Talk:PlanetPhysics/Heaviside Formula

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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 zeroes $a_1,\,a_2,\,\ldots,\,a_n$ 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 \htmladdnormallink{formula}{http://planetphysics.us/encyclopedia/Formula.html} (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}

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