implement symbolic lower incomplete gamma function

[rwst]

This is actually a defect because we leave a result from Maxima undefined:

sage: hypergeometric([1],[b],x).simplify_hypergeometric()
(b - 1)*x^(-b + 1)*e^x*gamma_greek(b - 1, x)
sage: gamma_greek
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-18-6e90901bc5cb> in <module>()
----> 1 gamma_greek

NameError: name 'gamma_greek' is not defined

https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function

Mathematica seems to have Gamma[a,z] for upper and Gamma[a,0,z] for lower; Maple seems to have upper Gamma. gamma_inc (the upper one in Sage) gets immediately converted to gamma(a,x). The symbolic functions gamma_inc==incomplete_gamma are converted and never returned to the user as expression:

sage: gamma_inc(x,x,hold=True)
gamma(x, x)
sage: incomplete_gamma(x,x,hold=True)
gamma(x, x)
sage: assume(x>0)
sage: integral(t^(s-1)*e^(-t),t,0,x)
-gamma(s, x) + gamma(s)

This ticket should deprecate "incomplete_gamma" and add the symbolic function gamma_inc_lower, leaving open the question of the global alias for and the displayed name of Function_gamma_inc.

Previous part of description:

---

1. Provide all three "user input interfaces" gamma_inc, incomplete_gamma and lower_incomplete_gamma, and convert the Maxima gamma_greek to -gamma(a, x) + gamma(a)
2. Provide all three "user input interfaces" gamma_inc, incomplete_gamma and lower_incomplete_gamma, and convert to gamma(a,x) and gamma(a,0,x)n on output
3. Provide all three "user input interfaces" gamma_inc, incomplete_gamma and lower_incomplete_gamma, and have incomplete_gamma and lower_incomplete_gamma as result instead of `gamma(...)'
4. Change the user interface completely to lower_incomplete_gamma and upper_incomplete_gamma
5. Change the user interface completely to gamma_inc_lower and gamma_inc_upper
6. Change the user interface completely to gamma(a,x) and gamma(a,0,x)

Depends on #20185

CC: @kcrisman @nbruin

Component: symbolics

Keywords: gamma, incomplete, special, functions

Author: Ralf Stephan

Branch/Commit: 157b268

Reviewer: Buck Evan, Paul Masson

Issue created by migration from https://trac.sagemath.org/ticket/16697

[nbruin]

comment:2

We don't need all of gamma, gamma_inc, incomplete_gamma as symbolic function if they all mean the same. Some of them are probably there for user convenience or because of legacy. I think we want only one spelling for gamma_upper, whatever it may be. The gamma(a,x) convention seems fairly prominent in computer algebra, but is a poor translation of math notation, where the difference between upper and lower is made with case (and indeed, the complete gamma function tends to be capital gamma. I don't think I've ever seen different).

Because of tab completion, you definitely want a binding available that starts with gamma. I'd be fine with gamma_lower but if you want to go with gamma_lower_incomplete, that's fine with me too. So, I'd propose

8. Maintain status quo with respect to upper incomplete gamma, i.e., don't change sage's behaviour for gamma(a,x) and gamma_inc(a,x). Just implement a symbolic function gamma_lower (or modified spelling) to correspond to maxima's gamma_greek.

[rwst]

comment:3

I don't underswtand. With "leave as is" you would keep gamma(a,x) as returned expression for the upper function? Or would you make gamma_inc fully symbolic in that it gets no longer converted to gamma(a,x)?

[rwst]

comment:4

The reason why gamma_inc(x,y) gets output as gamma(x,y) is not conversion as I thought but simple naming super in the constructor. It would be just as easy to make that gamma_inc and change a few doctests such that then:

         sage: f = exp_integral_e(-1,x)
         sage: f.integrate(x)
-        Ei(-x) - gamma(-1, x)
+        Ei(-x) - gamma_inc(-1, x)

             sage: gamma_inc(3,2)
-            gamma(3, 2)
+            gamma_inc(3, 2)

             sage: gamma(3,2)
             gamma_inc(3, 2)

             sage: gamma(x,0)
             gamma(x)

             sage: gamma(x, 0, hold=True)
-            gamma(x, 0)
+            gamma_inc(x, 0)

I'm appending the patch that enables this here, so it is preserved.

[rwst]

Attachment: 16697-gammainc-alt-patch.gz

see comment

[rwst]

Branch: u/rws/implement_symbolic_lower_incomplete_gamma_function

[sagetrac-git]

Commit: 93c2f94

[sagetrac-git]

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

93c2f94

16697: deprecate "incomplete_gamma"

[rwst]

Description changed:

--- 
+++ 
@@ -13,7 +13,7 @@
 ```
 https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function
 
-Mathematica seems to have Gamma[a,z] for upper and Gamma[a,0,z] for lower; Maple seems to have upper Gamma. `gamma_inc` (the upper one in Sage) gets immediately converted to `gamma(a,x)`, so Sage has already went the Mathematica path of `gamma(a)=gamma(a,x)+gamma(a,0,x)`. The symbolic functions `gamma_inc==incomplete_gamma` are converted and never returned to the user as expression:
+Mathematica seems to have Gamma[a,z] for upper and Gamma[a,0,z] for lower; Maple seems to have upper Gamma. `gamma_inc` (the upper one in Sage) gets immediately converted to `gamma(a,x)`. The symbolic functions `gamma_inc==incomplete_gamma` are converted and never returned to the user as expression:
 
 ```
 sage: gamma_inc(x,x,hold=True)
@@ -24,7 +24,11 @@
 sage: integral(t^(s-1)*e^(-t),t,0,x)
 -gamma(s, x) + gamma(s)
 ```
-So, what's the plan?
+This ticket should deprecate "incomplete_gamma" and add the symbolic function `gamma_inc_lower`, leaving open the question of the global alias for and the displayed name of `Function_gamma_inc`.
+
+Previous part of description:
+
+---
 
 1. Provide all three "user input interfaces" `gamma_inc`, `incomplete_gamma` and `lower_incomplete_gamma`, and convert the Maxima `gamma_greek` to `-gamma(a, x) + gamma(a)`
 2. Provide all three "user input interfaces" `gamma_inc`, `incomplete_gamma` and `lower_incomplete_gamma`, and convert to `gamma(a,x)` and `gamma(a,0,x)`n on output

[sagetrac-git]

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

5c1dd67

16697: implement gamma_inc_lower

[sagetrac-git]

Changed commit from 93c2f94 to 5c1dd67