Let
and
be polynomials with the degree of the former less than the degree of the latter.
- If all complex zeroes
of
are simple, then
- If the different zeroes
of
have the multiplicities
, respectively, we denote\,
;\, then
![{\displaystyle {\begin{matrix}{\mathcal {L}}^{-1}\left\{{\frac {P(s)}{Q(s)}}\right\}\;=\;\sum _{j=1}^{n}e^{a_{j}t}\sum _{k=0}^{m_{j}-1}{\frac {F_{j}^{(k)}(a_{j})t^{m_{j}\!-\!1\!-\!k}}{k!(m_{j}\!-\!1\!-\!k)!}}.\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe1f6a232fce193956e0b537274d6ea3ef93b1d)
A special case of the Heaviside formula (1) is
Failed to parse (syntax error): {\displaystyle \mathcal{L}^{-1}\left\{\frac{Q'(s)}{Q(s)}\right\} \;=\; \sum_{j=1}^ne^{a_jt}.\\}
Example. \, Since the zeros of the binomial
are\,
,\, we obtain
\\
Proof of (1). \, Without hurting the generality, we can suppose that
is monic.\, Therefore
For\,
,\, denoting
one has\,
.\, We have a partial fraction expansion of the form
![{\displaystyle {\begin{matrix}{\frac {P(s)}{Q(s)}}\;=\;{\frac {C_{1}}{s\!-\!a_{1}}}+{\frac {C_{2}}{s\!-\!a_{2}}}+\ldots +{\frac {C_{n}}{s\!-\!a_{n}}}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33116472e03c67d655cf9abbdcba8badfff429b6)
with constants
.\, According to the linearity and the formula 1 of the parent entry,
one gets
![{\displaystyle {\begin{matrix}{\mathcal {L}}^{-1}\left\{{\frac {P(s)}{Q(s)}}\right\}\;=\;\sum _{j=1}^{n}C_{j}e^{a_{j}t}.\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55fb8996beea9b1c6cc79e0d54b25e069d810327)
For determining the constants
, multiply (3) by
.\, It yields
Setting to this identity \,
\, gives the value
![{\displaystyle {\begin{matrix}C_{j}\;=;{\frac {P(a_{j})}{Q_{j}(a_{j})}}.\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68ce74763f5ecba00a509dbbd5b42250e59db343)
But since\,
,\, we see that\,
;\, thus the equation (5) may be written
![{\displaystyle {\begin{matrix}C_{j};=\;{\frac {P(a_{j})}{Q'(a_{j})}}.\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/152cbeacac449de3af1b7d2f5486d2f9b43cee9c)
The values (6) in (4) produce the formula (1).