Laplace Transform Definitions and Results
Explore the Laplace transform definitions and results, including its properties and applications. Learn about transforming functions using Laplace transforms and understand the fundamental principles behind this powerful mathematical tool.
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LAPLACE LAPLACE TRANSFORMS TRANSFORMS
DEFINITION If a function f(t) is defined for all positive values of the variable 0 st ( ) e f t dt t and if exists and is equal to F(s) , then F(s) is called the Laplace transform of f(t) and is denoted by the symbol L{f(t)}. Hence The operator L that transform f(t) into 0 = = st { ( )} ( ) ( ). L f t e f t dt F s F(s) is called the Laplace transform operator.
RESULTS RESULTS
L{f(t)+g(t)} = L{f(t)}+L{g(t)} (i) we have L{f(t)+g(t)} = 0 + st [ ( ) ( )] e f t g t dt 0 + st st ( ) ( ) e f t dt e g t dt = 0 = L{f(t)}+L{g(t)}.
(ii) L{cf(t)} = cL{f(t)} , where c is a constant. 0 we have L{cf(t)} = st ( dt t ) e cf = 0 st ( dt t ) c e f = cL{f(t)}.
(iii) L{f(t)} =sL{f(t)}- f(0). 0 st ( ' ) e f t dt we have L{f (t)} = = (on integration by parts) 0 0 st st ( )[ ] ( )( ) f t e f t s e dt 0 = + st ) 0 ( ( ) f s e f t dt = sL{f(t)}-f(0) .
(iv) ( ' ' { t f L = 2 )} { ( )} ) 0 ( 0 ( ' ). s L f t sf f L{f (t)} = L{F (t)} where F(t)=f (t) = sL{F(t)}-F(0) = sL{f (t)}-f (0) = s[sL{f(t)}-f(0)]-f (0) 2s = L{f(t)}-sf(0)-f (0).
(v) By extending the previous result , we get = 1 1 2 1 n n n n n n { ( )} { ( )} ) 0 ( ) 0 ( 0 ( ' )...... 0 ( ). L f t s L f t s f s f s f f (vi) If L{f(t)} = F(s) (a) lim t = ( ) lim s ( ). f t sF s 0 (b) = lim t ( ) lim s ( ). f t sF s 0
(vii) L 1 = at ( ) e + s a 0 = at st at ( ) L e e e dt 0 + = ( ) s a e dt + ( ) s a t e = + ( ) s a 0 1 = s + a 1 Similarly = at ( ) L e s a
(viii) L s + = (cos ) . at 2 2 s a 0 = st (cos ) cos . L at e atdt + st ( cos s sin ) e s at a at = + 2 2 a 0 s + = . 2 2 s a (ix) similarly a + = (sin ) . L at 2 2 s a
(x) ) ( t L + ( ) 1 1 + n = n . n s 0 = n st n We have . ( t L ) e t dt n 1 s x by putting st=x. = n x ( ) . L t e dx s 0 1 n . 0 = n x x e dx + 1 s + ( ) 1 1 + n = . n s
Thank you Prepared by D.PHILOMINE JEEVITHA Department of Mathematics St.Joseph s college trichy
