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ExpandDenominator

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

ExpandDenominator[expr]
    expands out products and powers that appear as denominators in expr.
ExpandDenominator works only on negative integer powers.
ExpandDenominator applies only to the top level in expr.
ExpandDenominator leaves the numerator unexpanded.
ExpandDenominator automatically threads over lists, as well as equations,
    inequalities, and logic functions.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= ExpandDenominator[(x-1)(x-2)/((x-3)(x-4))]
Out[1]= ((-2 + x) (-1 + x))/(12 - 7 x + x^2)

In[2]:= ExpandDenominator[1/(x+1) + 2/(x+1)^2 + 3/(x+1)^3]
Out[2]= 1/(1 + x) + 2/(1 + 2 x + x^2) + 3/(1 + 3 x + 3 x^2 + x^3)

In[3]:= ExpandDenominator[(a+b)(a-b)/((c+d)(c-d))]
Out[3]= ((a + b) (a - b))/(c^2 - d^2)

Implementation notes

Algorithm. builtin_expand_denominator (in src/expand.c) calls expr_expand_denominator, the mirror of ExpandNumerator: it expands only the denominator. For a Times it collects the negative-integer-power factors (is_negative_int_power), rebuilds them as positive powers, multiplies them into a single denominator product, runs expr_expand on that product, and re-inverts it as Power[expandedDen, -1] multiplied back against the untouched numerator factors. A bare Power[base, -k] is handled by expanding base^k and re-inverting. Threading over List, equations, inequalities, logic heads and Plus matches ExpandNumerator.

Data structures. Parallel Expr** buffers for numerator factors and positive-power denominator factors; only the denominator product is expanded, the numerator is copied through.

  • Protected.
  • Acts only on factors with negative integer exponents (the "denominator part" of expr).
  • Combines all denominator factors of a top-level Times into a single expanded polynomial wrapped in Power[..., -1].
  • Applies only to the top level in expr; it does not descend into function bodies.
  • Leaves the numerator factors unchanged.
  • Threads over List, Equal, Unequal, Less, LessEqual, Greater, GreaterEqual, And, Or, Not, and Plus.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= ExpandDenominator[(a + b)^2 / (c + d)^2]
Out[1]= (a + b)^2/(c^2 + 2 c d + d^2)

In[1]:= ExpandDenominator[1 / ((x + 1)(x + 2)(x + 3))]
Out[1]= 1/(6 + 11 x + 6 x^2 + x^3)

In[1]:= ExpandDenominator[((x + 1)(x + 2)) / (y (y + 1))]
Out[1]= ((1 + x) (2 + x))/(y + y^2)

Notes

ExpandDenominator multiplies out only the denominator (the negative integer powers), leaving the numerator factored — the mirror image of ExpandNumerator. The product of three linear factors above expands to the cubic 6 + 11 x + 6 x^2 + x^3 in the denominator. It threads over lists, equations, inequalities, and logic functions.