MATH 251 Lecture Notes - Algebraic Equation, Dirac Delta Function, Integral Transform
Document Summary
Definition and properties of laplace transform, piecewise continuous functions, the laplace transform method of solving initial value problems. The method of laplace transforms is a system that relies on algebra (rather than calculus-based methods) to solve linear differential equations. While it might seem to be a somewhat cumbersome method at times, it is a very powerful tool that enables us to readily deal with linear differential equations with discontinuous forcing functions. Definition: let f (t) be defined for t 0. The laplace transform of f (t), denoted by f(s) or l{f (t)}, is an integral transform given by. Provided that this (improper) integral exists, i. e. that the integral is convergent. For functions continuous on [0, ), the above transformation is one-to-one. That is, different continuous functions will have different transforms. Example: let f (t) = 1, then sf. 1 s s > 0. st t e. The integral is divergent whenever s 0.
