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Harlamova29_29 [7]
3 years ago
7

Definition 7.1.1 laplace transform let f be a function defined for t ≥ 0. then the integral {f(t)} = ∞ e−stf(t) dt 0 is said to

be the laplace transform of f, provided that the integral converges. to find {f(t)}. f(t) = cos t, 0 ≤ t < π 0, t ≥ π
Mathematics
1 answer:
andreev551 [17]3 years ago
5 0
\mathcal L\{f(t)\}=\displaystyle\int_{t=0}^{t\to\infty}f(t)e^{-st}\,\mathrm dt

Given that

f(t)=\begin{cases}\cos t&\text{for }0\le t

the Laplace transform of f(t) is given by the definite integral

\displaystyle\int_{t=0}^{t\to\infty}f(t)e^{-st}\,\mathrm dt=\int_{t=0}^{t=\pi}\cos t\,e^{-st}\,\mathrm dt+\int_{t=\pi}^{t\to\infty}0\,\mathrm dt
=\displaystyle\int_0^\pi\cos t\,e^{-st}\,\mathrm dt
=\dfrac{(1-e^{-\pi s})s}{s^2+1}

(which you can find by integrating by parts twice)
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