Heaviside formula
HeavisideFormula
2009-04-17 19:25:37
2009-04-17 19:25:37
Definition
Legacy L2H is simply more forgiving of that malformed layout.
I regenerated the article and also cleaned several source errors I noticed:
removed both stray \\ display breaks
changed referenced displays to numbered equation environments
corrected
(s-a_1)\cdots(s-s_n)
to
(s-a_1)\cdots(s-a_n)
corrected the malformed equality
C_j \;=;
and
C_j ;=\;
to ordinary =
removed the legacy \PMlinkname around “zeroes”
cleaned the long example into an aligned block
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Let $P(s)$ and $Q(s)$ be polynomials with the degree of the former less than the degree of the latter.
\begin{itemize}
\item If all complex \PMlinkname{zeroes}{Zero} $a_1,\,a_2,\,\ldots,\,a_n$ of $Q(s)$ are simple, then
\begin{align}
\mathcal{L}^{-1}\left\{\frac{P(s)}{Q(s)}\right\} \;=\; \sum_{j=1}^n\frac{P(a_j)}{Q'(a_j)}e^{a_jt}.
\end{align}
\item If the different zeroes $a_1,\,a_2,\,\ldots,\,a_n$ of $Q(s)$ have the multiplicities $m_1,\,m_2,\,\ldots,\,m_n$, respectively, we denote\, $F_j(s) := (s\!-\!a_j)^{m_j}P(s)/Q(s)$;\, then
\begin{align}
\mathcal{L}^{-1}\left\{\frac{P(s)}{Q(s)}\right\} \;=\;
\sum_{j=1}^ne^{a_jt}\sum_{k=0}^{m_j-1}\frac{F_j^{(k)}(a_j)t^{m_j\!-\!1\!-\!k}}{k!(m_j\!-\!1\!-\!k)!}.
\end{align}
\end{itemize}
A special case of the {\em Heaviside formula} (1) is
$$\mathcal{L}^{-1}\left\{\frac{Q'(s)}{Q(s)}\right\} \;=\; \sum_{j=1}^ne^{a_jt}.\\$$
\textbf{Example.}\, Since the zeros of the binomial $s^4\!+\!4a^4$ are\, $s = (\pm1\!\pm\!i)a$,\, we obtain
$$\mathcal{L}^{-1}\left\{\frac{s^3}{s^4\!+\!4a^4}\right\}
\;=\; \frac{1}{4}\mathcal{L}^{-1}\left\{\frac{4s^3}{s^4\!+\!4a^4}\right\} \;=\; \frac{1}{4}\sum_\pm e^{(\pm 1\pm i)at}
\;=\; \frac{e^{at}+e^{-at}}{2}\cdot\frac{e^{iat}+e^{-iat}}{2}
\;=\; \cosh{at}\,\cos{at}.$$\\
{\em Proof of (1).}\, Without hurting the generality, we can suppose that $Q(s)$ is monic.\, Therefore
$$Q(s) \;=\; (s\!-\!a_1)(s\!-\!a_2)\cdots(s\!-\!s_n).$$
For\, $j = 1,\,2,\;\ldots,\,n$,\, denoting
$$Q(s) \;:=\; (s\!-\!a_j)Q_j(s),$$
one has\, $Q_j(a_j) \neq 0$.\, We have a partial fraction expansion of the form
\begin{align}
\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{align}
with constants $C_j$.\, According to the linearity and the formula 1 of the parent entry,
one gets
\begin{align}
\mathcal{L}^{-1}\left\{\frac{P(s)}{Q(s)}\right\} \;=\; \sum_{j=1}^nC_je^{a_jt}.
\end{align}
For determining the constants $C_j$, multiply (3) by $s\!-\!a_j$.\, It yields
$$\frac{P(s)}{Q_j(s)} = C_j+(s\!-\!a_j)\sum_{\nu \neq j}\frac{C_\nu}{s\!-\!a_\nu}.$$
Setting to this identity \,$s := a_j$\, gives the value
\begin{align}
C_j \;=; \frac{P(a_j)}{Q_j(a_j)}.
\end{align}
But since\, $Q'(s) = \frac{d}{ds}((s\!-\!a_j)Q_j(s)) = Q_j(s)\!+\!(s\!-\!a_j)Q_j'(s)$,\, we see that\, $Q'(a_j) = Q_j(a_j)$;\, thus the equation (5) may be written
\begin{align}
C_j ;=\; \frac{P(a_j)}{Q'(a_j)}.
\end{align}
The values (6) in (4) produce the formula (1).
\begin{thebibliography}{9}
\bibitem{K.V.}{\sc K. V\"ais\"al\"a:} {\em Laplace-muunnos}.\, Handout Nr. 163.\quad Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).
\end{thebibliography}