Partial Fraction Decomposition Tool

Decompose a strictly proper rational function whose monic denominator is caller supplied product of distinct simple real (x−pole) factors; residues computed by p(pole)/q′(pole), not a general factoring claim.

Description

Simple-pole partial fractions express a strictly proper rational function as a sum of residues divided by x−pole.

Simple-pole partial fractions express a strictly proper rational function as a sum of residues divided by x−pole. Here the denominator is the monic product of the caller's supplied real poles.^[1]

When to use Partial Fraction Decomposition Tool

Use this tool to decompose a strictly proper rational function whose monic denominator is caller supplied product of distinct simple real (x−pole) factors; residues computed by p(pole)/q′(pole), not a general factoring claim.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Partial Fraction Decomposition Tool and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter numeratorCoefficients, suppliedSimpleRealPoles using the field table and calculation rules, then run the tool.
Read simplePoleResidues. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
numeratorCoefficients	Numerator coefficients	list of number	Required
suppliedSimpleRealPoles	Supplied simple real poles	list of number	Required

Provide 1–32 poles and a nonempty constant-first numerator coefficient list with at most as many entries as poles. Poles must be separated by more than 10⁻¹² times max(1,|each pole|). Evaluate residues in the supplied pole order.

residue at pole_i = numerator(pole_i)/∏[j≠i](pole_i−pole_j)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
simplePoleResidues	Simple pole residues	list of number

json

{
  "input": {
    "numeratorCoefficients": [
      1
    ],
    "suppliedSimpleRealPoles": [
      1,
      2
    ]
  },
  "output": {
    "simplePoleResidues": [
      -1,
      1
    ]
  }
}

Limitations and assumptions

Repeated poles, polynomial parts, general denominator factoring and complex poles are unsupported.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Padé Approximant Calculator: Solve bounded [m/n] Padé approximant from exactly m+n+1 caller Taylor coefficients, denominator constant one; singular/unresolved systems refused rather than invent a unique approximation. For a complementary task, use Sylvester Resultant Calculator: Compute exact integer resultant by fraction-free Sylvester determinant for constant-first integer polynomials of degree ≤12; zero polynomial returns zero resultant. Nonzero constants have no common roots.

References

Partial Fraction Decomposition Tool — tool.io

Similar or alternative tools

Padé Approximant Calculator
Solve bounded [m/n] Padé approximant from exactly m+n+1 caller Taylor coefficients, denominator constant one; singular/unresolved systems refused rather than invent a unique approximation.
Sylvester Resultant Calculator
Compute exact integer resultant by fraction-free Sylvester determinant for constant-first integer polynomials of degree ≤12; zero polynomial returns zero resultant. Nonzero constants have no common roots.