Difference between revisions of "Laplace Transforms Table" - Rhea

Revision as of 16:56, 8 November 2010

IT SEEMS LIKE THE VARIABLES s AND t WERE INTERCHANGED BELOW.

Laplace Transform Pairs and Properties
Definition
Laplace Transform	$F(s)=\int_{-\infty}^\infty f(t) e^{-st}dt, \ s\in {\mathbb C} \$
Inverse Laplace Transform	add formula here
Properties of the Laplace Transform
	function $f(s) \$	Laplace transform $F(t) \$	ROC $$ R $$
	$af_1(s)+bf_2(s) \$	$aF_1(t)+bF_2(t) \$
	$f\left( \frac{s}{a} \right)$	$aF(at) \$
	$f(s-a) \$	$e^{at}F(t) \$
	$e^{-as}f(s) \$	$u(t-a) = \begin{cases} F(t-a) & t>a \\ 0 & t<a \end{cases}$
	$sf(s)-F(0) \$	$F'(t) \$
	$s^2f(s)-sF(0)-F'(0) \$	$F''(t)\$
	$s^{n}f(s)-\sum_{k=1}^ns^{n-k}F^{(k)}(0)\$	$F^{(n)}(t)\$
	$f'(s)\$	$-tF(t)\$
	$f''(s)\$	$t^2F(t)\$
	$f^{(n)}(s)\$	$(-1)^{(ntn)}F(t)\$
	$\frac{f(s)}s\$	$\int_{0}^{t} F(u) du \$
	$\frac{f(s)}{s^n} \$	$\int_{0}^{t}...\int_{0}^{t}F(u)du^n = \int_{0}^{t}\frac{{(t-u)}^{n-1}}{(n-1)!} F(u)du\$
	$f(s)g(s)\$	$\int_{0}^{t}F(u)G(t-u)du\$
	$\int_{s}^{\infty}f(u)du \$	$\frac{F(t)}t \$
	$\frac1{1-e^{-sT}}\int_{0}^{T}e^{-su}F(u)du \$	$F(t)=F(t+T)\$
	$\frac{f(\sqrt{s})}s\$	$\frac{1}{\sqrt{{\pi}t}}\int_{0}^{\infty}e^{-\frac{u^2}4t}F(u)du$
	$\frac1sf\left(\frac1s\right)\$	$\int_{0}^{\infty}J_0(2\sqrt{ut})F(u)du\$
	$\frac1{g^{n+1}}f\left(\frac1s\right)\$	$t^{\frac{n}2}\int_{0}^{\infty}u^{-\frac{n}2}J_n(2\sqrt{ut})F(u)du \$
	$\frac{f(s+\frac1s)}{s^2+1}\$	$\int_{0}^{t}J_0(2\sqrt{u(t-u)})F(u)du\$
	$\frac1{2\sqrt\pi}\int_{0}^{\infty}u^{-\frac32}e^{-\frac{s^2}{4u}}f(u)du\$	$F(t^2)\$
	$\frac{f(\ln s)}{s\ln s}\$	$\int_{0}^{\infty}\frac{t^uF(u)}{\Gamma(u+1)}du\$
	$\frac{P(s)}{Q(s)}\$	$\sum_{k=1}^N \frac{P(\alpha_k)}{Q'(\alpha_k)}e^{\alpha_kt}\$
	$\frac1s\$	$1\$
	$\frac1{s^2}\$	$t(s)\$
	$\frac1{s^n} n=1,2,3,...\$	$\frac{t^{n-1}}{(n-1)!}, 0!=1\$
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Laplace Transform Pairs
notes	Signal	Laplace Transform	ROC
unit impulse/Dirac delta	$\,\!\delta(t)$	$1$	$\text{All}\, s \in {\mathbb C}$
unit step function	$\,\! u(t)$	$\frac{1}{s}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
	$\,\! -u(-t)$	$\frac{1}{s}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0$
	$\frac{t^{n-1}}{(n-1)!}u(t)$	$\frac{1}{s^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
	$-\frac{t^{n-1}}{(n-1)!}u(-t)$	$\frac{1}{s^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0$
	$\,\!e^{-\alpha t}u(t)$	$\frac{1}{s+\alpha}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
	$\,\! -e^{-\alpha t}u(-t)$	$\frac{1}{s+\alpha}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha$
	$\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)$	$\frac{1}{(s+\alpha )^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
	$-\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t)$	$\frac{1}{(s+\alpha )^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha$
	$\,\!\delta (t - T)$	$\,\! e^{-sT}$	$\text{All}\,\, s\in {\mathbb C}$
	$\,\cos( \omega_0 t)u(t)$	$\frac{s}{s^2+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
	$\, \sin( \omega_0 t)u(t)$	$\frac{\omega_0}{s^2+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
	$\,e^{-\alpha t}\cos( \omega_0 t) u(t)$	$\frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
	$\, e^{-\alpha t}\sin( \omega_0 t)u(t)$	$\frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
	$u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}}$	$\,\!s^{n}$	$All\,\, s$
	$u_{-n}(t) = \underbrace{u(t) \dots u(t)}_{n\,\,times}$	$\frac{1}{s^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$

