#16697: implement symbolic lower incomplete gamma function
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Reporter: rws | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.3
Component: symbolics | Keywords: gamma, incomplete, special,
Merged in: | functions
Reviewers: | Authors:
Work issues: | Report Upstream: N/A
Commit: | Branch:
Stopgaps: | Dependencies:
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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)`, 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:
{{{
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)
}}}
So, what's the plan?
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)`
--
Ticket URL: <http://trac.sagemath.org/ticket/16697>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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