Source code for diofant.polys.rationaltools

"""Tools for manipulation of rational expressions. """

from ..core import Add, gcd_terms, sympify

__all__ = 'together',

[docs]def together(expr, deep=False):
"""
Denest and combine rational expressions using symbolic methods.

This function takes an expression or a container of expressions
and puts it (them) together by denesting and combining rational
subexpressions. No heroic measures are taken to minimize degree
of the resulting numerator and denominator. To obtain completely
reduced expression use :func:~diofant.polys.polytools.cancel. However, :func:together
can preserve as much as possible of the structure of the input
expression in the output (no expansion is performed).

A wide variety of objects can be put together including lists,
tuples, sets, relational objects, integrals and others. It is
also possible to transform interior of function applications,
by setting deep flag to True.

By definition, :func:together is a complement to :func:~diofant.polys.partfrac.apart,
so apart(together(expr)) should return expr unchanged. Note
however, that :func:together uses only symbolic methods, so
it might be necessary to use :func:~diofant.polys.polytools.cancel to perform algebraic
simplification and minimise degree of the numerator and denominator.

Examples
========

>>> together(1/x + 1/y)
(x + y)/(x*y)
>>> together(1/x + 1/y + 1/z)
(x*y + x*z + y*z)/(x*y*z)

>>> together(1/(x*y) + 1/y**2)
(x + y)/(x*y**2)

>>> together(1/(1 + 1/x) + 1/(1 + 1/y))
(x*(y + 1) + y*(x + 1))/((x + 1)*(y + 1))

>>> together(exp(1/x + 1/y))
E**(1/y + 1/x)
>>> together(exp(1/x + 1/y), deep=True)
E**((x + y)/(x*y))

>>> together(1/exp(x) + 1/(x*exp(x)))
E**(-x)*(x + 1)/x

>>> together(1/exp(2*x) + 1/(x*exp(3*x)))
E**(-3*x)*(E**x*x + 1)/x

"""
def _together(expr):
if expr.is_Atom or (expr.is_Function and not deep):
return expr
elif expr.is_Pow:
base = _together(expr.base)

if deep:
exp = _together(expr.exp)
else:
exp = expr.exp

return expr.__class__(base, exp)
else:
return expr.__class__(*[_together(arg) for arg in expr.args])

return _together(sympify(expr))