Page 1 Laplace Transform And Its Applications Page 2 Laplace Transform And Its Applications Topics ? Definition of Laplace Transform ? Linearity of the Laplace Transform ? Laplace Transform of some Elementary Functions ? First Shifting Theorem ? Inverse Laplace Transform ? Laplace Transform of Derivatives & Integral ? Differentiation & Integration of Laplace Transform ? Evaluation of Integrals By Laplace Transform ? Convolution Theorem ? Application to Differential Equations ? Laplace Transform of Periodic Functions ? Unit Step Function ? Second Shifting Theorem ? Dirac Delta Function Page 3 Laplace Transform And Its Applications Topics ? Definition of Laplace Transform ? Linearity of the Laplace Transform ? Laplace Transform of some Elementary Functions ? First Shifting Theorem ? Inverse Laplace Transform ? Laplace Transform of Derivatives & Integral ? Differentiation & Integration of Laplace Transform ? Evaluation of Integrals By Laplace Transform ? Convolution Theorem ? Application to Differential Equations ? Laplace Transform of Periodic Functions ? Unit Step Function ? Second Shifting Theorem ? Dirac Delta Function Definition of Laplace Transform ? Let f(t) be a given function of t defined for all then the Laplace Transform ot f(t) denoted by L{f(t)} or or F(s) or is defined as provided the integral exists,where s is a parameter real or complex. 0 ? t ) (s f ) (s ? dt t f e s s F s f t f L st ) ( ) ( ) ( ) ( )} ( { 0 ? ? ? ? ? ? ? ? Page 4 Laplace Transform And Its Applications Topics ? Definition of Laplace Transform ? Linearity of the Laplace Transform ? Laplace Transform of some Elementary Functions ? First Shifting Theorem ? Inverse Laplace Transform ? Laplace Transform of Derivatives & Integral ? Differentiation & Integration of Laplace Transform ? Evaluation of Integrals By Laplace Transform ? Convolution Theorem ? Application to Differential Equations ? Laplace Transform of Periodic Functions ? Unit Step Function ? Second Shifting Theorem ? Dirac Delta Function Definition of Laplace Transform ? Let f(t) be a given function of t defined for all then the Laplace Transform ot f(t) denoted by L{f(t)} or or F(s) or is defined as provided the integral exists,where s is a parameter real or complex. 0 ? t ) (s f ) (s ? dt t f e s s F s f t f L st ) ( ) ( ) ( ) ( )} ( { 0 ? ? ? ? ? ? ? ? Linearity of the Laplace Transform ? If L{f(t)}= and then for any constants a and b ) (s f ) ( )] ( [ s g t g L ? )] ( [ )] ( [ )] ( ) ( [ t g bL t f aL t bg t af L ? ? ? )] ( [ )] ( [ )} ( ) ( { ) ( ) ( )] ( ) ( [ )} ( ) ( { Definition -By : Proof 0 0 0 t g bL t f aL t bg t af L dt t g e b dt t f e a dt t bg t af e t bg t af L st st st ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Page 5 Laplace Transform And Its Applications Topics ? Definition of Laplace Transform ? Linearity of the Laplace Transform ? Laplace Transform of some Elementary Functions ? First Shifting Theorem ? Inverse Laplace Transform ? Laplace Transform of Derivatives & Integral ? Differentiation & Integration of Laplace Transform ? Evaluation of Integrals By Laplace Transform ? Convolution Theorem ? Application to Differential Equations ? Laplace Transform of Periodic Functions ? Unit Step Function ? Second Shifting Theorem ? Dirac Delta Function Definition of Laplace Transform ? Let f(t) be a given function of t defined for all then the Laplace Transform ot f(t) denoted by L{f(t)} or or F(s) or is defined as provided the integral exists,where s is a parameter real or complex. 0 ? t ) (s f ) (s ? dt t f e s s F s f t f L st ) ( ) ( ) ( ) ( )} ( { 0 ? ? ? ? ? ? ? ? Linearity of the Laplace Transform ? If L{f(t)}= and then for any constants a and b ) (s f ) ( )] ( [ s g t g L ? )] ( [ )] ( [ )] ( ) ( [ t g bL t f aL t bg t af L ? ? ? )] ( [ )] ( [ )} ( ) ( { ) ( ) ( )] ( ) ( [ )} ( ) ( { Definition -By : Proof 0 0 0 t g bL t f aL t bg t af L dt t g e b dt t f e a dt t bg t af e t bg t af L st st st ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Laplace Transform of some Elementary Functions a s if a - s 1 ) ( e . ) e ( Definition -By : Proof a - s 1 ) L(e (2) ) 0 ( , s 1 1 . ) 1 ( Definition -By : Proof s 1 L(1) (1) 0 ) ( 0 ) ( 0 at at at 0 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? a s e dt e dt e L s s e dt e L t a s t a s st st stRead More

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