https://icci.ink/categories/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF/Recent content in 拉普拉斯 on ICCIHugozh-cnFri, 18 Apr 2025 00:00:00 +0000https://icci.ink/study/theory/math-%E6%8B%89%E6%B0%8F%E5%8F%98%E6%8D%A2%E7%9A%84%E6%80%A7%E8%B4%A8/Fri, 18 Apr 2025 00:00:00 +0000https://icci.ink/study/theory/math-%E6%8B%89%E6%B0%8F%E5%8F%98%E6%8D%A2%E7%9A%84%E6%80%A7%E8%B4%A8/<p>拉氏变化的性质</p> <h2 id="拉氏变化的性质表">拉氏变化的性质表<a class="anchor" href="#%e6%8b%89%e6%b0%8f%e5%8f%98%e5%8c%96%e7%9a%84%e6%80%a7%e8%b4%a8%e8%a1%a8">#</a></h2> <p>表5-1 单边拉普拉斯变换的性质</p> <table> <thead> <tr> <th>名称</th> <th>时域 (f(t)\leftrightarrow F(s))</th> <th>s域</th> </tr> </thead> <tbody> <tr> <td>定义</td> <td>(f(t)=\frac{1}{2\pi j}\int_{\sigma-j\omega}^{\sigma+j\omega}F(s)e^{st}ds)</td> <td>(F(s)=\int_{-\infty}^{\infty}f(t)e^{-st}dt,\sigma&gt;\sigma_0)</td> </tr> <tr> <td>线性</td> <td>(a_1f_1(t)+a_2f_2(t))</td> <td>(a_1F_1(s)+a_2F_2(s),\sigma&gt;\max(\sigma_1,\sigma_2))</td> </tr> <tr> <td>尺度变换</td> <td>(f(at))</td> <td>(\frac{1}{a}F\left(\frac{s}{a}\right),\sigma&gt;a\sigma_0)</td> </tr> <tr> <td>时移</td> <td>(f(t-t_0)\varepsilon(t-t_0))</td> <td>(e^{-s_0t}F(s),\sigma&gt;\sigma_0)</td> </tr> <tr> <td>复频移</td> <td>(f(t)e^{at})</td> <td>(F(s-a),\sigma&gt;\sigma_0+a)</td> </tr> <tr> <td>时域微分</td> <td>(f^{(n)}(t))</td> <td>(s^nF(s)-\sum_{m=0}^{n-1}s^{n-m-1}f^{(m)}(0_-))</td> </tr> <tr> <td>时域积分</td> <td>(\int_{0^-}^{t}f(x)dx)</td> <td>(\frac{1}{s}F(s),\sigma&gt;\max(\sigma_0,0))</td> </tr> <tr> <td>时域积分</td> <td>(f^{(-1)}(t))</td> <td>(\frac{1}{s}F(s)+\frac{1}{s}f^{(-1)}(0_-))</td> </tr> <tr> <td>时域积分</td> <td>(f^{(-n)}(t))</td> <td>(\frac{1}{s^n}F(s)+\sum_{m=1}^{n}\frac{1}{s^{n-m+1}}f^{(-m)}(0_-))</td> </tr> </tbody> </table> <p>注：(\varepsilon) 为单位阶跃函数，(a) 和 (b) 为正实常数，分别称为收敛域的内、外半径。</p> <h2 id="参考资料">参考资料<a class="anchor" href="#%e5%8f%82%e8%80%83%e8%b5%84%e6%96%99">#</a></h2> <p><a href="https://zh.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0">维基百科</a><br> <a href="https://blog.csdn.net/GiantCrocodile/article/details/109400459">参考博客</a></p>