transformations

transformations - Lecture Notes on Laplace and z-transforms...

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Unformatted text preview: Lecture Notes on Laplace and z-transforms Ali Sinan Sert¨oz [email protected] http://www.fen.bilkent.edu.tr/˜sertoz 11 April 2003 1 Introduction These notes are intended to guide the student through problem solving using Laplace and z-transform techniques and is intended to be part of MATH 206 course. These notes are freely composed from the sources given in the bibli- ography and are being constantly improved. Check the date above to see if this is a new version. You are welcome to contact me through e-mail if you have any comments on these notes such as praise, criticism or suggestions for further improvements. 1 MATH 206 Complex Calculus and Transform Techniques [11 April 2003] 2 2 Laplace Transformation The main application of Laplace transformation for us will be solving some differential equations. A differential equation will be transformed by Laplace transformation into an algebraic equation which will be solvable, and that solution will be transformed back to give the actual solution of the DE we started with. We define the Laplace Transform of a function f : [0 , ∞ ) → C as L ( f ( t )) = Z ∞ e- st f ( t ) dt for s ∈ C We sometimes use F ( s ) to denote L ( f ( t )) if there is no confusion. But beware of conflicting notation in the literature. Euler 1 was the first one to use this transformation to solve certain differential equations in 1737. Later Laplace 2 independently used it in his book Th´ eorie Analytique de Probabilit´ es in 1812, [6, p285]. 2.1 Existence of Laplace Transformation It is clear that L ( f ) does not exist for every function f . For example it can be easily verified that L ( e t 2 ) does not exist, i.e. the associated integral clearly diverges. However L exists for a large class of functions. For example consider the following class of functions: A function f : [0 , ∞ ) → C is said to be of exponential order a if there are positive real constants M , T and a such that | f ( t ) | ≤ Me at for all t ≥ T . L ( f ) exists if f is integrable on [0 ,b ] for every b > 0 and f is of exponential order a for some a > 0. In this case F ( s ) is defined if and only if Re s > a . Moreover observe from the definition that lim Re s →∞ F ( s ) = 0. A word of relief: We will basically be using Laplace transform techniques to 1 Leonhard Euler 1707-1783. 2 Pierre-Simon Laplace 1749-1827. MATH 206 Complex Calculus and Transform Techniques [11 April 2003] 3 solve differential equations. Most differential equations with initial values will have a unique solution, see for example [7, p498-Thm 10.6 and p501-Thm 10.8]. We therefore formally apply Laplace transform techniques, without checking for validity, and if in the end the function we find solves the differ- ential equation then it is the solution. For this reasons most tables of Laplace transforms do not give the range of validity and are therefore wrong per se but perfectly acceptable given the overall purpose....
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transformations - Lecture Notes on Laplace and z-transforms...

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