7.1 notes - 2 − k 2 , L{ cosh kt } = s s 2 − k 2 . 1 3....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 7. The laplace Transform § 7.1 DeFnition of the Laplace Transform 1. Review of Integral with InFnite Intervals: Let f be a function deFned on [ a, ) , then ° a f ( t ) dt = lim b →∞ ° b a f ( t ) dt provided that the limit exists and Fnite. 2. Laplace Transform: Let f be a function deFned for t 0 . Then the integral L{ f ( t ) } = ° 0 e st f ( t ) dt is said to be the Laplace transform of f , provided that the integral converges. In general, L{ f ( t ) } = F ( s ) , L{ g ( t ) } = G ( s ) . Ex: ±ind the Laplace transform. L{ 1 } , L{ t } , L{ e 2 t } , L{ sin 3 t } L is a linear transform: i.e. L{ α f ( t )+ β g ( t ) } = α F ( s )+ β G ( s ) So L{ 5 + 4 sin 3 t 7 e 2 t } = Theorem: Transforms of some basic functions. 1. L{ 1 } = 1 s . 2. L{ t n } = n ! s n +1 ,n =1 , 2 , 3 ,.... 3. L{ e at } = 1 s a . 4. L{ sin kt } = k s 2 + k 2 , L{ cos kt } = s s 2 + k 2 . 5. L{ sinh kt } = k s
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2 − k 2 , L{ cosh kt } = s s 2 − k 2 . 1 3. Su ffi cient Conditions for Existence of L{ f ( t ) } . Def: A function f is said to be of exponential order c if there exist constants c, M > , and T > 0 such that | f ( t ) | ≤ Me ct for all t ≥ T. Theorem: If f is piecewise continuous on [0 , ∞ ) and of exponential order c , then L{ f ( t ) } exists for s > c. Theorem: If f is piecewise continuous on [0 , ∞ ) and of exponential order and F ( s ) = L{ f ( t ) } , then lim s →∞ F ( s ) = 0 . 2...
View Full Document

This note was uploaded on 01/05/2011 for the course MATH 3310 taught by Professor Dr.du during the Fall '08 term at Kennesaw.

Ask a homework question - tutors are online