Heaviside formula

Let P(s) and Q(s) be polynomials with the degree of the former less than the degree of the latter.

• If all complex zeroes a₁, a₂, …, a_n of Q(s) are simple, then

  (1)

• If the different zeroes a₁, a₂, …, a_n of Q(s) have multiplicities m₁, m₂, …, m_n, respectively, define
  

  Then

  (2)

A special case of the Heaviside formula (1) is

(3)

Example. Since the zeroes of the binomial s⁴ + 4a⁴ are s = (±1 ± i)a, we obtain

(4)

Proof of (1). Without loss of generality, suppose that Q(s) is monic. Therefore

(5)

For j = 1, 2, …, n, write

(6)

so that Q_j(a_j)≠0.

We have a partial fraction expansion of the form

(7)

with constants C_j. By linearity of the inverse Laplace transform,

(8)

To determine the constants C_j, multiply the partial fraction expansion by s − a_j. This gives

(9)

Setting s = a_j gives

(10)

Since

(11)

we have Q′(a_j) = Q_j(a_j). Hence

(12)

Substituting these values into the inverse Laplace expansion yields formula (1).

References