1、Chapter 3: The Laplace Transform,3.1. Definition and Basic Properties 。 Objective of Laplace transform- Convert differential into algebraic equations Definition 3.1: Laplace transforms.t. convergess, t : independent variables,1, Representation:,。Example 3.2:,2,Consider,3,* Not every function has a L
2、aplace transform.In general, can not converge 。Example 3.1:,4, Definition 3.2.: Piecewise continuity (PC)f is PC on if there are finite pointss.t.and are finite,5,i.e., f is continuous on a, b except at finite points, at each of which f has finite one-sided limits,6,If f is PC on 0, k, then so is an
3、dexists, Theorem 3.2: Existence off is PC onIfProof:,7,8,* Theorem 3.2 is a sufficient but not a necessarycondition., There may be different functions whose Laplace transforms are the samee.g., andhave the same Laplace transform Theorem 3.3: Lerchs Theorem Table 3.1 lists Laplace transforms of funct
4、ions,9, Theorem 3.1: Laplace transform is linearProof: Definition 3.3:. Inverse Laplace transforme.g., Inverse Laplace transform is linear,10,3.2 Solution of Initial Value Problems Using Laplace Transform Theorem 3.5: Laplace transform off: continuous on: PC on 0, kThen, -(3.1),11,Proof:Let,12, Theo
5、rem 3.6: Laplace transform of: PC on 0, kfor s 0, j = 1,2 , n-1,13,。 Example 3.3:From Table 3.1, entries (5) and (8),14,15, Laplace Transform of Integral,16,From Eq. (3.1),3.3. Shifting Theorems and Heaviside Function 3.3.1.The First Shifting Theorem Theorem 3.7: Example 3.6: Given,17, Example 3.8:,
6、18,3.3.2. Heaviside Function and Pulses f has a jump discontinuity at a, ifexistand are finite but unequal Definition 3.4: Heaviside function,19,。 Shifting,20,。 Laplace transform of heaviside function,3.3.3 The Second Shifting Theorem Theorem 3.8:Proof:,21, Example 3.11:Rewrite,22, The inverse versi
7、on of the second shifting theorem Example 3.13:,23,rewritten as,where,24,25,3.4. Convolution,26, Theorem 3.9: Convolution theoremProof:,27, Theorem 3.10: Exmaple 3.18,28, Theorem 3.11:Proof :, Example 3.19:,29,3.5 Impulses and Dirac Delta Function Definition 3.5: Pulse Impulse: Dirac delta function:
8、30,A pulse of infinite magnitude over an infinitely short duration, Laplace transform of the delta function Filtering (Sampling) Theorem 3.12: f : integrable and continuous at a,31,32,Proof:,33,by Hospitals rule, Example 3.20:,3.6 Laplace Transform Solution of Systems Example 3.22Laplace transformS
9、olve for,34,Partial fractions decompositionInverse Laplace transform,35,3.7. Differential Equations with Polynomial Coefficient Theorem 3.13:Proof: Corollary 3.1:,36, Example 3.25:Laplace transform,37,Find the integrating factor,Multiply (B) by the integrating factor,38,Inverse Laplace transform,39,
10、 Apply Laplace transform toalgebraic expression for YApply Laplace transform toDifferential equation for Y,40, Theorem 3.14: PC on 0, k,41, Example 3.26:Laplace transform-(A)-(B),42,Finding an integrating factor,Multiply (B) by ,43,In order to have,44,Formulas: Laplace Transform: Laplace Transform of Derivatives: Laplace Transform of Integral:,45,Shifting Theorems: Convolution:Convolution Theorem: ,46,47,
