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Handbook of mathematics for engineers and scienteists part 68

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 68', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 11.2. Laplace Transform 437 11.2.1-2. Inverse Laplace transform. Given the transform f p the function f x can be found by means of the inverse Laplace transform 1 rc ix f x f p epx dp i -1 11.2.1.2 2 1 J c-iX where the integration path is parallel to the imaginary axis and lies to the right of all singularities of f p which corresponds to c a0. The integral in inversion formula 11.2.1.2 is understood in the sense of the cauchy principal value cc ix rc iw _ f p epx dp lim f p epx dp. I w X I Jc-iX Jc-iw In the domain x 0 formula 11.2.1.2 gives f x 0. Formula 11.2.1.2 holds for continuous functions. If f x has a finite jump discontinuity at a point x x0 0 then the left-hand side of 11.2.1.2 is equal to f x0-0 f x0 0 at this point for x0 0 the first term in the square brackets must be omitted . For brevity we write the Laplace inversion formula 11.2.1.2 as follows f x L-1 p or f x L-1 p x . There are tables of direct and inverse Laplace transforms see Sections T3.1 and T3.2 which are handy in solving linear differential and integral equations. 11.2.2. Main Properties of the Laplace Transform. Inversion Formulas for Some Functions 11.2.2-1. Convolution theorem. Main properties of the Laplace transform. 1 . The convolution of two functions f x and g x is defined as an integral of the form x J0 f t g x -1 dt and is usually denoted by f x g x or x f x g x f t g x -1 dt. 0 By performing substitution x -1 u we see that the convolution is symmetric with respect to the convolved functions f x g x g x f x . The convolution theorem states that L f x g x L f x L g x and is frequently applied to solve Volterra equations with kernels depending on the difference of the arguments. 2 . The main properties of the correspondence between functions and their Laplace transforms are gathered in Table 11.1. 3 . The Laplace transforms of some functions are listed in Table 11.2 for more detailed tables see Section T3.1 and the list of references at the end of this chapter. 438 Integral .