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86 lines
2.8 KiB
86 lines
2.8 KiB
5 months ago
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"""Tools for manipulation of rational expressions. """
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from sympy.core import Basic, Add, sympify
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from sympy.core.exprtools import gcd_terms
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from sympy.utilities import public
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from sympy.utilities.iterables import iterable
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@public
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def together(expr, deep=False, fraction=True):
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"""
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Denest and combine rational expressions using symbolic methods.
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This function takes an expression or a container of expressions
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and puts it (them) together by denesting and combining rational
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subexpressions. No heroic measures are taken to minimize degree
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of the resulting numerator and denominator. To obtain completely
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reduced expression use :func:`~.cancel`. However, :func:`~.together`
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can preserve as much as possible of the structure of the input
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expression in the output (no expansion is performed).
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A wide variety of objects can be put together including lists,
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tuples, sets, relational objects, integrals and others. It is
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also possible to transform interior of function applications,
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by setting ``deep`` flag to ``True``.
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By definition, :func:`~.together` is a complement to :func:`~.apart`,
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so ``apart(together(expr))`` should return expr unchanged. Note
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however, that :func:`~.together` uses only symbolic methods, so
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it might be necessary to use :func:`~.cancel` to perform algebraic
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simplification and minimize degree of the numerator and denominator.
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Examples
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========
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>>> from sympy import together, exp
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>>> from sympy.abc import x, y, z
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>>> together(1/x + 1/y)
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(x + y)/(x*y)
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>>> together(1/x + 1/y + 1/z)
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(x*y + x*z + y*z)/(x*y*z)
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>>> together(1/(x*y) + 1/y**2)
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(x + y)/(x*y**2)
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>>> together(1/(1 + 1/x) + 1/(1 + 1/y))
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(x*(y + 1) + y*(x + 1))/((x + 1)*(y + 1))
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>>> together(exp(1/x + 1/y))
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exp(1/y + 1/x)
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>>> together(exp(1/x + 1/y), deep=True)
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exp((x + y)/(x*y))
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>>> together(1/exp(x) + 1/(x*exp(x)))
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(x + 1)*exp(-x)/x
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>>> together(1/exp(2*x) + 1/(x*exp(3*x)))
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(x*exp(x) + 1)*exp(-3*x)/x
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"""
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def _together(expr):
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if isinstance(expr, Basic):
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if expr.is_Atom or (expr.is_Function and not deep):
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return expr
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elif expr.is_Add:
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return gcd_terms(list(map(_together, Add.make_args(expr))), fraction=fraction)
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elif expr.is_Pow:
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base = _together(expr.base)
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if deep:
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exp = _together(expr.exp)
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else:
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exp = expr.exp
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return expr.__class__(base, exp)
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else:
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return expr.__class__(*[ _together(arg) for arg in expr.args ])
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elif iterable(expr):
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return expr.__class__([ _together(ex) for ex in expr ])
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return expr
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return _together(sympify(expr))
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