Partial Fraction Calculator: Split a Rational Function with Linear Factors and Invert the Laplace Transform

Partial fractions with repeated factors and f(t).

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose the number of factors and enter the numerator coefficients.

Enter the root of each factor. A factor s + 1 has the root -1. A repeated factor has the same root twice.

The calculator gives each coefficient and the inverse Laplace transform.

Formulas used

  • \frac{N(s)}{(s - p_1)(s - p_2)} = \frac{A}{s - p_1} + \frac{B}{s - p_2}
  • \frac{1}{(s - p)^k} \to \frac{t^{k-1}}{(k-1)!}e^{pt}

Worked example

With the default numbers in the calculator above:

  1. The fraction to split: (1 s + 5) / (s + 1)(s + 2)(s + 3)
  2. Term 1: the number over (s + 1): 2
  3. Term 2: the number over (s + 2): -3
  4. Term 3: the number over (s + 3): 1
  5. The split fraction: 2 / (s + 1) - 3 / (s + 2) + 1 / (s + 3)
  6. Inverse Laplace transform f(t) at the time you entered: 0.37954

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Numerator: coefficient of s squared 1; Numerator: coefficient of s 0; Numerator: constant term 0; Root of factor 2 -1; Root of factor 3 (ignored for 2 factors) -2.

  1. The fraction to split: (1 s^2) / (s + 1)^2(s + 2)
  2. Term 1: the number over (s + 1): -3
  3. Term 2: the number over (s + 1)^2: 1
  4. Term 3: the number over (s + 2): 4
  5. The split fraction: -3 / (s + 1) + 1 / (s + 1)^2 + 4 / (s + 2)
  6. Inverse Laplace transform f(t) at the time you entered: -0.194418

Common questions

How do partial fractions work?

You write the numerator as the sum of each partial fraction times the whole denominator, match the coefficients of each power of s, and solve for the unknown numbers.

How is a repeated factor handled?

It needs one term for each power, so a factor (s + 1) squared gives a term over (s + 1) and a term over (s + 1) squared.

What does a positive root mean?

The term grows with time, so the system it describes is unstable.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any engineering mathematics text (partial fractions, the inverse Laplace transform).

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