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### Laplace Dönüşüm Formülleri

$\mathcal{L}[f(t)]=F(s)=\int^{\infty}_{0}{f(t)e^{-st}dt}\quad;\quad \mathcal{L}[g(t)]=G(s)=\int^{\infty}_{0}{g(t)e^{-st}dt}$

1.
$\mathcal{L}(1)(s)=\frac{1}{s}\quad;\quad s>0$

2.
$\mathcal{L}(e^{at})(s)=\frac{1}{s-a}\quad;\quad s>a$

3.
$\mathcal{L}(e^{t^n})(s)=\frac{n!}{s^{n+1}}\quad; \quad s>0 , n \;\; \text{ pozitif tam say{\i}}$

4.
$\mathcal{L}(e^{t^p})(s)=\frac{\Gamma(p+1)}{s^{p+1}}\quad; \quad s>0 , p>-1$

5.
$\mathcal{L}(\sin{at})(s)=\frac{a}{s^2+a^2}\quad; \quad s>0$

6.
$\mathcal{L}(\cos{at})(s)=\frac{s}{s^2+a^2}\quad; \quad s>0$

7.
$\mathcal{L}(e^{at}.\sin{bt})(s)=\frac{b}{(s-a)^2+b^2}\quad; \quad s>a$

8.
$\mathcal{L}(e^{at}.\cos{bt})(s)=\frac{s-a}{(s-a)^2+b^2}\quad; \quad s>a$

9.
$\mathcal{L}(t^{n}.e^{at})(s)=\frac{n!}{(s-a)^{n+1}}\quad; \quad s>a$

10.
$\mathcal{L}(t^{n}.f(t))(s)=(-1)^n\frac{d^n}{ds^n}(\mathcal{L}(f(t)))$

11.
$\mathcal{L}(f'(t))(s)=s\mathcal{L}(f(t))-f(0)=sF(s)-f(0)$

12.
$\mathcal{L}(f''(t))(s)=s^2\mathcal{L}(f(t))-sf(0)-f'(0)=s^2F(s)-sf(0)-f'(0)$

13.
\begin{align*} \mathcal{L}(f^{(n)}(t))(s) &=s^n\mathcal{L}(f(t))-s^{n-1}f(0)-...-f^{(n-1)}(0)&=s^nF(s)-s^{n-1}f(0)- \\ & & \cdots -f^{(n-1)}(0) \end{align*}

14.
$\mathcal{L}(af(t)+bg(t))(s)=aF(s)+bG(s)$

15.
$\mathcal{L}(f(at))(s)=\frac{1}{|a|}F(\frac{s}{a})$

16.
$\mathcal{L}(e^{at}f(t))(s)=F(s-a)$

17.
$\mathcal{L}(f(t-a)u(t-a))(s)=e^{-as}F(s)\quad; \quad u(t-a)=\left\{\begin{matrix} 1 & t\geqslant a \\ 0 & t< a \end{matrix}\right.$

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