limit(Z=1000)
Ei(-1000)
sage: (Ei(-Z)).limit(Z=1000).n()
-5.07089306023517e-438

Tests look good.

[Karl-Dieter Crisman]

In theory, fixing this doesn't need #11143, but might as well do it on that since it's about the same thing.

Oops, still need to add doctests...

[Karl-Dieter Crisman]

Not that I'm guaranteeing this didn't mess anything else up. But it seems like a better solution overall than keeping the old hack.

[Benjamin Jones]

Looks good to me, and nice catch. I think it makes sense to have this ticket depend on 11143 so the doctests can be added to the right place. Hopefully that ticket will make it through the gauntlet soon!

I noticed missing indentation in the doctest block you added. I'll post a quick reviewer patch. If it looks ok to you, I'll set this to positive review.

[Benjamin Jones]

added indentation to new doctest block

[Karl-Dieter Crisman]

I noticed missing indentation in the doctest block you added. I'll post a quick reviewer patch. If it looks ok to you, I'll set this to positive review.

Please do. I can't believe I indented them wrong - that's what happens when you're doing a patch at 11 PM. Thanks for catching this.