{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

laplace-table

# laplace-table - S Boyd EE102 Table of Laplace Transforms...

This preview shows pages 1–3. Sign up to view the full content.

S. Boyd EE102 Table of Laplace Transforms Remember that we consider all functions (signals) as defined only on t 0. General f ( t ) F ( s ) = Z 0 f ( t ) e - st dt f + g F + G αf ( α R ) αF df dt sF ( s ) - f (0) d k f dt k s k F ( s ) - s k - 1 f (0) - s k - 2 df dt (0) - · · · - d k - 1 f dt k - 1 (0) g ( t ) = Z t 0 f ( τ ) G ( s ) = F ( s ) s f ( αt ), α > 0 1 α F ( s/α ) e at f ( t ) F ( s - a ) tf ( t ) - dF ds t k f ( t ) ( - 1) k d k F ( s ) ds k f ( t ) t Z s F ( s ) ds g ( t ) = ( 0 0 t < T f ( t - T ) t T , T 0 G ( s ) = e - sT F ( s ) 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Specific 1 1 s δ 1 δ ( k ) s k t 1 s 2 t k k ! , k 0 1 s k +1 e at 1 s - a cos ωt s s 2 + ω 2 = 1 / 2 s - + 1 / 2 s + sin ωt ω s 2 + ω 2 = 1 / 2 j s - - 1 / 2 j s + cos( ωt + φ ) s cos φ - ω sin φ s 2 + ω 2 e - at cos ωt s + a ( s + a ) 2 + ω 2 e - at sin ωt ω ( s + a ) 2 + ω 2 2
Notes on the derivative formula at t = 0 The formula L ( f 0 ) = sF ( s ) - f (0 - ) must be interpreted very carefully when f has a discon- tinuity at t = 0. We’ll give two examples of the correct interpretation. First, suppose that f is the constant 1, and has no discontinuity at t = 0. In other words, f is the constant function with value 1. Then we have f 0 = 0, and f (0 - ) = 1 (since there is no jump in f at t = 0). Now let’s apply the derivative formula above. We have F ( s ) = 1 /s , so the formula reads L ( f 0 ) = 0 = sF ( s ) - 1 which is correct. Now, let’s suppose that
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern