sageMath符号产品无法替代和无限

问题描述

该软件为SageMath版本9.1，并且代码主要涉及Sympy package's product function，因为SageMath以sympy为核心。

预期表达是

的乘积

（1-q）...（1-q ^ n）

n的范围是1到无穷大。

首先，显示出the product did recognize the infinity symbol

product(1/x^2,x,1,oo)

返回正确的结果0。

错误1 但是，代码

product(1-a^x,oo)

返回的错误

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-11-d7561515fdb5> in <module>()
----> 1 product(Integer(1)-a**x,Integer(1),oo)

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/misc/functional.py in symbolic_prod(expression,*args,**kwds)
    629     from .misc_c import prod as c_prod
    630     if hasattr(expression,'prod'):
--> 631         return expression.prod(*args,**kwds)
    632     elif len(args) <= 1:
    633         return c_prod(expression,*args)

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.prod (build/cythonized/sage/symbolic/expression.cpp:63909)()
  12375         """
  12376         from sage.calculus.calculus import symbolic_product
> 12377         return symbolic_product(self,**kwds)
  12378 
  12379     def integral(self,**kwds):

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/calculus/calculus.py in symbolic_product(expression,v,a,b,algorithm,hold)
    867 
    868     if algorithm == 'maxima':
--> 869         return maxima.sr_prod(expression,b)
    870 
    871     elif algorithm == 'mathematica':

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/interfaces/maxima_lib.py in sr_prod(self,*args)
    902         """
    903         try:
--> 904             return max_to_sr(maxima_eval([[max_ratsimp],[[max_simplify_prod],([max_prod],[sr_to_max(SR(a)) for a in args])]]));
    905         except RuntimeError as error:
    906             s = str(error)

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/interfaces/maxima_lib.py in max_to_sr(expr)
   1683             op=max_op_dict[op_max]
   1684         max_args=cdr(expr)
-> 1685         args=[max_to_sr(a) for a in max_args]
   1686         return op(*args)
   1687     elif expr.symbolp():

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/interfaces/maxima_lib.py in <listcomp>(.0)
   1683             op=max_op_dict[op_max]
   1684         max_args=cdr(expr)
-> 1685         args=[max_to_sr(a) for a in max_args]
   1686         return op(*args)
   1687     elif expr.symbolp():

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/interfaces/maxima_lib.py in max_to_sr(expr)
   1684         max_args=cdr(expr)
   1685         args=[max_to_sr(a) for a in max_args]
-> 1686         return op(*args)
   1687     elif expr.symbolp():
   1688         if not(expr in max_sym_dict):

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/integer.pyx in sage.rings.integer.Integer.__pow__ (build/cythonized/sage/rings/integer.c:14994)()
   2222             return (<Integer>left)._pow_(right)
   2223         elif isinstance(left,Element):
-> 2224             return coercion_model.bin_op(left,right,operator.pow)
   2225         # left is a non-Element: do the powering with a Python int
   2226         return left ** int(right)

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/coerce.pyx in sage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:10143)()
   1209             self._record_exception()
   1210         else:
-> 1211             return PyObject_CallObject(op,xy)
   1212 
   1213         if op is mul:

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.__pow__ (build/cythonized/sage/structure/element.c:14059)()
   2034         cdef int cl = classify_elements(left,right)
   2035         if HAVE_SAME_PARENT(cl):
-> 2036             return (<Element>left)._pow_(right)
   2037         if BOTH_ARE_ELEMENT(cl):
   2038             return coercion_model.bin_op(left,pow)

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression._pow_ (build/cythonized/sage/symbolic/expression.cpp:25373)()
   4092                            relational_operator(self._gobj))
   4093         else:
-> 4094             x = g_pow(self._gobj,nexp._gobj)
   4095         return new_Expression_from_GEx(self._parent,x)
   4096 

ValueError: power::eval(): pow(1,Infinity) is not defined.

错误2 还定义了一个不确定的表达式

expres1=product(1-a^x,N) 

返回了

-(-1)^N*product(a^x - 1,N)

但是，用替换

h=expres1.substitute({N:10})

返回了

-product(a^x - 1,10)

符号表达式h不能扩展

(h).series(a,10).expand()

返回的错误

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-17-e9b5fcf66965> in <module>()
      1 #help(prod)
      2 #(1/eta_1).series(q,10).expand()
----> 3 (h).series(a,Integer(10)).expand()

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.series (build/cythonized/sage/symbolic/expression.cpp:27193)()
   4484         sig_on()
   4485         try:
-> 4486             x = self._gobj.expand(0).series(symbol0._gobj,cprec,options)
   4487             nex = SymbolicSeries.__new__(SymbolicSeries)
   4488             nex._parent = self._parent

ValueError: power::eval(): division by zero

问题

您能告诉我上述代码出了什么问题吗？为什么sageMath不能识别无限积？为什么替换无效，并且程序无法评估功能？

解决方法

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