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Coefficient

Status: Stable

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

Description

Coefficient[expr, form]
    gives the coefficient of form^1 in expr.  form is matched
    structurally against the bases of products in the expanded form
    of expr.
Coefficient[expr, form, n]
    gives the coefficient of form^n.  n may be a non-negative integer
    or (for Laurent / Puiseux expressions) a rational.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Coefficient[(x+1)^3, x, 2]
Out[1]= 3

In[2]:= Coefficient[(x+y)^4, x y^3]
Out[2]= 4

In[3]:= Coefficient[x^s x, x^s]
Out[3]= x

Implementation notes

Algorithm. builtin_coefficient (in src/poly/poly.c) extracts the coefficient of form^n (default n = 1) from expr. It first runs expr_expand to a flat sum of monomials, then decomposes both the target form and each summand into base–exponent pairs via decompose_to_bp. The helper get_k computes the integer power at which the target's base(s) divide each term; terms with k == n contribute, with the matching base factors stripped out (get_k handles multi-factor monomial forms like x y). For n == 0 the whole term is taken. Surviving residual factors are reassembled with internal_times/internal_plus.

Data structures. BPList — a list of {base, exp} pairs (initialised with bp_init, freed with bp_free) — is the core representation for both the target form and each term.

  • Protected, Listable.
  • Coefficient[expr, form, 0] picks out terms that do NOT contain form.
  • Works whether or not expr is explicitly given in expanded form (it automatically expands internally).
  • Treats distinct transcendental powers as algebraically unrelated (e.g., x^s is treated as a separate base from x).

Attributes: Listable, 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]:= Coefficient[x^2 + 3 x + 2, x]
Out[1]= 3

In[1]:= Coefficient[x^2 + 3 x + 2, x, 2]
Out[1]= 1

In[1]:= Coefficient[a x^2 + b x + c, x, 0]
Out[1]= c

In[1]:= Coefficient[3 x^2 y + 2 x y, x, 2]
Out[1]= 3 y

In[1]:= Coefficient[(1 + x)^50, x, 25]
Out[1]= 126410606437752

In[1]:= Coefficient[(a + b)^4, a^2 b^2]
Out[1]= 6

Notes

Coefficient[expr, form] returns the coefficient of form^1 after expanding expr, while the three-argument Coefficient[expr, form, n] extracts the coefficient of form^n. Using n = 0 recovers the term free of form, e.g. the constant c in a x^2 + b x + c. The extracted coefficient retains any other variables, so the x^2 coefficient of 3 x^2 y + 2 x y is 3 y. The form is matched structurally against the bases of products, and n may be a non-negative integer (or a rational for Laurent/Puiseux expressions). It scales to high degrees — extracting the central binomial coefficient C(50, 25) = 126410606437752 from (1 + x)^50 directly — and form need not be a single variable: passing the monomial a^2 b^2 picks out its coefficient 6 in (a + b)^4.