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ExpandNumerator

Status: Stable

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

Description

ExpandNumerator[expr]
    expands out products and powers that appear in the numerator of expr.
ExpandNumerator works on terms that have positive integer exponents.
ExpandNumerator applies only to the top level in expr.
ExpandNumerator does not separate the fraction; Expand does.
ExpandNumerator leaves the denominator unexpanded.
ExpandNumerator 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]:= ExpandNumerator[(x-1)(x-2)/((x-3)(x-4))]
Out[1]= (2 - 3 x + x^2)/((-4 + x) (-3 + x))

In[2]:= ExpandNumerator[(a+b)^2/x + (c+d)(c-d)/y]
Out[2]= (a^2 + 2 a b + b^2)/x + (c^2 - d^2)/y

In[3]:= ExpandNumerator[x == (a+b)^2/c && y >= (a-b)^2/c]
Out[3]= x == (a^2 + 2 a b + b^2)/c && y >= (a^2 - 2 a b + b^2)/c

Implementation notes

Algorithm. builtin_expand_numerator (in src/expand.c) calls expr_expand_numerator, which separates an expression's numerator from its denominator and expands only the numerator. For a Times, it partitions factors into denominator factors (those of the form Power[base, k] with k a negative integer, detected by is_negative_int_power) and the rest; the non-denominator product is run through expr_expand, then recombined with the untouched denominator factors. A bare Power with negative integer exponent is a pure denominator and is returned unchanged; a positive/symbolic power is expanded at the top level. It threads over List, equations, inequalities, And/Or/Not, and Plus (the is_thread_head set), expanding per-summand.

Data structures. Separate Expr** accumulators for numerator and denominator factors; the denominator is preserved verbatim while only the numerator passes through expr_expand.

  • Protected.
  • Acts only on factors with positive integer exponents (the "numerator part" of expr).
  • Applies only to the top level in expr; it does not descend into function bodies.
  • Leaves the denominator factors (those with negative integer exponents) unchanged.
  • Does not separate the fraction into a sum of fractions; only Expand does that.
  • Threads over List, Equal, Unequal, Less, LessEqual, Greater, GreaterEqual, And, Or, Not, and Plus (so each summand of a sum-of-fractions is processed independently).

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

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

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

Notes

ExpandNumerator distributes products and positive integer powers in the numerator only, leaving the denominator factored. Unlike Expand, it does not split the fraction into separate terms. It threads over lists and relations, so both sides of the equation above have their numerators expanded independently.