Back to Collective Table

@@ Line 26: / Line 26: @@
 | align="right" style="padding-right: 1em;" |
 | <math> f\left( \frac{s}{a} \right) </math>
-| <math> aF(at)</math>
+| <math> aF(at) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> f(s-a)</math>
+| <math> f(s-a) \</math>
-|<math> e^{at}F(t) </math>
+|<math> e^{at}F(t) \</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> e^{-as}f(s)</math>
+| <math> e^{-as}f(s) \</math>
 | <math> u(t-a) =
 \begin{cases}
@@ Line 41: / Line 41: @@
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> sf(s)-F(0) </math>
+| <math> sf(s)-F(0) \</math>
-| <math> F'(t)</math>
+| <math> F'(t) \ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> s^2f(s)-sF(0)-F'(0) </math>
+| <math> s^2f(s)-sF(0)-F'(0) \</math>
-| <math> F''(t)</math>
+| <math> F''(t)\</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> s^{n}f(s)-\sum_{k=1}^ns^{n-k}F^{(k)}(0)</math>
+| <math> s^{n}f(s)-\sum_{k=1}^ns^{n-k}F^{(k)}(0)\</math>
-| <math> F^{(n)}(t)</math>
+| <math> F^{(n)}(t)\</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> f'(s)</math>
+| <math> f'(s)\</math>
-| <math> -tF(t)</math>
+| <math> -tF(t)\</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>f''(s) </math>
+| <math>f''(s)\ </math>
-| <math> t^2F(t) </math>
+| <math> t^2F(t)\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> f^{(n)}(s)</math>
+| <math> f^{(n)}(s)\</math>
-| <math> (-1)^{(ntn)}F(t)</math>
+| <math> (-1)^{(ntn)}F(t)\</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{f(s)}s</math>
+| <math> \frac{f(s)}s\</math>
-| <math> \int_{0}^{t} F(u) du </math>
+| <math> \int_{0}^{t} F(u) du \</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{f(s)}{s^n} </math>
+| <math> \frac{f(s)}{s^n} \</math>
-| <math> \int_{0}^{t}...\int_{0}^{t}F(u)du^n = \int_{0}^{t}\frac{{(t-u)}^{n-1}}{(n-1)!} F(u)du</math>
+| <math> \int_{0}^{t}...\int_{0}^{t}F(u)du^n = \int_{0}^{t}\frac{{(t-u)}^{n-1}}{(n-1)!} F(u)du\</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>f(s)g(s)</math>
+| <math>f(s)g(s)\</math>
-| <math> \int_{0}^{t}F(u)G(t-u)du </math>
+| <math> \int_{0}^{t}F(u)G(t-u)du\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \int_{s}^{\infty}f(u)du </math>
+| <math> \int_{s}^{\infty}f(u)du \</math>
-| <math> \frac{F(t)}t</math>
+| <math> \frac{F(t)}t \</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{1-e^{-sT}}\int_{0}^{T}e^{-su}F(u)du </math>
+| <math> \frac1{1-e^{-sT}}\int_{0}^{T}e^{-su}F(u)du \</math>
-| <math> F(t)=F(t+T)</math>
+| <math> F(t)=F(t+T)\</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{f(\sqrt{s})}s</math>
+| <math> \frac{f(\sqrt{s})}s\</math>
 | <math> \frac{1}{\sqrt{{\pi}t}}\int_{0}^{\infty}e^{-\frac{u^2}4t}F(u)du </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1sf(\frac1s)</math>
