Property B For rational Laplace transforms the ROC does not contain any poles. This property simply recognizes that the Laplace transform goes to infinity at a pole so the Laplace transform integral will not converge at that point and hence it cannot be in the ROC. Property C If the Laplace transform of x(t) is rational then the ROC is the K. Webb MAE 3401 7 Laplace Transforms -Motivation We'll use Laplace transforms to solve differential equations Differential equations in the time domain difficult to solve Apply the Laplace transform Transform to the s‐domain Differential equations becomealgebraic equations easy to solve Transform the s‐domain solution back to the time domain Integral Transform - Laplace Transform -Definition ³ E D F (s) k (s,t) f (t)dt • Tool for solving linear diff. eq. -Integral transform k(s,t) - The kernel of the transformation n D, E( D f ; E f ) f F Transform • Laplace transform whenever this improper integral converges ^ ` ³ f 0 L f (t) F (s) e st f (t)dt l k(s,t) e st ME375 Laplace - 4 Definition • Laplace Transform - One Sided Laplace Transform where s is a complex variable that can be represented by s = σ +j ω and f (t) is a continuous function of time that equals 0 when t < 0. - Laplace Transform converts a function in time t into a function of a complex variable s. • Inverse Laplace Transform [] 0 The Laplace Transform is Linear If a is a constant and f and gare functions, then For example, by the above property (1) As an another example, by property (2) L(e5t+cos(3t)) = L(e5t)+L(cos(3t)) = 1 s−5 + s s2+9 ,s>5. L(3t5)=3L(t5)=3 5! s6 = 360 s6 ,s>0. L(af)=aL(f) (1) L(f +g)=L(f)+L(g) (2) 6 An example where both (1) and (2) are used, The Laplace transform can be interpreted as a transforma- tion from the time domain where inputs and outputs are functions of time to the frequency domain where inputs and outputs are functions of complex angular frequency. In order for any function of time f(t) to be Laplace transformable, it must satisfy the following Dirichlet con- ditions [1]: LaPlace Transform in Circuit Analysis Recipe for Laplace transform circuit analysis: 1. Redraw the circuit (nothing about the Laplace transform changes the types of elements or their interconnections). 2. Any voltages or currents with values given are Laplace-transformed using the functional and operational tables. 3. Chapter 4 Laplace Transforms Notes Proofread by Yunting Gao and corrections made on 03/30/2021 4 Introduction 4.1 Definition and the Laplace transform of simple functions Given f, a function of time, with value f(t) at time t, the Laplace transform of fwhich is denoted by L(f) (or F) is defined by L(f)(s) = F(s) = Z 1 0 e stf(t)dt s>0: (1 The Inverse Laplace Transform of a Product 1. Solving initial value problems ay00 +by0 +cy=f with Laplace transforms leads to a transform Y =F·R(s)+···. 2. If the Laplace transform F of f is not easily computed or if the inverse transform of the product is hard, it would be nice to have a direct formula for the inverse transform of a product. 1.1 Laplace Transformation Laplace transformation belongs to a class of analysis methods called integral transformation which are studied in the eld of operational calculus. These methods include the Fourier transform, the Mellin transform, etc. In each method, the idea is to transform a di cult problem into an easy problem. For laplace transform is defined over a portion of complex plane. If L{f(t)} exists for s real and then L{f(t)} exists in h