Complex Analysis: Theory and Applications · Problem 11.5.1

Fourier Series and the Laplace Transform — Howell & Matthews

Mathematics — Complex Analysis · laplace-transform-definition-properties

Problem

Exercises for Section 11.5: 1. Show that L(1)=1s\mathcal{L}(1) = \frac{1}{s} by using the integral definition for the Laplace transform. Assume that Re(s)>0\mathrm{Re}(s) > 0. 2. Show that L(t2)=2s3\mathcal{L}(t^2) = \frac{2}{s^3} by using the integral definition for the Laplace transform. Assume that Re(s)>0\mathrm{Re}(s) > 0. 3. Find L(f(t))\mathcal{L}\bigl(f(t)\bigr) for each of the following. ((a)) U(t)={tfor 0≤t<c,0otherwise.U(t) = \begin{cases} t & \text{for } 0 \leq t < c, \\ 0 & \text{otherwise.} \end{cases} ((b)) U(t)={1for 1<t<2,0otherwise.U(t) = \begin{cases} 1 & \text{for } 1 < t < 2, \\ 0 & \text{otherwise.} \end{cases} ((c)) U(t)={eatfor 0≤t<1,0otherwise.U(t) = \begin{cases} e^{at} & \text{for } 0 \leq t < 1, \\ 0 & \text{otherwise.} \end{cases} ((d)) U(t)={sin⁡tfor 0≤t≤π,0otherwise.U(t) = \begin{cases} \sin t & \text{for } 0 \leq t \leq \pi, \\ 0 & \text{otherwise.} \end{cases} 4. Use the linearity of Laplace transform and Table 11.2 to find: ((a)) L(3t2−4t+5)\mathcal{L}(3t^2 - 4t + 5). ((b)) L(2cos⁡4t)\mathcal{L}(2\cos 4t). ((c)) L(e2t−3)\mathcal{L}(e^{2t-3}). ((d)) L(6e−t+3sin⁡5t)\mathcal{L}(6e^{-t} + 3\sin 5t). ((e)) L((t+1)4)\mathcal{L}((t+1)^4). ((f)) L(cosh⁡2t)\mathcal{L}(\cosh 2t). 5. Use the linearity of the inverse Laplace transform and Table 11.3 to find: ((a)) L−1 ⁣(1s2+25)\mathcal{L}^{-1}\!\left(\frac{1}{s^2+25}\right). ((b)) L−1 ⁣(42−6s2)\mathcal{L}^{-1}\!\left(\frac{4}{2} - \frac{6}{s^2}\right). ((c)) L−1 ⁣(1+s2−ss4)\mathcal{L}^{-1}\!\left(\frac{1+s^2-s}{s^4}\right). ((d)) L−1 ⁣(2s+9s2+9)\mathcal{L}^{-1}\!\left(\frac{2s+9}{s^2+9}\right). ((e)) L−1 ⁣(6ss2−4)\mathcal{L}^{-1}\!\left(\frac{6s}{s^2-4}\right). ((f)) L−1 ⁣(2s+1s(s+1))\mathcal{L}^{-1}\!\left(\frac{2s+1}{s(s+1)}\right). 6. Write a report on how complex analysis is used in the study of Laplace transforms. Include ideas and examples that are not mentioned in the text.

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