f ( t ) = L -1 {F ( s )} 1. 3. 5. 7. 9. 11. 1 t n , n = 1, 2,3,K t sin ( at ) t sin ( at ) sin ( at ) - at cos ( at ) cos ( at ) - at sin ( at ) sin ( at + b ) sinh ( at ) e at sin ( bt ) e at sinh ( bt ) t ne at , n = 1, 2,3,K uc ( t ) = u ( t - c ) Heaviside Function

F ( s ) = L { f ( t )} 1 s n! s n +1

Table of Laplace Transforms

f ( t ) = L -1 {F ( s )}

F ( s ) = L { f ( t )} 1 s-a G ( p + 1) s p +1 1 × 3 × 5L ( 2n - 1) p 2n s 2 s 2 s + a2 s2 - a2 2 n+ 1

2. 4. 6. 8.
2

e at t p , p > -1 t
n- 1 2

p
2s a 2 s + a2 2as
2
3 2

, n = 1, 2,3,K

cos ( at ) t cos ( at ) sin ( at ) + at cos ( at ) cos ( at ) + at sin ( at ) cos ( at + b ) cosh ( at ) e at cos ( bt ) e at cosh ( bt ) f ( ct )

s n F ( s ) - s n-1 f ( 0 ) - s n- 2 f ¢ ( 0 )L - sf ( n- 2) ( 0 ) - f ( n-1) ( 0 )

Table Notes
1. This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas. 2. Recall the definition of hyperbolic functions. et + e - t et - e - t cosh ( t ) = sinh ( t ) = 2 2 3. Be careful when using “normal” trig function vs. hyperbolic functions. The only difference in the formulas is the “+ a2” for the “normal” trig functions becomes a “- a2” for the hyperbolic functions! 4. Formula #4 uses the Gamma function which is defined as G ( t ) = ò e - x x t -1 dx 0 ¥

If n is a positive integer then,

G ( n + 1) = n !

The Gamma function is an extension of the normal factorial function. Here are a couple of quick facts for the Gamma function G ( p + 1) = pG ( p ) p ( p + 1)( p + 2 )L ( p + n - 1) = æ1ö Gç ÷ = p è2ø G ( p + n) G ( p)

... we propose a new notion of privacy called “closeness.” We first present the base model closeness, which requires that the distribution of a sensitive attribute in any equivalence class is close to the distribution of the attribute in the overall table (i.e., the distance between the two distributions should be no more than a threshold t). We then propose a more flexible privacy model called closeness that offers higher utility. We describe our desiderata for designing a...

...======== 101 Ideas for Great Table Topics ========
Compiled by Mark LaVergne, DTM, Past International Director of Toastmasters International
Phone: 718-457-8390; Email: MarkLaVergne@aol.com Address: 38-11 Ditmars Boulevard, Astoria, NY 11105 Updated: May 2006 – Background and Acknowledgements on Page 14
----------------------------------------------------------------------------------------------------------------------------------==> TABLE TOPIC IDEA #1...

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...Chapter 7
Laplace Transform
The Laplace transform can be used to solve diﬀerential equations. Besides being a diﬀerent and eﬃcient alternative to variation of parameters and undetermined coeﬃcients, the Laplace method is particularly advantageous for input terms that are piecewise-deﬁned, periodic or impulsive. The direct Laplace transform or the Laplace integral of a function f (t) deﬁned for 0 ≤ t < ∞ is the ordinary...

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TerminologySolving the transformDiscontinuous FunctionsUsing Properties of Laplace Transforms
Edited by Caidoz, Flickety, Zareen, Garshepp and 4 others
The Laplace transform is an integral transform which allows a differential equation to be converted into a (hopefully) simpler algebraic equation, making it easier to solve.
While you can use tables of Laplace...

...Laplace Transformation Laplace transformation is a Mathematical tool which can be used to solve several problems in science and engineering. The transformed was first introduced by Pierre-Simon Laplace a French Mathematician, in the year 1790 in his work on probability theorem. Application of Laplace Transform The Laplace transform technique is applicable in many fields of science and technology such as: Control...

...Q-A. Find the Laplace transform of the following functions 1. f (t) = t − 1, 0 < t < 3; 7, t > 3. 2. f (t) = cos t − 0,
2π 3
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2π ; 3
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