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22p. Partial fraction expansion _printable_

22p. Partial fraction expansion _printable_ - The Partial...

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1 ©2009 by L. Lagerstrom The Partial Fraction Expansion For an introduction to the basic concepts, see the associated video clip Motivation via the Laplace Transform Proper and improper rational functions Basic form of the partial fraction expansion (PFE) Finding the residues and roots Using the residue function Repeated roots and the PFE Improper rational functions and the PFE ©2009 by L. Lagerstrom Motivation via the Laplace Transform In certain types of engineering analysis, such as the analysis of circuits, a mathematical technique involving something called the Laplace transform is very helpful. The details of the Laplace transform are beyond this course, but the results that sometimes occur when it's used are important for our current topic (the "partial fraction expansion"). In particular: When using the Laplace transform to analyze a circuit, we get an integral of the following form: where A(x) and B(x) are polynomials. In general, this is not an easy integral to solve. But by using this "partial fraction expansion" technique, we can solve it relatively easily. dx x B x A ) ( ) ( ©2009 by L. Lagerstrom Proper and Improper Rational Functions Before we introduce the partial fraction expansion (PFE), we need the concepts of a "proper rational function" and an "improper rational function". Consider the ratio of two polynomials, A(x) and B(x): We say that H(x) is a proper rational function if the order (degree) of the numerator polynomial, A(x), is less than the order of the denominator polynomial, B(x). So, for example, if A(x) is a quadratic (second-order) polynomial and B(x) is a cubic (third-order) polynomial, then H(x) is proper. If the order of A(x) is greater than or equal to the order of B(x), then we say that H(x) is an improper rational function. This matters because the partial fraction expansion only works if H(x) is a proper rational function. ) ( ) ( ) ( x B x A x H = ©2009 by L. Lagerstrom Basic Form of the PFE Let's consider a proper rational function H(x) = A(x)/B(x). Although we will leave the proof to your math class, it turns out that it's possible to expand a proper rational function as a sum of terms thusly: In this expression the c's are constants known as "residues" and the r's are the roots of the denominator polynomial B(x). (The variable x is of course the independent variable.)
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