Difference between revisions of "Laplace Transforms Table" - Rhea

Revision as of 15:56, 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\$
please continue	place formula here
please continue	place formula here
please continue	place formula here
please continue	place formula here
please continue	place formula here
please continue	place formula here
please continue	place formula here

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 15:56, 8 November 2010

Alumni Liaison