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# ラプラス変換公式集

ラプラス変換公式集です。

## 公式

f(t) \rightarrow F(s)

f(t) \rightarrow \int_0^\infty f(t)e^{-st} dt

1 \rightarrow \frac{ 1 }{ s }

t^n \rightarrow \frac{ 1 }{ s^{n+2} }\Gamma(n+1)   (s \geq 0)

t^a \rightarrow \frac{ 1 }{ s^{a +2} }\Gamma(a +1)   (s \geq 0 ,a \gt -1)

e^{at} \rightarrow \frac{ s }{ s-a }    (s-a \gt 0)

\cos a t \rightarrow \frac{ s }{ s^2 + a^2 }    (s \gt 0)

\sin a t \rightarrow \frac{ a }{ s^2 + a^2 }    (s \gt 0)

\cosh a t \rightarrow \frac{ s }{ s^2 - a^2 }    (s \gt |a|)

\sinh a t \rightarrow \frac{ a }{ s^2 - a^2 }    (s \gt |a|)

\delta (t) \rightarrow 1

u (t) \rightarrow \frac{ 1 }{ s }

u (t-a) \rightarrow \frac{ 1 }{ s } e^{-as}  (a\geq 0)

f(t-a)u(t-a) \rightarrow e^{-as}F(s)  (a\geq 0)

e^{-at}f(t) \rightarrow F(s-a)

周期Tの周期関数f(t) \rightarrow \frac{ 1 }{ 1- e^{-st}}F_0(s)

space{60pt}\rightarrow \frac{ 1 }{ 1- e^{-st}} \int_0^T f(t)e^{-st} dt

erf(\sqrt{at}) \rightarrow \frac{ \sqrt{a} }{ s\sqrt{s+a} }

J_0(at) \rightarrow \frac{ 1 }{ \sqrt{s^2+a^2} }

f'(t) \rightarrow sF(s)-f(0)

f''(t) \rightarrow s^2F(s)-sf(0)-f'(0)

\int_0^t f(u) du \rightarrow \frac{ 1 }{ s }F(s)

t^nf(t) \rightarrow (-1)^nF(s)^{(n)}

\frac{ f(t) }{ t^n } \rightarrow \int_s^\infty ...\int_s^\infty F(s)(ds)^n

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