Додаток B: Таблиця перетворень Лапласа

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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\) "> $\int_{s}^{\infty} F(\sigma )d\sigma$