+| <math> \frac1sf\left(\frac1s\right)\</math>
-| <math> \int_{0}^{\infty}J_0(2\sqrt{ut})F(u)du</math>
+| <math> \int_{0}^{\infty}J_0(2\sqrt{ut})F(u)du\</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{g^{n+1}}f(\frac1s)</math>
+| <math> \frac1{g^{n+1}}f\left(\frac1s\right)\</math>
-| <math> t^{\frac{n}2}\int_{0}^{\infty}u^{-\frac{n}2}J_n(2\sqrt{ut})F(u)du </math>
+| <math> t^{\frac{n}2}\int_{0}^{\infty}u^{-\frac{n}2}J_n(2\sqrt{ut})F(u)du \</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{f(s+\frac1s)}{s^2+1}</math>
+| <math> \frac{f(s+\frac1s)}{s^2+1}\</math>
-| <math> \int_{0}^{t}J_0(2\sqrt{u(t-u)})F(u)du</math>
+| <math> \int_{0}^{t}J_0(2\sqrt{u(t-u)})F(u)du\</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac1{2\sqrt\pi}\int_{0}^{\infty}u^{-\frac32}e^{-\frac{s^2}{4u}}f(u)du</math>
+| <math> \frac1{2\sqrt\pi}\int_{0}^{\infty}u^{-\frac32}e^{-\frac{s^2}{4u}}f(u)du\</math>
-| <math> F(t^2)</math>
+| <math> F(t^2)\</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{f(\ln s)}{s\ln s}</math>
+| <math> \frac{f(\ln s)}{s\ln s}\</math>
-| <math> \int_{0}^{\infty}\frac{t^uF(u)}{\Gamma(u+1)}du</math>
+| <math> \int_{0}^{\infty}\frac{t^uF(u)}{\Gamma(u+1)}du\</math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math> \frac{P(s)}{Q(s)}</math>
+| <math> \frac{P(s)}{Q(s)}\</math>
-| <math> \sum{k=1}^N \frac{P(\alpha_k)}{Q'(\alpha_k)}e^{\alpha_kt}</math>
+| <math> \sum_{k=1}^N \frac{P(\alpha_k)}{Q'(\alpha_k)}e^{\alpha_kt}\</math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac1s\</math>
+| <math> 1\</math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac1{s^2}\</math>
+| <math> t(s)\</math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> \frac1{s^n}  n=1,2,3,...\</math>
+| <math> \frac{t^{n-1}}{(n-1)!},  0!=1\</math>
+|-
+| align="right" style="padding-right: 1em;" | please continue
+| place formula here
+|-<math> af_1(s)+bf_2(s)</math>
+|-
+| align="right" style="padding-right: 1em;" | please continue
+| place formula here
+|-<math> af_1(s)+bf_2(s)</math>
+|-
+| align="right" style="padding-right: 1em;" | please continue
+| place formula here
+|-<math> af_1(s)+bf_2(s)</math>
+|-
+| align="right" style="padding-right: 1em;" | please continue
+| place formula here
+|-<math> af_1(s)+bf_2(s)</math>
+|-
+| align="right" style="padding-right: 1em;" | please continue
+| place formula here
+|-<math> af_1(s)+bf_2(s)</math>
 |-
 | align="right" style="padding-right: 1em;" | please continue
@@ Line 120: / Line 152: @@
 |-<math> af_1(s)+bf_2(s)</math>
 |-
 |}
 {|

Difference between revisions of "Laplace Transforms Table" - Rhea

Revision as of 16:56, 8 November 2010

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