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fix(ch7): correct convolution example sign, fix cos^2 transform

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2026-06-04 16:47:36 -05:00
parent 6f2b094291
commit ceefd3147d
1 changed files with 6 additions and 6 deletions
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chapters/ch07_laplace_transforms.tex
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@@ -130,7 +130,7 @@ The following table collects the most commonly used Laplace transforms. Each ent
$e^{at}\sin(bt)$ & $\dfrac{b}{(s-a)^2 + b^2}$ & $s > a$ \\[10pt] $e^{at}\sin(bt)$ & $\dfrac{b}{(s-a)^2 + b^2}$ & $s > a$ \\[10pt]
$e^{at}\cos(bt)$ & $\dfrac{s-a}{(s-a)^2 + b^2}$ & $s > a$ \\[10pt] $e^{at}\cos(bt)$ & $\dfrac{s-a}{(s-a)^2 + b^2}$ & $s > a$ \\[10pt]
$\sin^2(bt)$ & $\dfrac{2b^2}{s(s^2 + 4b^2)}$ & $s > 0$ \\[10pt] $\sin^2(bt)$ & $\dfrac{2b^2}{s(s^2 + 4b^2)}$ & $s > 0$ \\[10pt]
$\cos^2(bt)$ & $\dfrac{s^2}{(s^2 + 4b^2)(s)}$ & $s > 0$ \\[10pt] $\cos^2(bt)$ & $\dfrac{s^2+2b^2}{s(s^2 + 4b^2)}$ & $s > 0$ \\[10pt]
$t\sin(bt)$ & $\dfrac{2bs}{(s^2 + b^2)^2}$ & $s > 0$ \\[10pt] $t\sin(bt)$ & $\dfrac{2bs}{(s^2 + b^2)^2}$ & $s > 0$ \\[10pt]
$t\cos(bt)$ & $\dfrac{s^2 - b^2}{(s^2 + b^2)^2}$ & $s > 0$ \\[10pt] $t\cos(bt)$ & $\dfrac{s^2 - b^2}{(s^2 + b^2)^2}$ & $s > 0$ \\[10pt]
$u_c(t)$ \quad (Heaviside step) & $\dfrac{e^{-cs}}{s}$ & $s > 0$ \\[10pt] $u_c(t)$ \quad (Heaviside step) & $\dfrac{e^{-cs}}{s}$ & $s > 0$ \\[10pt]
@@ -843,15 +843,15 @@ The inner integral is $G(s)$, and the remaining integral is $F(s)$. Thus the res
\[ \[
\begin{aligned} \begin{aligned}
\int_0^t \tau\,\sin(t-\tau)\,\diff\tau \int_0^t \tau\,\sin(t-\tau)\,\diff\tau
&= \Bigl[-\tau\cos(t-\tau)\Bigr]_0^t + \int_0^t \cos(t-\tau)\,\diff\tau \\ &= \Bigl[\tau\cos(t-\tau)\Bigr]_0^t - \int_0^t \cos(t-\tau)\,\diff\tau \\
&= \bigl(-t\cos(0) - 0\bigr) + \Bigl[-\sin(t-\tau)\Bigr]_0^t \\ &= \bigl(t\cos(0) - 0\bigr) - \Bigl[-\sin(t-\tau)\Bigr]_0^t \\
&= -t + \bigl(-\sin(0) + \sin(t)\bigr) \\ &= t - \bigl(-\sin(0) + \sin(t)\bigr) \\
&= \sin(t) - t. &= t - \sin(t).
\end{aligned} \end{aligned}
\] \]
Therefore: Therefore:
\[ \[
\mathcal{L}^{-1}\left\{\frac{1}{s^2(s^2+1)}\right\} = \sin(t) - t. \mathcal{L}^{-1}\left\{\frac{1}{s^2(s^2+1)}\right\} = t - \sin(t).
\] \]
\end{workedexample} \end{workedexample}