Додаток B: Таблиця перетворень Лапласа
- Page ID
- 61759
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Функція\(u\) є функцією Хевісайд,\(\delta\) є дельта-функцією Дірака, і
\[\Gamma(t)=\int_{0}^{\infty}e^{-\tau}\tau^{t-1}d\tau ,\qquad\text{erf}(t)=\frac{2}{\sqrt{\pi}}\int_{0}^{t}e^{-\tau^{2}}d\tau , \qquad\text{erfc}(t)=1-\text{erf}(t). \nonumber \]
Таблиця\(\PageIndex{1}\)
| \(f(t)\) | \(F(s)=\mathcal{L}\{f(t)\}=\int_{0}^{\infty}e^{-st}f(t)dt\) |
|---|---|
| \ (f (t)\) ">\(C\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{C}{s}\) |
| \ (f (t)\) ">\(t\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{1}{s^{2}}\) |
| \ (f (t)\) ">\(t^{2}\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{2}{s^{3}}\) |
| \ (f (t)\) ">\(t^{n}\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{n!}{s^{n+1}}\) |
| \ (f (t)\) ">\(t^{p}\quad (p>0)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{\Gamma (p+1)}{s^{p+1}}\) |
| \ (f (t)\) ">\(e^{-at}\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{1}{s+a}\) |
| \ (f (t)\) ">\(\sin(\omega t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{\omega}{s^{2}+\omega ^{2}}\) |
| \ (f (t)\) ">\(\cos(\omega t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{s}{s^{2}+\omega^{2}}\) |
| \ (f (t)\) ">\(\sinh (\omega t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{\omega}{s^{2}-\omega ^{2}}\) |
| \ (f (t)\) ">\(\cosh (\omega t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{s}{s^{2}-\omega^{2}}\) |
| \ (f (t)\) ">\(u(t-a)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{e^{-as}}{s}\) |
| \ (f (t)\) ">\(\delta (t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(1\) |
| \ (f (t)\) ">\(\delta (t-a)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(e^{-as}\) |
| \ (f (t)\) ">\(\text{erf}\left(\frac{t}{2a}\right)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{1}{s}e^{(as)^{2}}\text{erfc}(as)\) |
| \ (f (t)\) ">\(\frac{1}{\sqrt{\pi t}}\text{exp}\left(\frac{-a^{2}}{4t}\right)\quad (a\geq 0)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{e^{-as}}{\sqrt{s}}\) |
| \ (f (t)\) ">\(\frac{1}{\sqrt{\pi t}}-ae^{a^{2}t}\text{erfc}(a\sqrt{t})\quad (a>0)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{1}{\sqrt{s}+a}\) |
| \ (f (t)\) ">\(af(t)+bg(t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(aF(s)+bG(s)\) |
| \ (f (t)\) ">\(f(at)\quad (a>0)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{1}{a}F\left(\frac{s}{a}\right)\) |
| \ (f (t)\) ">\(f(t-a)u(t-a)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(e^{-as}F(s)\) |
| \ (f (t)\) ">\(e^{-at}f(t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(F(s+a)\) |
| \ (f (t)\) ">\(g'(t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(sG(s)-g(0)\) |
| \ (f (t)\) ">\(g''(t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(s^{2}G(s)-sg(0)-g'(0)\) |
| \ (f (t)\) ">\(g'''(t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(s^{3}G(s)-s^{2}g(0)-sg'(0)-g''(0)\) |
| \ (f (t)\) ">\(g^{(n)}(t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(s^{n}G(s)-s^{n-1}g(0)-\cdots -g^{(n-1)}(0)\) |
| \ (f (t)\) ">\((f\ast g)(t)=\int_{0}^{t} f(\tau )g(t-\tau )d\tau \) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(F(s)G(s)\) |
| \ (f (t)\) ">\(tf(t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(-F'(s)\) |
| \ (f (t)\) ">\(t^{n}f(t)\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\((-1)^{n}F^{(n)}(s)\) |
| \ (f (t)\) ">\(\int_{0}^{t}f(\tau )d\tau \) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\frac{1}{s}F(s)\) |
| \ (f (t)\) ">\(\frac{f(t)}{t}\) | \ (F (s) =\ математичний {L}\ {f (t)\} =\ int_ {0} ^ {\ infty} e^ {-st} f (t) дт\) ">\(\int_{s}^{\infty} F(\sigma )d\sigma\) |