applying full_simplify() to gamma functions causes an error

[tomc]

Applying full_simplify() to the gamma function sometimes causes an error. This example works:

sage: gamma(4/3).full_simplify()
1/3*gamma(1/3)

but this example does not:

sage: gamma(1/3).full_simplify()
ERROR: An unexpected error occurred while tokenizing input
The following traceback may be corrupted or invalid
The error message is: ('EOF in multi-line statement', (1254, 0))

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)

/Users/tomc/sage-4.4.1/<ipython console> in <module>()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/symbolic/expression.so in sage.symbolic.expression.Expression.simplify_full (sage/symbolic/expression.cpp:21549)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/symbolic/expression.so in sage.symbolic.expression.Expression.simplify_factorial (sage/symbolic/expression.cpp:22240)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/structure/parent.so in sage.structure.parent.Parent.__call__ (sage/structure/parent.c:6332)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/structure/coerce_maps.so in sage.structure.coerce_maps.NamedConvertMap._call_ (sage/structure/coerce_maps.c:4053)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/interfaces/maxima.pyc in _symbolic_(self, R)
   1810             sqrt(2)
   1811         """
-> 1812         return R(self._sage_())
   1813 
   1814     def __complex__(self):

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/interfaces/maxima.pyc in _sage_(self)
   1791         import sage.calculus.calculus as calculus
   1792         return calculus.symbolic_expression_from_maxima_string(self.name(),
-> 1793                 maxima=self.parent())
   1794 
   1795     def _symbolic_(self, R):

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/calculus/calculus.pyc in symbolic_expression_from_maxima_string(x, equals_sub, maxima)
   1524         # evaluation of maxima code are assumed pre-simplified
   1525         is_simplified = True
-> 1526         return symbolic_expression_from_string(s, syms, accept_sequence=True)
   1527     except SyntaxError:
   1528         raise TypeError, "unable to make sense of Maxima expression '%s' in Sage"%s

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/calculus/calculus.pyc in symbolic_expression_from_string(s, syms, accept_sequence)
   1692             global _augmented_syms
   1693             _augmented_syms = syms
-> 1694             return parse_func(s)
   1695         finally:
   1696             _augmented_syms = {}

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/misc/parser.so in sage.misc.parser.Parser.parse_sequence (sage/misc/parser.c:3855)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/misc/parser.so in sage.misc.parser.Parser.parse_sequence (sage/misc/parser.c:3747)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/misc/parser.so in sage.misc.parser.Parser.p_sequence (sage/misc/parser.c:4376)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/misc/parser.so in sage.misc.parser.Parser.p_tuple (sage/misc/parser.c:5032)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/misc/parser.so in sage.misc.parser.Parser.p_eqn (sage/misc/parser.c:5145)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/misc/parser.so in sage.misc.parser.Parser.p_expr (sage/misc/parser.c:5465)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/misc/parser.so in sage.misc.parser.Parser.p_term (sage/misc/parser.c:5690)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/misc/parser.so in sage.misc.parser.Parser.p_factor (sage/misc/parser.c:6053)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/misc/parser.so in sage.misc.parser.Parser.p_power (sage/misc/parser.c:6264)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/symbolic/function.so in sage.symbolic.function.GinacFunction.__call__ (sage/symbolic/function.cpp:6321)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/symbolic/expression.so in sage.symbolic.expression.Expression.factorial (sage/symbolic/expression.cpp:20595)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/symbolic/pynac.so in sage.symbolic.pynac.py_factorial (sage/symbolic/pynac.cpp:9156)()

/Users/tomc/sage-4.4.1/local/lib/python2.6/site-packages/sage/rings/arith.pyc in factorial(n, algorithm)
    403     """
    404     if n < 0:
--> 405         raise ValueError, "factorial -- must be nonnegative"
    406     if algorithm == 'gmp':
    407         return ZZ(n).factorial()

ValueError: factorial -- must be nonnegative

I am running Sage 4.4.1 on Mac OS X version 10.6 (Snow Leopard), built from source. But the second example also fails on Sage 4.3.5 on 64-bit Linux, built from source. Looking at the source code suggests that the second example will fail on all platforms.

The problem occurs because full_simplify() here runs the following commands in Maxima:

(%i1) minfactorial(factcomb(makefact(gamma(1/3))));
                                       2
(%o1)                               (- -)!
                                       3

and then the Maxima interface converts this to Sage as factorial(-2/3). This causes an error. For Sage, factorial(x) is only defined if x is a non-negative integer, whereas for Maxima factorial(x) is equivalent to gamma(1+x) and so makes sense whenever x is not in {-1, -2, -3, ...}

[tomc]

based on Sage 4.4.1