Calculator form
Enter numerator coefficients in descending powers of s. Then select the denominator factor pattern you want to decompose.
Formula used
For a rational Laplace expression F(s) = N(s) / D(s), the calculator assumes a denominator factorization pattern and writes a matching partial-fraction form.
It then multiplies both sides by D(s), expands polynomial identities, and equates coefficients of equal powers of s.
The unknown constants are solved with Gaussian elimination. This gives coefficients such as A, B, C, and D for the selected structure.
Example: N(s)/[(s+a)(s^2+bs+c)] = A/(s+a) + (Bs+C)/(s^2+bs+c)
How to use this calculator
- Select the denominator factor pattern that matches your Laplace-domain expression.
- Enter the numerator coefficients in descending powers of s.
- Fill the required denominator parameters, such as a, b, and quadratic terms.
- Press Submit to display the decomposition above the form.
- Use the CSV option for a quick coefficient record or PDF for a printable worksheet.
Example data table
| Case | Numerator N(s) | Denominator D(s) | Sample decomposition |
|---|---|---|---|
| Distinct linear | 2s + 7 | (s+1)(s+4) | 5/(s+1) - 3/(s+4) |
| Repeated linear | s² + 3s + 2 | (s+1)²(s+3) | 1/(s+1) + 0/(s+1)² + 0/(s+3) |
| Linear with quadratic | s² + 3s + 5 | (s+2)(s²+4s+13) | 0.307692/(s+2) + (0.692308s + 1)/(s²+4s+13) |
Frequently asked questions
1. What does this calculator solve?
It decomposes selected rational Laplace expressions into simpler partial fractions. The output helps with inverse transforms, coefficient checking, and hand-solution verification.
2. Can it handle repeated poles?
Yes. It includes a repeated linear-pole model and a repeated linear plus quadratic model, which are both common in transform tables and differential-equation work.
3. Why is there a quadratic numerator term like Bs + C?
An irreducible quadratic factor needs a first-degree numerator. That structure preserves generality and makes coefficient matching complete for real-valued decompositions.
4. What if my denominator does not match these cases?
Use a factorization pattern that matches your expression first. If the denominator has more factors or higher multiplicity, extend the identity using the same coefficient-matching method.
5. Can I use negative shift values?
Yes. Negative values are allowed, so factors like s-3 can be entered using a = -3 because the calculator works with the form s+a.
6. Why do some results show zero coefficients?
A zero coefficient means that term is unnecessary for the current numerator and denominator pair. The full assumed structure remains valid, but one or more constants may vanish.
7. How accurate are the coefficient values?
Values are calculated numerically and displayed in rounded decimal form. For exact symbolic work, you can still use the coefficients as a strong verification reference.
8. Does this also compute the inverse Laplace transform?
Not directly. It prepares the decomposition so you can apply standard inverse-transform pairs term by term with much less algebraic effort.